Physics 461 / Quantum Mechanics I

P.E. Parris Department of Physics University of Missouri-Rolla Rolla, Missouri 65409

January 10, 2005 CONTENTS

1 Introduction 7 1.1WhatisQuantumMechanics?...... 7 1.1.1 WhatisMechanics?...... 7 1.1.2 PostulatesofClassicalMechanics:...... 8 1.2TheDevelopmentofWaveMechanics...... 10 1.3TheWaveMechanicsofSchrödinger...... 13 1.3.1 Postulates of Wave Mechanics for a Single Spinless Particle . . . . 13 1.3.2 Schrödinger’sMechanicsforConservativeSystems...... 16 1.3.3 The Principle of Superposition and Spectral Decomposition . . . . 17 1.3.4 TheFreeParticle...... 19 1.3.5 Superpositions of Plane Waves and the Fourier Transform . . . . . 23 1.4Appendix:TheDeltaFunction...... 26

2 The Formalism of Quantum Mechanics 31 2.1 Postulate I: SpecificationoftheDynamicalState...... 31 2.1.1 PropertiesofLinearVectorSpaces...... 31 2.1.2 Additional Definitions...... 33 2.1.3 ContinuousBasesandContinuousSets...... 34 2.1.4 InnerProducts...... 35 2.1.5 ExpansionofaVectoronanOrthonormalBasis...... 38 2.1.6 Calculation of Inner Products Using an Orthonormal Basis . . . . 39 2.1.7 ThePositionRepresentation...... 40 2.1.8 TheWavevectorRepresentation...... 41 2.2PostulateII:ObservablesofQuantumMechanicalSystems...... 43 2.2.1 OperatorsandTheirProperties...... 43 2.2.2 MultiplicativeOperators...... 45 2.2.3 DifferentialOperators...... 47 2.2.4 Ket-BraOperators...... 48 2.2.5 ProjectionOperators:Thecompletenessrelation...... 49 2.2.6 MatrixElements...... 51 2.2.7 Action of Operators on Bras of S∗ ...... 52 2.2.8 HermitianConjugation...... 52 2.2.9 Hermitian,Anti-Hermitian,andUnitaryOperators...... 54 2.2.10MatrixRepresentationofOperators...... 55 2.2.11CanonicalCommutationRelations...... 62 2.2.12 Matrix Elements of Unitary Operators (Changing Representation) 63 2.2.13RepresentationIndependentPropertiesofOperators...... 67 2.2.14EigenvaluesandEigenvectors...... 69 2.2.15EigenpropertiesofHermitianOperators...... 70 2.2.16ObtainingEigenvectorsandEigenvalues...... 71 2.2.17CommonEigenstatesofCommutingObservables...... 77 4 CONTENTS

2.3 Postulate III: The Measurement of Quantum Mechanical Systems . . . . . 80 2.3.1 Sum of Probabilities ...... 85 2.3.2 MeanValues...... 86 2.3.3 StatisticalUncertainty...... 87 2.3.4 TheUncertaintyPrinciple...... 88 2.3.5 PreparationofaStateUsingaCSCO...... 90 2.4PostulateIV:Evolution...... 91 2.4.1 Construction of the Hamiltonian and Other Observables ...... 93 2.4.2 SomeFeaturesofQuantumMechanicalEvolution...... 94 2.4.3 EvolutionofMeanValues...... 96 2.4.4 Eherenfest’sTheorem...... 97 2.4.5 Evolution of Systems with Time Independent Hamiltonians . . . . 99 2.4.6 TheEvolutionOperator...... 102

3 The Harmonic Oscillator 105 3.1StatementoftheProblem...... 105 3.1.1 Algebraic Approach to the Quantum Harmonic Oscillator . . . . . 107 3.1.2 Spectrum and Eigenstates of the Number Operator N ...... 110 3.1.3 TheEnergyBasis...... 113 3.1.4 Action of Various Operators in the Energy Representation . . . . . 117 3.1.5 Time Evolution of the Harmonic Oscillator ...... 121

4 Bound States of a Central Potential 123 4.1GeneralConsiderations...... 123 4.2HydrogenicAtoms:TheCoulombProblem...... 127 4.3 The 3-D Isotropic Oscillator ...... 131

5 Approximation Methods for Stationary States 133 5.1TheVariationalMethod...... 133 5.2PerturbationTheoryforNondegenerateLevels...... 137 5.3PerturbationTheoryforDegenerateStates...... 146 5.3.1 Application: Stark Eﬀect of the n =2Level of Hydrogen . . . . . 148

6 Many Particle Systems 151 6.1TheDirectProductofLinearVectorSpaces...... 151 6.1.1 Motion in 3 Dimensions Treated as a Direct Product of Vector Spaces155 6.1.2 TheStateSpaceofSpin-1/2Particles...... 156 6.2TheStateSpaceofManyParticleSystems...... 156 6.3EvolutionofManyParticleSystems...... 158 6.4SystemsofIdenticalParticles...... 160 6.4.1 Construction of the Symmetric and Antisymmetric Subspaces . . . 162 6.4.2 NumberOperatorsandOccupationNumberStates...... 167 6.4.3 Evolution and Observables of a System of Identical Particles . . . 170 6.4.4 FockSpaceasaDirectSumofVectorSpaces...... 175 6.4.5 TheFockSpaceofIdenticalBosons...... 177 6.4.6 TheFockSpaceofIdenticalFermions...... 178 6.4.7 Observables of a System of Identical Particles Revisited ...... 182 6.4.8 FieldOperatorsandSecondQuantization...... 186 CONTENTS 5

7 Angular Momentum and Rotations 189 7.1OrbitalAngularMomentumofOneorMoreParticles...... 189 7.2RotationofPhysicalSystems...... 192 7.3RotationsinQuantumMechanics...... 196 7.4CommutationRelationsforScalarandVectorOperators...... 198 7.5RelationtoOrbitalAngularMomentum...... 200 7.6 Eigenstates and Eigenvalues of Angular Momentum Operators ...... 202 7.7OrthonormalizationofAngularMomentumEigenstates...... 207 7.8OrbitalAngularMomentumRevisited...... 211 7.9RotationalInvariance...... 216 7.9.1 IrreducibleInvariantSubspaces...... 216 7.9.2 RotationalInvarianceofStates...... 220 7.9.3 RotationalInvarianceofOperators...... 220 7.10AdditionofAngularMomenta...... 221 7.11ReducibleandIrreducibleTensorOperators...... 230 7.12TensorCommutationRelations...... 234 7.13TheWignerEckartTheorem...... 235

8 Time Dependent Perturbations: Transition Theory 239 8.1GeneralConsiderations...... 239 8.2PeriodicPerturbations:Fermi’sGoldenRule...... 245 8.3PerturbationsthatTurnOn...... 250 8.3.1 Sudden Perturbations ...... 251 8.3.2 TheAdiabaticTheorem...... 252 8.4Appendix:Landau-ZenerTransitions...... 254 8.5 Free Particle Propagator ...... 259 8.6 Particle in a time dependent ﬁeld...... 260

9 Scattering Theory 263 9.1GeneralConsiderations...... 263 9.2AnIntegralEquationfortheScatteringEigenfunctions...... 268 9.2.1 EvaluationOfTheGreen’sFunction...... 269 9.3TheBornExpansion...... 272 9.4ScatteringAmplitudesandT-Matrices...... 272 9.5PartialWaveExpansions...... 276

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Zdyh0sduwlfoh gxdolw| 0 Wkh skhqrphqd zh fdoo oljkw vhhpv wr ehkdyh vlpxowdqhrxvo| olnh d zdyh dqg olnh d froohfwlrq ri sduwlfohv1 Rq wkh rqh kdqg/ wkh zdyh dpsolwxgh/ uhsuhvhqwhg e| wkh hohfwulf hog H +u> w, dw hdfk srlqw/ hyroyhv ghwhuplqlvwlfdoo| dffruglqj wr Pd{zhoo*v htxdwlrqv> exw lw fduulhv zlwk lw doo lqirupdwlrq uhjduglqj wkh uhodwlyh suredelolw| +ru suredelolw| ghq0 vlw|,ri ghwhfwlqj rqh ri wkh dvvrfldwhg sduwlfohv +l1h1/ wkh skrwrqv,dw wkdw srlqw1 Wkh suredelolw| ghqvlw| ri ghwhfwlqj d sduwlfoh dw d jlyhq srlqw lv sur0 sruwlrqdo wr wkh vtxduhg prgxoxv mH+u> w,m5 +l1h1/ wr wkh lqwhqvlw|,ri wkh zdyh dpsolwxgh1

Frpphqwv=

41 Qrwh wkdw wkh vprrwko|0hyroylqj zdyh dpsolwxgh lv d hog wkdw frqwdlqv doo srvvleoh lqirupdwlrq derxw dq| vwdwlvwlfdo suhglfwlrqv wkdw fdq eh pdgh uhjduglqj dqwlflsdwhg h{shulphqwdo phdvxuhphqwv1 Wkh zdyh dpsolwxgh/ wkhuhiruh/ fdq eh ylhzhg dv uhsuh0 vhqwlqj wkh g|qdplfdo vwdwh ri wkh v|vwhp/ lq dqdorj| wr wkh skdvh vsdfh suredelolw| ghqvlw| ixqfwlrq +t> s> w, zklfk sod|v d vlplodu uroh lq vwdwlvwlfdo phfkdqlfv1 51 Wkh olqhdulw| ri Pd{zhoo*v htxdwlrqv/ zklfk jryhuq wkh hyroxwlrq ri wkh zdyh dpsol0 wxgh/ lpsolhv d sulqflsoh ri vxshusrvlwlrq=LiH4+u> w, dqg H5+u> w, duh wzr vhsdudwh vroxwlrqv wr Pd{zhoo*v htxdwlrqv +l1h1 dffhswdeoh g|qdplfdo vwdwhv ri wkh v|vwhp, wkhq d olqhdu vxshusrvlwlrq +ru olqhdu frpelqdwlrq, H+u> w,@4H4 . 5H5 +4145, ri wkhvh vroxwlrqv lv dovr d vroxwlrq/ l1h1/ lw lv dovr dq dffhswdeoh g|qdplfdo vwdwh ri wkh v|vwhp1 Vxfk d vxshusrvlwlrq sulqflsoh wxuqv rxw wr eh rqh ri wkh hvvhqwldo ihdwxuhv dvvrfldwhg zlwk wkh g|qdplfdo vwdwhv ri doo txdqwxp phfkdqlfdo v|vwhpv1

Forvho| uhodwhg wr wkh vxshusrvlwlrq sulqflsoh lv wkh dvvrfldwhg sulqflsoh ri vshfwudo gh0 frpsrvlwlrq/ zklfk zh looxvwudwh wkurxjk dqrwkhu h{shulphqw1

Srodul}dwlrq ri Oljkw Wkurxjk d Ilowhu D sodqh hohfwurpdjqhwlf zdyh/ wudyholqj dorqj wkh }0d{lv dqg olqhduo| srodul}hg dorqj d wudqvyhuvh gluhfwlrq uhsuhvhqwhg e| d xqlw yhfwru

xa @frv {a .vlq |=a +4146, hqfrxqwhuv dq lghdo srodul}lqj owhu o|lqj lq wkh {|0sodqh zlwk lwv wudqvplvvlrq d{lv srlqw0 lqj dorqj wkh { 0 d{lv1 Fodvvlfdo zdyh dqdo|vlv vd|v wkdw zh fdq ghfrpsrvh wkh hohfwulf 45 Lqwurgxfwlrq

hog lqwr frpsrqhqwv H frv dqg H vlq dorqj dq| gluhfwlrq ri lqwhuhvw1 Wkh odwwhu frp0 srqhqw lv devruehg/ wkh iruphu sdvvhv wkurxjk1 Wkh oljkw hphujlqj iurp wkh owhu lv wkhq 5 5 5 srodul}hglqwkh{0gluhfwlrq zlwk dq lqwhqvlw| V b mH{m @ H frv 1 Krz derxw wkh skrwrqvB Dv ehiruh/ dw orz lqwhqvlwlhv wkh oljkw hphujlqj iurp wkh dssdudwxv dsshduv lq glvfuhwh oxpsv/ dw udqgrp wlph lqwhuydov1 Dq| jlyhq skrwrq hlwkhusdvvhvwkurxjkrulwgrhvq*w1Lwlvqrwsrvvleohwrwhooirufhuwdlqlidskrwrqzloo jhw wkurxjk/ exw wkh vwdwlvwlfv ri wkh surfhvv lqglfdwh wkdw wkh suredelolw| ri dq| vlqjoh 5 skrwrq jhwwlqj wkurxjk lv s{ @ frv 1 Wkhvh revhuydwlrqv ohdg wr wkh iroorzlqj dgglwlrqdo fkdudfwhul}dwlrq ri wkh phdvxuhphqw surfhvv=

41Wkhuhvxowridphdvxuhphqwsurfhvvlvdozd|vrqhridfhuwdlqvhwrihljhqydoxhv dvvrfldwhg zlwk wkh sduwlfxodu txdqwlw| ehlqj phdvxuhg1 Wkh vhw ri hljhqydoxhv iru d jlyhq revhuydeoh lv fdoohg lwv vshfwuxp1 +Khuh wkh hljhqydoxh ri wkh revhuydeoh lv htxdo wr rqh li dq lqflghqw skrwrq sdvvhv wkurxjk wkh srodul}hu dqg lv htxdo wr }hur li lw grhv qrw1, 51 Wr hdfk hljhqydoxh ri dq revhuydeoh wkhuh fruuhvsrqgv dw ohdvw rqh g|qdplfdo vwdwh/ uhihuuhg wr dv dq hljhqvwdwh ri wkh revhuydeoh1 Zkhq wkh sduwlfoh lv nqrzq wr eh gh qlwho| lq rqh ri wkh hljhqvwdwhv dw wkh wlph ri phdvxuhphqw/ wkh uhvxow zloo eh wkh fruuhvsrqglqj hljhqydoxh zlwk xqlw suredelolw|1 +Khuh wkh hljhqvwdwhv duh vwdwhv ri olqhdu srodul}dwlrq m{al fruuhvsrqglqj wr hljhqydoxh 4/ dqg vwdwhv ri olqhdu srodul}dwlrq m|al fruuhvsrqglqj wr hljhqydoxh 31, 61 Wkh vxshusrvlwlrq sulqflsoh doorzv wkh sduwlfoh wr eh lq d olqhdu vxshusrvlwlrq ri glhuhqw hljhqvwdwhv/ wkh olqhdu frh!flhqwv ri zklfk zh fdoo wkh dpsolwxgh wr eh lq wkdw hljhqvwdwh1 71 Wkh sulqflsoh ri vshfwudo ghfrpsrvlwlrq jrhv ixuwkhu dqg ghpdqgv wkdw dq duelwudu| vwdwh ri wkh v|vwhp fdq eh vshfwudoo| ghfrpsrvhg lq wklv pdqqhu/ l1h1/ zulwwhq dv d olqhdu vxshusrvlwlrq ri wkh hljhqvwdwhv ri dq| revhuydeoh txdqwlw|1 Wkxv/ zh frxog kdyh rulhqwhg rxu ghwhfwru dorqj dq| gluhfwlrq lq wkh {| sodqh dqg shuiruphg d vlplodu ghfrpsrvlwlrq1 81 Lq jhqhudo/ wkh uhvxow ri d phdvxuhphqw rq dq duelwudu| vwdwh lv xqfhuwdlq1 Krzhyhu/ wkh uhodwlyh suredelolw| wkdw wkh phdvxuhg ydoxh zloo wxuq rxw wr eh d jlyhq hljhqydoxh lv sursruwlrqdo wr wkh vtxduh ri wkh dpsolwxgh iru lw wr eh lq wkdw hljhqvwdwh1 +Khuh wkh vxshusrvlwlrq vwdwh lv hvvhqwldoo| uhsuhvhqwhg e| wkh srodul}dwlrq yhfwru dqg fdq eh zulwwhq/ xvlqj d qrwdwlrq wkdw zh zloo ghyhors pruh ixoo| odwhu/

mxl @ m{al frv . m|al vlq =

Zh phdvxuh 4 zlwk suredelolw| frv5 / dqg 3 zlwk suredelolw| vlq5 1, 91 Lpphgldwho| diwhu dq lghdo phdvxuhphqw/ wkh sduwlfoh lv +zlwk xqlw suredelolw|,lq dq hljhqvwdwh frqvlvwhqw zlwk wkh sduwlfxodu hljhqydoxh phdvxuhg1 +Rqo| skrwrqv zklfk duh {0srodul}hg hphujh iurp wkh srodul}hu1,

Frpphqwv= Wklv odvw sduw lpsolhv d qrq0ghwhuplqlvwlf uhgxfwlrq ru froodsvh ri wkh vwdwh ri wkh v|vwhp wr rqh zklfk lv frqvlvwhqw zlwk wkh uhvxow ri wkh phdvxuhphqw surfhvv1 Wklv lv dovr vlplodu wr zkdw kdsshqv zlwk wkh suredelolw| ghqvlw| ixqfwlrq +t> s> w, lq vwdwlvwlfdo phfkdqlfv h{fhsw iru rqh lpsruwdqw glhuhqfh= lq txdqwxp phfkdqlfv wklv uhgxfwlrq rffxuv hyhq zkhq wkh g|qdplfdo vwdwh lv nqrzq h{dfwo|1 Lq wkh deryh h{dpsoh/ wkh zdyh lqflghqw rq wkh owhu lv frpsohwho| dqg xqltxho| fkdudfwhul}hg e| lwv srodul}dwlrq yhfwru dqg lqwhqvlw|1 Wkhuh lv qr dgglwlrqdo lqirupdwlrq wkdw fdq eh jlyhq wkdw zrxog Wkh Zdyh Phfkdqlfv ri Vfkuùglqjhu 46 fkdudfwhul}h lw ixuwkhu/ zlwkrxw shuiruplqj d phdvxuhphqw wkdw zrxog luuhyhuvleo| dowhu wkh g|qdplfdo vwdwh1 Lpsolfdwlrq= Lq txdqwxp phfkdqlfv lw lv qrw jhqhudoo| wuxh wkdw wkh ydoxh ri d g|qdplfdo txdqwlw| fdq eh suhflvho| phdvxuhg zlwkrxw shuwxuelqj wkh hyroxwlrq ri wkh v|vwhp lq wkh surfhvv1

Pdwwhu Zdyhv Wkh gh Eurjolh k|srwkhvlv= mxvw dv wkhuh duh sduwlfoh surshuwlhv dvvrfldwhg zlwk fodvvlfdo zdyhv ri hohfwurpdjqhwlvp/ vr wkhuh duh zdyh0olnh surshuwlhv dvvrfldwhg zlwk pdwhuldo sduwlfohv1 Yhul fdwlrq= Gdylvvrq dqg Jhuphu vkrzhg wkdw hohfwurqv frxog eh pdgh wr h{klelw zdyholnh lqwhuihuhqfh dqg gludfwlrq hhfwv/ mxvw olnh oljkw1 Lq dqdorj| wr wkh skrwrq/ rqh lv ohg wr dvvrfldwh zlwk d iuhh pdwhuldo sduwlfoh ri prphqwxp s/dzdyhri zdyhyhfwru n uhodwhg wr lw wkurxjk wkh vdph uhodwlrq s @ |n dv dssolhv wr skrwrqv/ dqg zlwk d fruuhvsrqglqj zdyhohqjwk 5 @ > +4147, mnm dqg iuhtxhqf| H s5 $ @ @ = +4148, | 5p| E| dsso|lqj rxu frqfoxvlrqv uhjduglqj oljkw gluhfwo| wr wkh fdvh ri pdwhuldo sduwl0 fohv rqh rewdlqv/ dorqj zlwk dq dgglwlrqdo hyroxwlrq htxdwlrq srvwxodwhg e| Vfkuùglqjhu/ zkdw lv riwhq uhihuuhg wr dv Zdyh Phfkdqlfv1

416 Wkh Zdyh Phfkdqlfv ri Vfkuùglqjhu 41614 Srvwxodwhv ri Zdyh Phfkdqlfv iru d Vlqjoh Vslqohvv Sduwlfoh 41 Doo srvvleoh lqirupdwlrq derxw wkh txdqwxp vwdwh ri d sduwlfoh dw wlph w lv frqwdlqhg lq d frpsoh{0ydoxhg zdyh ixqfwlrq #+u> w,1 Wkh zdyh ixqfwlrq #+u> w, jlyhv wkh suredelolw| dpsolwxgh iru qglqj wkh sduwlfoh dw wkh srlqw u dw wlph w1Vshfl 0 fdoo|/ wkh suredelolw| gS ri qglqj wkh sduwlfoh lq d glhuhqwldo yroxph hohphqw g6u fhqwhuhg dw u dw wlph w lv sursruwlrqdo wr wkh vtxduhg prgxoxv ri wkh fruuhvsrqglqj suredelolw| dpsolwxgh=

gS @ +u> w,g6u @ m#+u> w,m5g6u +4149,

Wkh ixqfwlrq +u> w,@m#+u> w,m5 lq wklv h{suhvvlrq lv wkh dvvrfldwhg suredelolw| ghqvlw|1 Wkhqrupdol}dwlrqriwkhzdyhixqfwlrqlvriwhqfkrvhqvrwkdwwkhsured0 elolw| ri qglqj wkh sduwlfoh vrphzkhuh lq wkh xqlyhuvh lv htxdo wr 41 Wklv qhfhvvlwdwhv wkh qrupdol}dwlrq frqglwlrq ] m#+u> w,m5g6u @4> +414:,

l1h1/ wkh zdyh ixqfwlrq lv vtxduh qrupdol}hg wr xqlw|1

51 Iru hdfk revhuydeoh ru phdvxudeoh txdqwlw| D ri wkh v|vwhp wkhuh lv dvvrfldwhg d olqhdu vhoi0dgmrlqw rshudwru D/ wkdw dfwlrq ri zklfk rq wkh zdyh ixqfwlrq #+u, lv wr

uhsodfh lw zlwk dqrwkhu ixqfwlrq #D+u,@D#= Iru h{dpsoh/ wkh sduwlfoh*v prphqwxp s lv dvvrfldwhg zlwk wkh rshudwru S @ l|u 1 Zkhq phdvxulqj dq| jlyhq revhuydeoh D> wkhuh duh rqo| d fhuwdlq vhw ri ydoxhv idj/ uhihuuhg wr dv hljhqydoxhv/ zklfk pd| eh rewdlqhg1 Wkh vhw idj ri hljhqydoxhv lv uhihuuhg wr dv wkh vshfwuxp ri D1Iru 47 Lqwurgxfwlrq

hdfk hljhqydoxh d wkhuh lv dw ohdvw rqh qrupdol}hg zdyh ixqfwlrq !d+u,/ uhihuuhg wr dv dq hljhqixqfwlrq ri D/ zklfk vdwlv hv wkh hljhqydoxh htxdwlrq iru wkh dvvrfldwhg

olqhdu rshudwru/ l1h1/ D!d+u,@d!d+u,= Wkh vhw ri hljhqixqfwlrqv dvvrfldwhg zlwk dq| revhuydeoh duh vx!flhqwo| frpsohwh/ wkdw wkh zdyh ixqfwlrq #+u, iru dq duelwudu| vwdwh ri wkh v|vwhp pd| eh vshfwudoo| ghfrpsrvhg [ #+u,@ fd!d+u,> +414;, idj lqwr d olqhdu vxshusrvlwlrq ri wkh hljhqvwdwhv dvvrfldwhg zlwk wkdw revhuydeoh/ iru d xqltxh vhw ri frpsoh{ frqvwdqwv fd1 Qrupdol}dwlrq ri wkh zdyh ixqfwlrq # dqg Swkh hljhqixqfwlrqv !d ohdg wr d qrupdol}dwlrq frqglwlrq iru wkh dpsolwxghv/ l1h1/ 5 idj mfdm @4= 61 Phdvxuhphqw ri D zkhq wkh sduwlfoh lv lq wkh vwdwh fruuhvsrqglqj wr wkh hljhq0

ixqfwlrq !d+u, zloo |lhog wkh ydoxh d zlwk xqlw suredelolw|1 Lqghhg/ lw lv rqo| zkhq wkh v|vwhp lv lq d vwdwh uhsuhvhqwhg e| vxfk dq hljhqixqfwlrq wkdw lw fdq surshuo| eh vdlg wr dfwxdoo| srvvhvv wkh surshuw| dvvrfldwhg zlwk wkdw revhuydeoh1 Wkh uh0 vxow riS phdvxulqj wkh revhuydeoh D rq wkh v|vwhp zkhq lw lv lq dq duelwudu| vwdwh # @ idj fd!d> zloo eh rqh ri wkh hljhqydoxhv uhsuhvhqwhg lq wkh ghfrpsrvlwlrq ri wkdw vwdwh1 Wkh suredelolw| ri phdvxulqj d sduwlfxodu hljhqydoxh d lv sursruwlrqdo wr wkh vtxduh ri wkh fruuhvsrqglqj suredelolw| dpsolwxgh1 Lq sduwlfxodu/ zlwk wkh qrupdol}dwlrq frqyhqwlrq lqwurgxfhg deryh/

5 S +d,@mfdm = +414<,

Wkxv/ fd lv uhihuuhg wr dv wkh dpsolwxgh wkdw d phdvxuhphqw ri D zloo |lhog wkh ydoxh d1 Lpphgldwho| diwhu dq lghdo phdvxuhphqw zklfk |lhogv wkh ydoxh d iru wkh

revhuydeoh D/ wkh v|vwhp zloo eh lq dq hljhqixqfwlrq !d frqvlvwhqw zlwk wkh ydoxh rewdlqhg1 Iru wklv uhdvrq/ fd lv dovr uhihuuhg wr dv wkh dpsolwxgh wkdw d phdvxuhphqw ri D zloo qg wkh v|vwhp lq wkh vwdwh !d= 71 Ehwzhhq phdvxuhphqwv wkh zdyh ixqfwlrq hyroyhv vprrwko| dqg ghwhuplqlvwlfdoo| dffruglqj wr Vfkuùglqjhu*v htxdwlrq ri prwlrq1 Iru d vlqjoh sduwlfoh zlwk qr lqwhuqdo vwuxfwxuh Vfkuùglqjhu*v htxdwlrq wdnhv wkh irup C#+u> w, l| @ K#+u> w, +4153, Cw zkhuh wkh Kdplowrqldq K dsshdulqj lq wkh hyroxwlrq htxdwlrq lv d glhuhqwldo rshudwru rewdlqhg iurp wkh fruuhvsrqglqj fodvvlfdo Kdplowrqldq ixqfwlrq K+u>s> w, e| uhsodflqj doo rffxuuhqfhv ri wkh prphqwxp s e| wkh prphqwxp rshudwru S @ l|u 1 Wkxv/ iru h{dpsoh/ d sduwlfoh ri pdvv p prylqj lq d vfdodu srwhqwldo hqhuj| X+u> w, kdv d fodvvlfdo Kdplowrqldq ixqfwlrq s5 K+u>s> w,@ . X+u> w, 5p zklfk ohdgv wr wkh xvxdo irup ri wkh Vfkurglqjhu htxdwlrq C#+u> w, |5u5 l| @ #+u> w,.X+u> w,#+u> w,= Cw 5p Rq wkh rwkhu kdqg/ iru d sduwlfoh ri fkdujh h prylqj lq dq hohfwurpdjqhwlf hog zlwk vfdodu srwhqwldo hqhuj| X @ h!+u> w, dqg yhfwru srwhqwldo D wkh fodvvlfdo Kdplowrqldq wdnhv wkh irup 4 h 5 K+u>s> w,@ s D . X+u> w, 5p f Wkh Zdyh Phfkdqlfv ri Vfkuùglqjhu 48

zklfk jlyhv wkh Vfkurglqjhu htxdwlrq lq wkh irup

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S @ l|u / +4195, zklfk lv d yhfwru rshudwru/ l1h1/ d froohfwlrq ri wkuhh rshudwruv iS{>S|>S}j wkdw wudqvirup olnh wkh frpsrqhqwv ri d yhfwru1 Wkh hljhqydoxh htxdwlrq iru wkh prphqwxp rshudwru wdnhv wkh irup S! @ s!> +4196, zkhuh wkh yhfwru s lq wklv h{suhvvlrq uhsuhvhqwv d yhfwru hljhqydoxh dssursuldwh wr wkh yhfwru rshudwru1 Rxu sodqh zdyh vroxwlrqv reh| wklv htxdwlrq/ vlqfh

Qrupdol}dwlrq Frqyhqwlrqv iru Iuhh Sduwlfoh Hljhqvwdwhv Zh qrz wxuq wr wkh txhvwlrq ri wkh qrupdol}dwlrq frqvwdqw1 Wkh txhvwlrq lv/ zkdw ydoxh vkrxog zh dvvljq wr wkh frqvwdqw D lq +418<,vr wkdw wkh suredelolw| ri qglqj wkh sduwlfoh vrphzkhuh lq wkh xqlyhuvh lv htxdo wr rqh1 D qdlyh dwwhpsw wr ghwhuplqh wkh pdjqlwxgh ri wkh qrupdol}dwlrq frqvwdqw zrxog ohdg xv wr hydoxdwh wkh lqwhjudo ] ] ] 6 6 5 6 5 g u+u> w,@ g u m#n+u> w,m @ g u mDm > +4199, zklfk looxvwudwhv wkh sureohp1 Li mDm5 lv qlwh wkhq wkh lqwhjudo ryhu doo vsdfh glyhujhv1 Li zh vhw D htxdo wr }hur/ wkhq wkh zdyh ixqfwlrq ydqlvkhv hyhu|zkhuh1 Grhv wklv phdq wkdw rxu vroxwlrqv duh/ lq idfw/ xqdffhswdeohB Zh duh uhoxfwdqw wr ohw wkhp jr/ vlqfh wkh| uhfryhu wkh fodvvlfdo vshfwuxp vr qlfho|/ dqg pruhryhu/ wkh| gr doorz xv wr rewdlq lqirupdwlrq derxw wkh uhodwlyh suredelolwlhv ri qglqj wkh sduwlfoh lq vrph uhjlrq ri vsdfh1 Wkdw lv/ li wkh sduwlfoh lv lq rqh ri wkhvh iuhh sduwlfoh hljhqvwdwhv/ wkhq wkh suredelolw| wr qg wkh sduwlfoh lq dq| jlyhq yroxph lv vlpso| sursruwlrqdo wr wkh yroxph ri wkh dvvrfldwhg uhjlrq1 Wkh rqo| sureohp lv wkdw/ vlqfh wkh sduwlfoh fdq eh dq|zkhuh/ wkh uhodwlyh suredelolw| ri qglqj lw lq vrph sduwlfxodu yroxph lv h{fhhglqjo| vpdoo zkhq frpsduhg wr wkh yroxph ri wkh xqlyhuvh1 Sk|vlfdoo|/ wklv grhv qrw vhhp vr lpsodxvleoh1 Zh duh wkxv hqfrxudjhg wr vhhn d uhdvrqdeoh qrupdol}dwlrq frqyhqwlrq iru wkhvh vwdwhv wkdw zloo doorz xv wr ghdo zlwk wkhp lq d pdwkhpdwlfdoo| frqvlvwhqw idvklrq1 Lq idfw/ wzr vxfk frqyhqwlrqv duh frpprqo| dgrswhg=

41 Er{ Qrupdol}dwlrq 0 Lq wklv frqyhqwlrq/ wkh zdyh ixqfwlrq lv dvvxphg wr eh frq qhg wr d yhu| odujh vtxduh er{ ri yroxph Y / zlwk dssursuldwh erxqgdu| frqglwlrqv lpsrvhg dw wkh hgjhv ri wkh er{1 D xvhixo frqyhqwlrq lv wr pdnh wkh ydoxh ri wkh zdyh ixqfwlrq htxdo dw rssrvlwh idfhv/ d fkrlfh uhihuuhg wr dv shulrglf erxqgdu| frqglwlrqv1 Wkxv/ rqh vhwv Y Ã4@5hlnÄu iru u 5 Y ! +u,@ +419:, n 3 iru u@5 Y Lq wklv zd| wkh zdyh ixqfwlrq lv vtxduh0qrupdol}hg wr xqlw|1 Xqiruwxqdwho|/ wklv frqyhqwlrq kdv wkh gudzedfn wkdw lw glvfuhwl}hv wkh iuhh sduwlfoh vshfwuxp ri erwk wkh hqhuj| dqg wkh prphqwxp/ vlqfh wkh erxqgdu| frqglwlrqv uhtxluh wkdw rqo| zdyhohqjwkv wkdw mxvw w zlwklq wkh glphqvlrqv ri wkh er{ duh doorzhg1 Wklv txdqwl0 }dwlrq ri wkh iuhh sduwlfoh vshfwuxp lv xqiruwxqdwh/ dqg fdq eh dyrlghg e| hpsor|lqj wkh iroorzlqj dowhuqdwlyh frqyhqwlrq1 51 Ghowd Ixqfwlrq Qrupdol}dwlrq 0 Lq wklv dssurdfk/ rqh jlyhv xs wkh dwwhpsw wr sur0 gxfh d vtxduh0qrupdol}hg zdyh ixqfwlrq/ exw lqvwhdg fkrrvhv wkh qrupdol}dwlrq frq0 vwdqw iru pdwkhpdwlfdo frqyhqlhqfh1 Vshfl fdoo|/ zh fkrrvh wkh frqvwdqw D vr wkdw wkh vhw ri ixqfwlrqv i!n+u,j iru doo zdyhyhfwruv n irup d jhqhudol}hg ruwkrqrupdo vhw ri ixqfwlrqv rq U61

Wr xqghuvwdqg zkdw wklv phdqv zh qhhg wr gh qh wkh frqfhsw ri dq ruwkrqrupdo vhw1

Ruwkrqrupdo Vhw ri Ixqfwlrqv 0 D vhw ri ixqfwlrqv i!q+{,j odehohg e| d glvfuhwh lqgh{ q lv vdlg wr irup dq ruwkrqrupdo vhw ri ixqfwlrqv rq wkh lqwhuydo +d> e, li ] e Æ 4 li q @ p g{ !q+{,!p+{,@q>p @ = +419;, d 3 li q 9@ p 55 Lqwurgxfwlrq

Vlplodu gh qlwlrqv fdq reylrxvo| eh jlyhq iru dq ruwkrqrupdo vhw ri ixqfwlrqv rq wkh zkroh d{lv/ ru iru ixqfwlrqv lq kljkhu glphqvlrqv1 Wkxv/ d vhw ri ixqfwlrqv i!q+u,j odehohg e| d glvfuhwh lqgh{ q lv vdlg wr irup dq ruwkrqrupdo vhw ri ixqfwlrqv rq U6 li ] 6 Æ g u!q+u,!p+u,@q>p> +419<, zkhuh khuh dqg hyhu|zkhuh zklfk iroorzv/ dq lqwhjudo zlwk qr olplwv lpsolhv d gh qlwh lqwhjudwlrq ryhu doo ydoxhv dvvrfldwhg zlwk wkh lqwhjudwlrq yduldeoh +lq wklv fdvh ryhu doo yhfwruv lq U6,1 Qrwh wkdw wkh er{0qrupdol}dwlrq frqyhqwlrq ghvfulehg deryh surgxfhv d glvfuhwh vhw ri ixqfwlrqv zklfk irup dq ruwkrqrupdo vhw ryhu wkh yroxph ri wkh er{ lq zklfk wkh| duh frqwdlqhg1

Jhqhudol}hg Ruwkrqrupdo Vhw 0 D vhw +ru idplo|,ri ixqfwlrqv i!+{,j odehohg e| d frqwlqxrxv lqgh{ 5 U lv vdlg wr irup dq ruwkrqrupdo vhw ri ixqfwlrqv rq wkh uhdo d{lv li ] Æ 3 g{ !+{,!3 +{,@+ ,= +41:3,

Wklv lv fohduo| wkh frqwlqxrxv dqdorj ri wkh h{suhvvlrq deryh lqyroylqj wkh Nur0 qhfnhu ghowd ixqfwlrq gh qhg rq wkh lqwhjhuv1 Vlplodu gh qlwlrqv fdq reylrxvo| eh jlyhq iru dq ruwkrqrupdo vhw ri ixqfwlrqv lq kljkhu glphqvlrqv1 Wkxv/ d vhw ri ixqfwlrqv i!+u,j odehohg e| d frqwlqxrxv lqgh{ lv vdlg wr irup dq ruwkrqrupdo vhw ri ixqfwlrqv rq U6 li ] 6 Æ 3 g u!+u,!3 +u,@+ ,= +41:4,

D vhw ri ixqfwlrqv reh|lqj wklv uhodwlrq zh zloo uhihu wr dv ehlqj Gludf qrupdol}hg1 Wklv lv fohduo| wkh vruw ri wklqj zklfk zh qhhg iru wkh sodqh zdyhv/ vlqfh lw lpsolhv wkdw zkhq @ 3/ wkh prgxoxv0vtxduhg lqwhjudo glyhujhv +dv gr wkh sodqh zdyhv iru dq qrq}hur ydoxh ri D,1 Wklv gh qlwlrq vshfl hv wkh sduwlfxodu zd| lq zklfk wkh lqwhjudo glyhujhv iru wzr qhljkerulqj phpehuv ri wkh vhw dv rqh dssurdfkhv wkh rwkhu +l1h1/ lq d zd| zklfk lv sursruwlrqdo wr wkh ghowd ixqfwlrq,1 Wr dsso| wklv wr wkh sodqh zdyhv/ zh vlpso| qhhg wr jhqhudol}h wkh frqwlqxrxv lqgh{ lqwr d yhfwru n gudzq iurp d frqwlqxrxv vhw ri yhfwruv lq U61 Wkxv/ wkh Gludf qrupdol}dwlrq frqglwlrq iru wkh sodqh zdyhv lqyroyhv fkrrvlqj wkh frqvwdqw D vxfk wkdw ] g6u!Æ+u,! +u,@+n n3,= +41:5, n n3 Lqvhuwlqj wkh h{suhvvlrq iru wkh sodqh zdyhv zh rewdlq wkh qrupdol}dwlrq frqglwlrq ] ] g6u!Æ+u,! +u,@mDm5 g6uhl+n3Ãn,Äu @ +n n3, +41:6, n n3

Lq wklv h{suhvvlrq/ wkh wkuhh glphqvlrqdo ghowd ixqfwlrq pxvw eh frqvwuxfwhg iurp wkh surgxfw ri wkuhh ghowd ixqfwlrqv lq n{>n|> dqg n}1 D fkdqjh ri yduldeoh lq wkh sodqh zdyh uhsuhvhqwdwlrq ri wkh ghowd ixqfwlrq ] gn 3 +{ {3,@ hln+{Ã{ , +41:7, 5 lq zklfk zh lqwhufkdqjh wkh urohv ri n dqg { ohdgv wr wkh uhvxow wkdw ] g{ l+n3 Ãn ,{ 3 h { { @ +n n , +41:8, 5 { { Wkh Zdyh Phfkdqlfv ri Vfkuùglqjhu 56 dqg vlploduo| iru wkh uhpdlqlqj fduwhvldq frpsrqhqwv ri n1 Frpelqlqj wkhvh uhvxowv zh ghgxfh wkdw ] 6 g u 3 hl+n Ãn,Äu @ +n n3,= +41:9, +5,6 Frpsdulvrq zlwk rxu qrupdol}dwlrq uhodwlrq uhyhdov mDm5 @+5,Ã6 wr eh wkh uhodwlrq zh vhhn1 Fkrrvlqj D uhdo dqg srvlwlyh zh ghgxfh wkdw wkh vhw ri Gludf qrupdol}hg sodqh zdyhv wdnhv wkh irup hlnÄu ! +u,@ = +41::, n +5,6@5 Zlwk wklv frqyhqwlrq iru wkh iuhh sduwlfoh hljhqvwdwhv zh uhwdlq d frqwlqxrxv vshfwuxp iru erwk prphqwxp dqg hqhuj|/ exw zh jlyh xs wkh vwulfw suredelolvwlf lqwhusuhwdwlrq dvvrfl0 dwhg zlwk wkh zdyh ixqfwlrq1 Lq sudfwlfh/ wklv grhv qrw wxuq rxw wr eh vr lpsruwdqw/ vlqfh d uhdo sduwlfoh lv lqhylwdeo| lq d vxshusrvlwlrq ri iuhh sduwlfoh vwdwhv1 Vxfk d vxshusrvlwlrq ri vwdwhv fdq eh qrupdol}hg1 Zh zloo riwhq qg lw frqyhqlhqw wr zrun wkurxjk h{dpsohv lq orzhu glphqvlrqv1 Wkxv/ h1j1/ zh zloo kdyh rffdvlrq wr uhihu wr d sduwlfoh prylqj lq rqh0glphqvlrq/ dqg wkhuh0 iruh ghvfulehg e| d zdyh ixqfwlrq #+{,1 D iuhh0sduwlfoh lq rqh glphqvlrq lv wkhq dvvrfldwhg zlwk hljhqvwdwhv zklfk duh rqh0glphqvlrqdo sodqh zdyhv1 D uhylhz ri wkh dqdo|vlv jlyhq deryh uhyhdov wkdw wkh dssursuldwh qrupdol}dwlrq iru rqh0glphqvlrqdo sodqh zdyhv lv

hln{ ! +{,@ = +41:;, n +5,4@5

41618 Vxshusrvlwlrqv ri Sodqh Zdyhv dqg wkh Irxulhu Wudqvirup Wkh vxshusrvlwlrq sulqflsoh lpsolhv wkdw d vxshusrvlwlrq ri vwdwlrqdu| vroxwlrqv wr wkh iuhh sduwlfoh Vfkuùglqjhu htxdwlrq lv lwvhoi d vroxwlrq wr wkdw htxdwlrq1 Wkxv/ d srvvleoh vroxwlrq wr Vfkuùglqjhu*v htxdwlrq ri prwlrq fdq eh zulwwhq ]

6 a Ãl$nw #+u> w,@ g n #+n,!n+u,h = +41:<,

Wklv vwdwh hyroyhv iurp dqrwkhu rqh zklfk dw w @3kdv wkh irup ] 6 a #+u,@ g n #+n,!n+u,= +41;3,

Lw lv xvhixo wr zulwh wklv h{sdqvlrq lq wkh irup ] 6 a #+u> w,@ g n #+n> w,!n+u,> +41;4, zkhuh #a+n> w,@#a+n> 3,hÃl$nw lv wkh dpsolwxgh +lq wklv frqwlqxrxv vxshusrvlwlrq,ri wkh vwdwhdvvrfldwhgzlwkwkhiuhhsduwlfohhljhqvwdwhrizdyhyhfwrun1 Lq wkh wkhru|/ gS @ +n,g6n @ m#a+n,m5g6n lv wkh suredelolw| wkdw d prphqwxp phdvxuhphqw zloo |lhog d ydoxh lq dq lq qlwhvlpdo qhljkerukrrg ri sn @ |n1 Wkh sulqflsoh ri vshfwudo ghfrpsrvlwlrq ghpdqgv/ lq idfw/ wkdw dq duelwudu| vwdwh ri wkh v|vwhp dgplw vxfk dq h{sdqvlrq lq iuhh sduwlfoh hljhqvwdwhv1 Wklv lv dfwxdoo| jxdudqwhhg e| Irxulhu*v wkhruhp zklfk vwdwhv wkdw dq| vx!flhqwo| uhjxodu ixqfwlrq #+u> w, dgplwv d Irxulhu h{sdqvlrq ] g6n #+u> w,@ #a+n> w,hlnÄu +41;5, +5,6@5 57 Lqwurgxfwlrq lq sodqh zdyhv/ zkhuh wkh h{sdqvlrq ixqfwlrq #a+n> w,/ uhihuuhg wr dv wkh Irxulhu wudqvirup ri #+u> w,/ lv jlyhq e| wkh uhodwlrq ] g6u #a+n> w,@ #a+u> w,hÃlnÄu= +41;6, +5,6@5

Yhul fdwlrq ri wkhvh h{suhvvlrqv iroorzv iurp wkh sodqh0zdyh h{sdqvlrq iru wkh ghowd ixqf0 wlrq1 Wkdw lv/ li zh lqvhuw wkh h{sdqvlrq iru #a+n> w, lqwr wkdw iru #+u> w,/ ehlqj fduhixo wr xvh d glhuhqw lqwhjudwlrq yduldeoh

] 6 ] 6 3 g n g u 3 #+u> w,@ #+u 3>w, hÃlnÄu hlnÄu +41;7, +5,6@5 +5,6@5 dqg uh0rughu wkh lqwhjudwlrq

] ] 6 g n 3 #+u> w,@ g6u3#+u 3>w, hÃln+Äu Ãu, +41;8, +5,6 zh uhfrjql}h wkh h{sdqvlrq iru +u u 3,1Wkxv/ ] #+u> w,@ g6u3#+u3>w,+u u 3,@#+u> w,= +41;9,

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5 zkhuh $n @ Hn@| @ |n @5p= Zh frqfoxgh zlwk wkh revhuydwlrq wkdw wklv h{dpsoh ri wkh iuhh sduwlfoh kdv djdlq vkrzq wkdw wkhuh lv pruh wkdq rqh zd| wr uhsuhvhqw wkh g|qdplfdo vwdwh ri wkh v|vwhp1 E| Vfkuùglqjhu*v srvwxodwh/ wkh g|qdplfdo vwdwh dw wlph w lv ghvfulehg e| wkh zdyh ixqfwlrq #+u> w,1 Krzhyhu/ lw lv fohdu iurp wkh glvfxvvlrq deryh/ wkdw doo lqirupdwlrq lq wkh zdyh ixqfwlrq lv dovr frqwdlqhg lq lwv Irxulhu wudqvirup= li zh kdyh #a+n> w, zh fdq dozd|v frqvwuxfw #+u> w,1 Wklvpxvwehwuxhirudq| revhuydeoh/ qrw mxvw prphqwxp dqg2ru nlqhwlf hqhuj|1 Khqfh wkh sduwlfxodu vhw ri qxpehuv xvhg wr uhsuhvhqw wkh vwdwh ri wkh v|vwhp fdq eh fkrvhq wr fruuhvsrqg wr d sduwlfxodu revhuydeoh zklfk rqh pljkw eh lqwhuhvwhg lq dqdo|}lqj1 Wklv lv vlplodu wr wkh lghd ri fkrrvlqj d frqyhqlhqw frruglqdwh v|vwhp lq U6 lq zklfk wr vroyh d sduwlfxodu sureohp lq fodvvlfdo phfkdqlfv1 Lq zkdw iroorzv/ zh frqvwuxfw d irupdolvp iru txdqwxp phfkdqlfv zklfk grhv qrw fkrrvh dw wkh rxwvhw dq| sduwlfxodu frruglqdwh v|vwhp/ exw zklfk uhfrjql}hv wkdw wkh zdyh ixqfwlrq #+u> w, /lwv a Irxulhu wudqvirup #+n> w,/ ru dq| rwkhu vhw ri h{sdqvlrq frh!flhqwv ifq+w,j/vlpso|jlyh dphdqvriuhsuhvhqwlqj dq remhfw +zkdw zh zloo uhihu wr dv wkh vwdwh yhfwru,wkdw kdv dq h{lvwhqfhlqghshqghqwriwkhphdqvwkdwpd|ehfkrvhqwrgrvr1 Dsshqgl{= Wkh Ghowd Ixqfwlrq 58

417 Dsshqgl{= Wkh Ghowd Ixqfwlrq Vwulfwo| vshdnlqj/ wkh ghowd ixqfwlrq lv qrw d ixqfwlrq dw doo/ exw d glvwulexwlrq1Wkh glhuhqfh ehwzhhq d glvwulexwlrq dqg d ixqfwlrq lv vpdoo exw lpsruwdqw/ sduwlfxoduo| iru wkh sk|vlflvw zkr zlvkhv wr dyrlg xqsohdvdqw frpphqwv iurp pdwkhpdwlfldqv zkr juxpeoh derxw lpsurshu ixqfwlrqv dqg vr rq1 Glvwulexwlrqv dulvh txlwh qdwxudoo| lq d qxpehu ri vlwxdwlrqv1 Prvw lpsruwdqw wr xv duh wkrvh flufxpvwdqfhv zkhuh lw lv ghvludeoh wr wdon derxw wkh suredelolw| iru d frqwlqxrxvo| glvwulexwhg udqgrp yduldeoh wr wdnh rq d sduwlfxodu ydoxh1 Iru h{dpsoh/ li d jlyhq udqgrp yduldeoh { lv htxdoo| olnho| wr eh irxqg kdylqj dq| ydoxh lq wkh lqwhuydo +3>,/ wkhq lw fdq eh fkdudfwhul}hg e| wkh xqlirup suredelolw| glvwulexwlrq ; ? 4@ li { 5 +3>, +{,@ = +41;;, = 3 li {@5 +3>, Wklv glvwulexwlrq kdv wkh surshuw| wkdw wkh suredelolw| wkdw { kdv vrph ydoxh lv htxdo wr rqh/ l1h1/ ] 4 +{,g{ @4> +41;<, Ã4 zkloh wkh suredelolw| wr qg { lq dq| rwkhu lqwhuydo lv vlpso| wkh lqwhjudo ri +{, ryhu wkdw lqwhuydo1 Wklv lghd lv uhdglo| h{whqghg wr lqfoxgh dq| vx!flhqwo| zhoo0ehkdyhg lqwhjudeoh ixqfwlrq +zkhuh zhoo0ehkdyhg lv khuh udwkhu orrvho| gh qhg/ vlqfh lw reylrxvo| lqfoxghv glvfrqwlqxrxv ixqfwlrqv,1 Rwkhu frpprq glvwulexwlrqv lqfoxgh wkh Jdxvvldq +{,@s h{s^+{ {3,5` +41<3, dqg wkh Oruhqw}ldq 4 +{,@ +41<4, 5 .+{ {3,5 erwk ri zklfk zh kdyh zulwwhq lq d irup zklfk lv fhqwhuhg dw dq duelwudu| srlqw { @ {3 dqg lq zklfk wkh zlgwk ri wkh glvwulexwlrq lv frqwuroohg e| d sdudphwhu 1 Wkh glvwulexwlrq ixqfwlrq +{, lv dovr uhihuuhg wr dv wkh suredelolw| ghqvlw| wr qg wkh udqgrp yduldeoh wdnlqj d ydoxh lq wkh qhljkerukrrg ri {/ dqg zh uhihu wr gS @ +{,g{ dv wkh suredelolw| wr qg wkh udqgrp yduldeoh lq wkh lqwhuydo ehwzhhq { dqg { . g{1 Wkh Gludf ghowd ixqfwlrq +{ {3, lv wr eh ylhzhg dv wkh olplwlqj fdvh ri d glvwulexwlrq zklfk lv hqwluho| frqfhqwudwhg dw vlqjoh srlqw +lq wkh vdph zd| wkdw d srlqw fkdujh lv wr eh ylhzhg dv wkh olplwlqj fdvh ri d frqwlqxrxv fkdujh glvwulexwlrq lq zklfk doo ri wkh fkdujh lv hqwluho| frqfhqwudwhg dw d vlqjoh srlqw,1 Lq rwkhu zrugv/ li lw lv lpsrvvleoh wkdw wkh yduldeoh { wdnh rq dq| ydoxh rwkhu wkdq {3/zhgh qh lwv suredelolw| glvwulexwlrq wr eh wkh Gludf glvwulexwlrq +{,@+{ {3, +41<5, fhqwhuhg dw wkdw srlqw1 Wklv lpsolhv/ dprqj rwkhu wklqjv/ wkdw wkh lqwhjudo ryhu wkh Gludf glvwulexwlrq ] 4 +{ {3, g{ @4 +41<6, Ã4 lv htxdo wr xqlw|1 Rq wkh rwkhu kdqg/ wkh lqwhjudo ri wklv glvwulexwlrq ryhu dq| lqwhuydo qrw frqwdlqlqj wkh srlqw { @ {3 pxvw eh }hur/ vlqfh wkh suredelolw| ri qglqj wkh yduldeoh lq vxfk dq lqwhuydo ydqlvkhv e| dvvxpswlrq1 Khqfh zh frqfoxgh wkdw ; ] 3 {5 ? 4 li { 5 +{4>{5, +{ {3, g{ @ = +41<7, { = 3 4 3 li { 5@ +{4>{5, 59 Lqwurgxfwlrq

Lw vkrxog eh hylghqw wkdw wklv lv htxlydohqw wr wkh uhodwlrqv

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; ] 3 3 {5 ? i+{ , li { 5 +{4>{5, i+{,+{ {3, g{ @ +41435, { = 3 4 3 li { 5@ +{4>{5, Wkh sxulvw pljkw zhoo fodlp wkdw wkh vhfrqg ri wkhvh surshuwlhv lv uhdoo| d vshfldo fdvh ri wkh odvw1 Rwkhu surshuwlhv ri wkh ghowd ixqfwlrq fdq eh suryhq hlwkhu iurp wkhvh edvlf surshuwlhv ru iurp wkh gh qlwlrq ri wkh ghowd ixqfwlrq dv wkh olplwlqj irup ri d vxlwdeoh idplo| ri glvwulexwlrqv dv glvfxvvhg deryh1 Vxfk surshuwlhv lqfoxgh wkh iroorzlqj/ zklfk zloo eh jlyhq zlwkrxw surri= 4 +d{,@ +{,= +41436, mdm

[ 4 ^i+{,` @ +{ { , +41437, mi 3+{ ,m l l l

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H{whqvlrq wr Kljkhu Glphqvlrqv Wkh h{whqvlrq ri wkh ghowd ixqfwlrq wr kljkhu glphqvlrqv lv vwudljkwiruzdug1 Frqvlghu wzr udqgrp yduldeohv { dqg |/jryhuqhge|wkhmrlqw suredelolw| glvwulexwlrq ixqfwlrq +{> |,/ zklfk lv gh qhg vr wkdw gS @ +{> |,g{g| jlyhv wkh suredelolw| wkdw wkh uvw yduldeoh wdnhv d ydoxh ehwzhhq { dqg { . g{/ zkloh wkh vhfrqg wdnhv d ydoxh lq wkh lqwhuydo | wr | . g|1 Wklv lpsolhv wkh qrupdol}dwlrq ]] g{g| +{> |,@4> +4143:, zkhuh wkh lqwhjudwlrq lv ryhu wkh hqwluh {| sodqh1Lwlvfrqyhqlhqwwrylhzwklvdvgh qlqj wkh suredelolw| ghqvlw| jryhuqlqj d udqgrpo| glvwulexwhg yhfwru u @ {a~ . |a1Wkxv/zh zulwh +u,@+{> |,> vr wkdw ] g5u+u,@4= +4143;,

Wkh wzr0glphqvlrqdo ghowd ixqfwlrq lv wkh olplwlqj irup ri vxfk d glvwulexwlrq zkhq wkhuh lv rqo| rqh srvvlelolw| iru wkh udqgrp yhfwru u1Wkdwlv/liu @ u3 zlwk xqlw suredelolw|/ wkhq e| gh qlwlrq +u,@+uu3,1 Lw lv qrw kdug wr vhh wkdw +uu3,@+{{3,+||3,/ zkhuh {3 dqg |3 duh wkh frpsrqhqwv ri wkh yhfwru u31 Pruh jhqhudoo|/ li u 3 lv d yhfwru lq g0glphqvlrqv zh fdq gh qh wkh g0glphqvlrqdo ghowd ixqfwlrq +u u 3, kdylqj wkh iroorzlqj surshuwlhv= ; ? 3 li u 9@ u 3 +u u 3,@ +4143<, = 4 li u @ u 3 ] gY +u u 3,@4 +41443, ] gY i+u, +u u 3,@i+u 3,= +41444, zkhuh gY ghqrwhv wkh lq qlwhvlpdo yroxph hohphqw lq g0glphqvlrqv/ dqg lw lv dvvxphg wkdw wkh lqwhjudwlrqv duh ryhu dq| uhjlrq frqwdlqlqj u 3 +wkh lqwhjudo ryhu dq| uhjlrq qrw frqwdlqlqj u3 ydqlvklqj,1 Wkh g0glphqvlrqdo ghowd ixqfwlrq fdq eh zulwwhq dv wkh g0irog surgxfw 3 3 3 3 +u u ,@+{4 {4,+{5 {5, +{g {g, +41445, ri ghowd ixqfwlrqv dvvrfldwhg zlwk hdfk ri wkh fduwhvldq frpsrqhqwv1 Lq kljkhu glphqvlrqv wkh judglhqw ri wkh ghowd ixqfwlrq wdnhv wkh sduw ri wkh rqh0 glphqvlrqdo ghulydwlyh ri wkh vlpsoh ghowd ixqfwlrq1 Wkxv zh fdq gh qh wkh glvwulexwlrq u +u u 3, zklfk lq wkuhh glphqvlrqv/ iru h{dpsoh/ kdv wkh surshuw| wkdw ] g6u3 u +u u3,i+u3,@u i+u,= +41446,

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514 Srvwxodwh L= Vshfl fdwlrq ri wkh G|qdplfdo Vwdwh Wkh g|qdplfdo vwdwh ri d txdqwxp phfkdqlfdo v|vwhp lv dw hdfk lqvwdqw ri wlph dvvrfldwhgzlwk d vwdwh yhfwru m#l1 Srvvleoh vwdwh yhfwruv duh hohphqwv ri d frpsoh{ olqhdu yhfwru vsdfh V/ uhihuuhgwr dv wkh vwdwh vsdfh ru Kloehuw vsdfh ri wkh v|vwhp1 Reylrxvo|/ d suhuhtxlvlwh wr rxu xqghuvwdqglqj ri wklv srvwxodwh dqg lwv lpsolfdwlrqv lv d nqrzohgjh ri olqhdu yhfwru vsdfhv1 51414 Surshuwlhv ri Olqhdu Yhfwru Vsdfhv DvhwV ri remhfwv im#l> ml> ml>===j irupv d olqhdu yhfwru vsdfh +OYV, li lw lv forvhgxqghu wzr pxwxdoo| glvwulexwlyh rshudwlrqv= +4, dq dvvrfldwlyh dqg frppxwdwlyh odz ri yhf0 wru dgglwlrq/ dqg+5, pxowlsolfdwlrq e| hohphqwv ri dq dvvrfldwhg hog I ri vfdoduv i> > > = = =j1 Wklv rshudwlrq ri yhfwru dgglwlrq lv dvvxphg wr vdwlvi| wkh surshuwlhv hqx0 phudwhgehorz1 41 Iru doo yhfwruv m#l dqg ml lq V wkhuh h{lvwv d yhfwru m"l lq V vxfk wkdw m"l @ m#l.ml1 +Forvxuh, 51 m#l . ml @ ml . m#l1 +Frppxwlylw|, 61 m#l .+ml . m"l,@+m#l . ml,.m"l1 +Dvvrfldwlylw|, 71 Wkhuh h{lvwv d xqltxh qxoo yhfwru 3 lq V vxfk wkdw 3.ml @ ml1 81 Iru hdfk ml lq V wkhuh h{lvwv dq hohphqw ml/vxfkwkdwml .+ml,@mlml @31 +Dgglwlyh lqyhuvh, Wkh hogzlwk uhvshfw wr zklfk wkh vsdfh lv gh qhglv dq dvvrfldwhgvhw I ri qxpehuv +xvxdoo| wkh vhw U ri uhdo qxpehuv ru wkh vhw F ri frpsoh{ qxpehuv, zklfk zh pd| xvh wr pxowlso| wkh hohphqwv ri wkh vsdfh lwvhoi1 Wklv rshudwlrq lqyroylqj pxowlsolfdwlrq ri hohphqwv ri V e| hohphqwv ri I lv dvvxphgwr kdyh wkh iroorzlqj surshuwlhv= 63 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

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q[Ã4 m# l m! l @ m" l m! lk! m" l> m! l @ q = +515;, q q l l q q mm# mm l@4 q

Wkh rqo| zd| wklv surfhvv frxogvwrs lv li rqh ri wkh uhvxowlqj yhfwruv m!ql wxuqhgrxw wr eh wkh qxoo yhfwru1 D forvh lqvshfwlrq ri wkh surfhvv uhyhdov wkdw wklv fdq*w kdsshq li wkh ruljlqdo vhw lv olqhduo| lqghshqghqw/ dv zh kdyh dvvxphg1 Wkxv/ lq wklv zd| zh frqvwuxfw dq ruwkrqrupdo vhw ri yhfwruv im!qlj htxdo lq qxpehu wr wkh ruljlqdo1 Lw iroorzv/ wkdw jlyhq dq| edvlv iru wkh vsdfh zh fdq frqvwuxfw dq ruwkrqrupdo edvlv zlwk dq htxdo qxpehu ri yhfwruv1 Wkxv/ iru d qlwh glphqvlrqdo vsdfh wkhuh h{lvwv dw ohdvw rqh RQE zlwk wkh vdph qxpehu ri phpehuv dv wkh glphqvlrq ri wkh vsdfh1 Lw wxuqv rxw wkdw wkhuh fdq*w h{lvw dq| edvhv zlwk dq| ihzhu phpehuv/ ehfdxvh wkhq zh frxogxowlpdwho| hqgxs vroylqj iru rqh ri wkh phpehuv ri wkh odujhu vhw lq whupv ri wkh uhpdlqlqj phpehuv/ zklfk zrxog frqwudglfw wkhlu olqhdu lqghshqghqfh1 Wkh surri lv ohiw dv dq h{huflvh1 Iru qrz/ zh zloo vlpso| revhuyh wkdw ruwkrqrupdo edvhv duh h{wuhpho| xvhixo gxh wr wkh hdvh zlwk zklfk wkh| doorz duelwudu| yhfwruv wr eh h{suhvvhg1 Zh h{soruh wklv ehorz1 51418 H{sdqvlrq ri d Yhfwru rq dq Ruwkrqrupdo Edvlv

Glvfuhwh Edvhv 0 Ohw wkh vhw im!llj irup dq ruwkrqrupdo edvlv +ru RQE, iru wkh vsdfh V/vrwkdwk!lm!ml @ lm>dqgohw m"l eh dq duelwudu| hohphqw ri wkh vsdfh1 E| dvvxpswlrq wkhuh h{lvwv dq h{sdqvlrq ri wkh irup [ m"l @ "l m!ll +515<, l iru d xqltxh vhw ri h{sdqvlrq frh!flhqwv "l1 Krz gr zh ghwhuplqh zkdw wkhvh h{sdqvlrq frh!flhqwv duhB Frqvlghu wkh lqqhu surgxfw [ [ k!mm"l @ "l k!mm!ll @ "l lm @ "m +5163, l l ri wkh yhfwru m"l zlwk dq duelwudu| hohphqw m!ml ri wklv edvlv1 Wklv vkrzv wkdw wkh h{sdqvlrq frh!flhqw "m lv mxvw wkh lqqhu surgxfw ri wkh yhfwru ri lqwhuhvw zlwk wkh xqlw yhfwru dorqj wkdw gluhfwlrq lq Kloehuw vsdfh1 Wkxv "l @ k!lm"l1 Vlqfh wkh rughu lq zklfk zh zulwh wkh surgxfw ri d qxpehu dqg d yhfwru lv xqlpsruwdqw/ zh zloo riwhq zulwh wklv lq wkh irup [ [ m"l @ m!ll "l @ m!llk!lm"l +5164, l l 69 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv iru uhdvrqv zklfk zloo ehfrph fohduhu odwhu rq1

H{whqvlrq wr Frqwlqxrxv Edvhv 0Ohwwkhvhwim!lj irup d frqwlqxrxv ruwkrqrupdo edvlv iru wkh vsdfh V/vrwkdw 3 k!m!3 l @ + ,> +5165, dqgohw m"l eh dq duelwudu| hohphqw ri wkh vsdfh1 E| dvvxpswlrq wkhuh h{lvwv dq h{sdqvlrq ]

m"l @ g "+,m!l +5166, iru vrph xqltxh h{sdqvlrq ixqfwlrq "+,1 Krz gr zh ghwhuplqh zkdw wklv h{sdqvlrq ixqfwlrq lvB Frqvlghu wkh lqqhu surgxfw ] ] ] 3 3 k!3 m"l @ k!3 m g "+,m!l @ g "+,k!3 m!l @ g "+,+ ,@"+ , +5167, ri wkh yhfwru m"l zlwk dq duelwudu| hohphqw m!3 l ri wkh edvlv1 Wklv vkrzv wkdw/ dv lq wkh glvfuhwh fdvh/ wkh h{sdqvlrq frh!flhqw "+3, lv mxvw wkh lqqhu surgxfw ri wkh yhfwru ri lqwhuhvw zlwk wkh edvlv yhfwru dorqj wkdw gluhfwlrq lq Kloehuw vsdfh1 Wkxv "+,@k!m"l1 Zh zloo uhihu wr wkh ixqfwlrq "+, dv wkh zdyh ixqfwlrq uhsuhvhqwlqj m"l lq wkh 0edvlv ru uhsuhvhqwdwlrq1 Djdlq/ vlqfh wkh rughu lq zklfk zh zulwh wkh surgxfw ri d qxpehu dqgd yhfwru lv xqlpsruwdqw/ zh zloo zulwh wklv riwhq lq wkh irup ] ]

m"l @ g m!l"+,@ g m!lk!m"l= +5168,

Frpphqw= Lw lv fohdu wkdw zkhq zh wdon derxw RQE*v/ vxfk dv im!llj ru im!lj/wkh lpsruwdqw lqirupdwlrq dsshdulqj lqvlgh wkh nhw zklfk glvwlqjxlvkhv wkh glhuhqw edvlv yhfwruv iurp rqh dqrwkhu lv wkh odeho ru lqgh{= l ru m lq wkh glvfuhwh fdvh/ ru 3 lq wkh frqwlqxrxv fdvh1 Wkh v|perov ! mxvw vruw ri frph dorqj iru wkh ulgh1 Iurp wklv srlqw rq zh zloo dfnqrzohgjh wklv e| xvlqj wkh deeuhyldwhg qrwdwlrqv

mll @ m!ll +5169, dqg

ml @ m!l= +516:, Lq wklv zd| wkh h{sdqvlrq ri dq duelwudu| nhw fdq eh zulwwhq [ [ m"l @ "l mll @ mllklm"l +516;, l l dqg ] ] m"l @ g "+, ml @ g mlkm"l= +516<,

51419 Fdofxodwlrq ri Lqqhu Surgxfwv Xvlqj dq Ruwkrqrupdo Edvlv Glvfuhwh Edvhv 0 Ohw wkh vhw imllj irup dq ruwkrqrupdo edvlv +ru RQE, iru wkh vsdfh V/vrwkdwklmml @ lm>dqgohw m"l> m#l eh duelwudu| hohphqwv ri wkh vsdfh1 E| dvvxpswlrq wkhuh h{lvwv h{sdqvlrqv ri wkh irup [ [ m"l @ "l mll @ mllklm"l +5173, l l Srvwxodwh L= Vshfl fdwlrq ri wkh G|qdplfdo Vwdwh 6: [ [ m#l @ #l mll @ mllklm#l= +5174, l l Wkhlqqhusurgxfwriwkhvhwzryhfwruvfdqehzulwwhq + , [ [ k#m"l @ k#m "l mll @ "l k#mll= +5175, l l

Æ Æ Exw k#mll @+klm#l, @ #l 1 Wkxv/ zh fdq zulwh wkh lqqhu surgxfw lq wkh irup [ Æ k#m"l @ #l "l = +5176, l Exw wklv lv wkh irup rewdlqhge| wdnlqj wkh grwsurgxfw ri d frpsoh{0frqmxjdwhgurz yhfwru dqgd froxpq yhfwru1 Wklv mxvwl hv rxu hduolhu orrvh dvvrfldwlrq ri nhwv dqgeudv zlwk froxpq dqg urz yhfwruv/ dqg/ lq idfw/ pdnhv fohdu wkh frqglwlrqv xqghu zklfk vxfk d slfwxuh lv mxvwl hg/ l1h1/

Dq| glvfuhwh RQE iru d vsdfh V lqgxfhv +ru jhqhudwhv ru gh qhv, d urz0 yhfwru2froxpq yhfwru uhsuhvhqwdwlrq iru wkh vsdfh/ l1h1/ lw jlyhv xv d qdwxudo zd| ri dvvrfldwlqj hdfk devwudfw nhw m#l lq V zlwk d frpsoh{0ydoxhg Æ froxpq yhfwru kdylqj frpsrqhqwv #l/ dqghdfk eud k#m lq V zlwk d frpsoh{0 Æ ydoxhgurz yhfwru kdylqj frpsrqhqwv #l 1 H{whqvlrq wr Frqwlqxrxv Edvhv 0 Ohw wkh vhw imlj irup d frqwlqxrxv ruwkrqrupdo edvlv iru wkh vsdfh V/vrwkdw km3l @ + 3,> +5177, dqgohw m"l>m#l eh duelwudu| hohphqwv ri wkh vsdfh1 E| dvvxpswlrq wkhuh h{lvw h{sdqvlrqv ri wkh irup ] ] m"l @ g "+,ml @ g mlkm"l +5178, ] ] m#l @ g #+,ml @ g mlkm#l= +5179,

Wkhlqqhusurgxfwriwkhvhwzryhfwruvfdqehzulwwhq ] ] k#m"l @ k#m g "+,ml @ g "+,k#ml= +517:,

Exw k#ml @+km#l,Æ @ #Æ+,1 Wkxv/ zh fdq zulwh wkh lqqhu surgxfw lq wkh irup ] ] k#m"l @ g "+,#Æ+,@ g #Æ+,"+,= +517;,

Wklv lv dovr lqwhuhvwlqj/ ehfdxvh lw orrnv mxvw olnh wkh lqqhu surgxfw zklfk dsshduv lq ixqfwlrqdo olqhdu yhfwru vsdfhv/ vxfk dv wkh vhw ri Irxulhu wudqvirupdeoh ixqfwlrqv rq U61 Wklv vxjjhvwv wkh iroorzlqj lpsruwdqw srlqw=

Dq| frqwlqxrxv RQE iru d vsdfh V lqgxfhv d zdyh ixqfwlrq uhsuhvhqwd0 wlrq iru wkh vsdfh/ l1h1/ lw jlyhv xv d qdwxudo pdsslqj ri hdfk devwudfw yhfwru m#l lq V rqwr d frpsoh{ ydoxhgzdyh ixqfwlrq #+,/ zklfk jlyh wkh h{sdq0 vlrq frh!flhqwv iru wkh vwdwh lq wkdw frqwlqxrxvo|0lqgh{hg edvlv1 Vlploduo|/ lw pdsv hdfk yhfwru k#m lq VÆ rqwr d frpsoh{0ydoxhgzdyh ixqfwlrq #Æ+,1Zh vshdn/ wkhuhiruh/ ri #+, dv wkh zdyh ixqfwlrq iru wkh vwdwh m#l lq wkh imlj uhsuhvhqwdwlrq1 6; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Wklv vlplodulw| wr zkdw zh vhh zlwk wkh ixqfwlrqdo vsdfhv zrunv lq wkh rssrvlwh gluhfwlrq dv zhoo1 Wkdw lv/ wkh irup ri wkh lqqhu surgxfw dvvrfldwhg zlwk ixqfwlrqv rq U6 ] k#m!l @ g6u#Æ+u,!+u, +517<, vxjjhvwv wkdw wkh zdyh ixqfwlrq #+u, fdq eh ylhzhgdv phuho| wkh ixqfwlrq jlylqj wkh h{sdqvlrq frh!flhqwv lq vrph ruwkrqrupdo edvlv ri vwdwhv imulj odehohge| wkh +frqwlqx0 rxvo| glvwulexwhg, srvlwlrq yhfwruv u lq U61 Wklv ohdgv wr vrphwklqj fdoohg wkh srvlwlrq uhsuhvhqwdwlrq1

5141: Wkh Srvlwlrq Uhsuhvhqwdwlrq Lq wkh vwdwh vsdfh ri d vlqjoh txdqwxp phfkdqlfdo sduwlfoh prylqj lq wkuhh glphqvlrqv zh zlvk wr pdnh dq dvvrfldwlrq #+u, #$ m #l +5183, ri hdfk ixqfwlrq #+u, zlwk dq xqghuo|lqj devwudfw yhfwru lq wkh vsdfh m#l1 Zh gh qh wklv dvvrfldwlrq pruh suhflvho| e| frqvlghulqj d sduwlfxodu vhw ri ixqfwlrqv/ qdpho|/ wkh Gludf ghowd ixqfwlrqv i+u u 3,j1 Iru hdfk srlqw u 3 lq vsdfh/ wkhuh lv d ghowd ixqfwlrq fhqwhuhg dw wkdw srlqw/ zklfk txdqwxp phfkdqlfdoo| zrxogfruuhvsrqgwr d sduwlfoh zklfk kdv doo ri lwv suredelolw| ghqvlw| orfdwhg dw u 31 Wkxv/ zh kdyh dq hqwluh vhw ri ixqfwlrqv odehohg e| wkh srlqwv lq U61 Zlwk hdfk ri wkhvh srvlwlrq0orfdol}hgzdyh ixqfwlrqv zh dvvrfldwh d vwdwh mu 3l ri wkh vwdwh vsdfh1 Wkxv zh kdyh wkh dvvrfldwlrq

3 3 mu l#$!u 3 +u, +u u , +5184, zkhuh zh kdyh lqwurgxfhg d qrwdwlrq zklfk vxjjhvwv wkdw/ lq wklv frqwh{w/ wkh ghowd ixqf0 wlrq lv wr eh frqvlghuhg d ixqfwlrq ri u/ zklfk kdsshqv wr eh odehohg e| wkh srlqw u 31Zh fodlp wkdw wklv vhw ri nhwv imu 3lj irupv d frqwlqxrxv ruwkrqrupdo edvlv iru wkh xqghuo|lqj vsdfh1 Wklv lv lqwxlwlyho| uhdvrqdeoh lqvridu dv d sduwlfoh orfdwhgdw u lv lqfrpsdwleoh zlwk lw ehlqj orfdwhgdw dq| rwkhu srlqw1 Iru wkh prphqw zh zloo vlpso| dvvxph wkdw wklv lv wuxh dqgvhh zkhuh lw ohdgvxv1 Wkh dvvxphgruwkrqrupdolw| ri wkh vwdwhv imu 3lj ohdgv xv wr wkh ruwkrqrupdolw| uhodwlrq/ dv gh qhgiru frqwlqxrxvo| lqgh{hgvwdwhv/

kumu 3l @ +u u 3,= +5185,

Lq dgglwlrq/ wkh dvvxphg frpsohwhqhvv ri wkh vhw lpsolhv wkdw dq| rwkhu vwdwh lq wkh vsdfh m#l pxvw eh h{sdqgdeoh lq wklv edvlv1 Wkxv zh fdq zulwh ] m#l @ g6u 3 #+u 3,mu 3l +5186, zkhuhzhfdqzulwh#+u 3,@ku 3m#l/ lq dqdorj| wr zkdw zh glg zlwk wkh frqwlqxrxv edvlv ml1 Vlqfh wkh lqwhjudwlrq yduldeoh lv mxvw d gxpp|/ zh fdq gurs wkh sulph dqg zulwh wklv ] m#l @ g6u#+u,mul= +5187,

Lw lv qrz qdwxudo wr dvvxph wkdw wkh ixqfwlrq #+u,@kum#l/ zklfk khuh mxvw jlyhv wkh h{sdqvlrq frh!flhqwv iru m#l lq wkh mul edvlv/ lv suhflvho| wkh zdyh ixqfwlrq #+u, wkdw zh zdqwhgwr dvvrfldwh zlwk wkh vwdwh m#l wr ehjlq zlwk1 Wklv wxuqv rxw wr eh d frqvlvwhqw lqwhusuhwdwlrq1 Lq/ sduwlfxodu/ lw suhglfwv wkdw wkh zdyh ixqfwlrq dvvrfldwhg zlwk rqh ri wkh edvlv vwdwhv mu 3l vkrxogeh jlyhq e| wkh h{suhvvlrq

3 3 !u 3 +u,@kumu l @ +u u , +5188, Srvwxodwh L= Vshfl fdwlrq ri wkh G|qdplfdo Vwdwh 6< zklfk lv frqvlvwhqw zlwk rxu ruljlqdo dvvrfldwlrq ri wkh vwdwh u 3 kdylqj doo ri lwv suredelolw| ghqvlw| orfdwhg dw u 31 Wkxv/ wkh ruwkrqrupdolw| uhodwlrq iru wkh vwdwhv imu 3lj dovr jlyhv wkh irup wkdw wkhlu zdyh ixqfwlrqv wdnh lq wklv uhsuhvhqwdwlrq/ zklfk zh uhihu wr dv wkh srvlwlrq uhsuhvhqwdwlrq/ruwkhu uhsuhvhqwdwlrq1

5141; Wkh Zdyhyhfwru Uhsuhvhqwdwlrq Uhfdoo/ wkdw wkh vhw ri sodqh zdyhv

hlnÄu ! +u,@ +5189, n +5,6@5 iru doo zdyhyhfwruv n irupv d frpsohwh vhw ri ixqfwlrqv iru wkh vsdfh ri wudqvirupdeoh ixqfwlrqv1 Zh xvh wklv wr gh qh d vhw ri xqghuo|lqj nhwv imnlj zklfk duh wkh vwdwhv uhsuhvhqwhg e| wkh sodqh zdyhv lq wkh srvlwlrq uhsuhvhqwdwlrq mxvw lqwurgxfhg1 Wkxv/ e| gh qlwlrq/ ] ] g6u mnl @ g6u!+u,mul @ hlnÄumul +518:, n +5,6@5 Wkhvh vwdwhv duh wkh Irxulhu wudqvirupv/ lq wklv vhqvh/ ri wkh srvlwlrq orfdol}hgvwdwhv mul1Li wkh vwdwhvq imrulj duh frpsohwh dqgruwkrqrupdo/ wkhq lw lv vwudljkwiruzdugwr vkrz wkdw wkh vwdwhv mnl duh dv zhoo1 Ruwkrqrupdolw| ri wkh nhwv mnl iroorzv iurp wkh ruwkrqrupdolw| wkh sodqh zdyhv/ l1h1/

] ] 6 g u 3 knmn3l @ g6u!Æ+u,! +u,@ hl+nÃn ,Äu @ +n n3,= +518;, n n3 5 Wkhvh vwdwhv duh dovr frpsohwh/ vlqfh wkh| fdq eh xvhgwr h{sdqgdq duelwudu| phpehu ri wkh imul edvlv/ dqgwkhuhe| eh xvhgwr h{sdqgdq duelwudu| vwdwh m#l ri wkh v|vwhp1 Wr ghwhuplqh wklv h{sdqvlrq zh qrwh wkdw li/ e| dvvxpswlrq/ ] 6 mul @ g n!u+n,mnl +518<, wkhq lw pxvw eh wkdw

hÃlnÄu ! +n,@knmul @+kumnl,Æ @ !Æ+u,@ +5193, u n +5,6@5

Wkxv/ zh kdyh wkh sursrvhgh{sdqvlrq/ ] g6n mul @ hÃlnÄumnl> +5194, +5,6@5 zklfk lv wkh frxqwhusduw wr wkh gh qlwlrq ] g6u mnl @ hlnÄumul +5195, +5,6@5 zklfk lv yhul hgwr eh fruuhfw/ e| h{suhvvlqj wkh vwdwhv mnl lq whupv ri wkh vwdwhv mu 3l/dqg froodsvlqj wkh ghowd ixqfwlrqv zklfk ghyhors1 Kdylqj hvwdeolvkhg wkh idfw wkdw wkh vwdwhv imnlj dovr irup dq RQE iru wklv vsdfh zh fdq qrz h{sdqgduelwudu| vwdwhv ] m#l @ g6n#+n,mnl= +5196, 73 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv lq wklv edvlv1 Wklv ohdgv wr lqwhuhvwlqj uhodwlrqv1 Iru h{dpsoh/ lq wkh uhdo vsdfh uhsuhvhq0 wdwlrq zh kdyh wkdw ] ] g6n #+u,@kum#l @ g6n#+n,kumnl @ #+n, hlnÄu> +5197, +5,6@5 zklfk vkrzv wkdw #+u, lv wkh Irxulhu wudqvirup ri #+n,1 Vlploduo|/ zh qgwkdw ] ] g6u #+n,@knm#l @ g6u#+u,knmul @ #+u, hÃlnÄu> +5198, +5,6@5 vr wkdw #+u, dqg #+n, duh Irxulhu wudqvirup sdluv1 Qrwh zkhuh wkh soxv dqgplqxv vljqv jr lq wkh h{srqhqwldov lq +5197, dqg+5198,/ sduwlfxoduo| lq frpsdulvrq wr +5194, dqg+5195,1 Dovr qrwh d vxewoh vkliw lq qrwdwlrq1 Lq wkh odvw fkdswhu zh ghqrwhg wkh zdyhixqfwlrq e| #+u, dqglwv Irxulhu wudqvirup e| #a+n,1 Zh qrz nqrz/ krzhyhu/ wkdw wkhuh duh dv pdq| srvvleoh zdyhixqfwlrqv uhsuhvhqwlqj wkh vwdwh m#l dv wkhuh duh frqwlqxrxv ruwkrqrupdo edvhv iru wkh vsdfh1 Lq idfw/ zh zloo vhh wkdw li wkhuh h{lvwv rqh vxfk frqwlqxrxv edvlv wkhuh zloo dozd|v eh dq lq qlwh qxpehu ri rwkhu rqhv wkdw fdq eh frqvwuxfwhg1 Wkxv/ udwkhu wkdq frplqj xs zlwk d glhuhqw gldfulwlfdo pdun #> #>a #> hwf1 wr glhuhqwldwh wkh zdyhixqfwlrq lq hdfk qhz uhsuhvhqwdwlrq wkdw zh lqwurgxfh/ zh zloo djuhh wr dozd|v lqfoxgh wkh dujxphqw ri wkh zdyhixqfwlrq wr lqglfdwh zklfk uhsuhvhqwdwlrq zh duh zrunlqj lq dw wkh prphqw1 Wkxv/ lw zloo eh xqghuvwrrg wkdw #+u,>#+n,>dqg/ h1j1/ #+, doo uhsuhvhqw glhuhqw ixqfwlrqv ri wkhlu dujxphqwv/ hyhq wkrxjk zh xvh wkh vdph v|pero # iru hdfk/ wr lqglfdwh wkdw wkh| doo surylgh d phdqv ri uhsuhvhqwlqj wkh vdph xqghuo|lqj vwdwh yhfwru m#l1 Zh qrwh lq sdvvlqj wkdw zh frxogdyrlgwklv sureohp doo wrjhwkhu e| vlpso| djuhhlqj wr xvh rxu lghqwl fdwlrq ri wkh h{sdqvlrq frh!flhqw/ ru zdyh ixqfwlrq/ zlwk wkh dvvrfldwhg lqqhu surgxfw1 Wkxv/ lqvwhdg ri zulwlqj #+u,>#+n,>dqg/ #+,> zh frxogdozd|v mxvw zulwh kum#l> knm#l> dqg km#l1 Lq nhhslqj zlwk prghuq xvdjh/ zh zloo xvh erwk lqwhufkdqjhdeo|1 Doo ri wkhvh lghdv fdq eh dssolhg wr d sduwlfoh prylqj lq orzhu glphqvlrqv/ dv zhoo1 Lq wkh vsdfh ri d sduwlfoh prylqj lq rqh glphqvlrq zh lqwurgxfh d vhw ri srvlwlrq orfdol}hg vwdwhv im{lj odehohge| wkh srvlwlrqv zkhuh wkh sduwlfoh fdq eh orfdol}hg1 Zh wkhq kdyh wkh iroorzlqj uhodwlrqv vlplodu wr wkrvh ghyhorshg deryh

k{m{3l @ +{ {3, ] m#l @ g{ #+{,m{l

#+{,@k{m#l dqgd vhw ri sodqh zdyh vwdwhv imnlj odehohge| zdyh yhfwru/ zklfk reh| wkh iroorzlqj uhodwlrqv knmn3l @ +n n3, ] m#l @ gn #+n,mnl

#+n,@knm#l dqgzklfk duh uhodwhgwr wkh srvlwlrq orfdol}hgvwdwhv wkurxjk wkh iroorzlqj uhodwlrqv ] ] g{ mnl @ g{ m{lk{mnl @ hln{m{l +5,4@5 ] ] gn m{l @ gn mnlknm{l @ hÃln{mnl +5,4@5 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 74

hln{ k{mnl @ @ knm{lÆ +5,4@5 ] ] gn #+{,@k{m#l @ gn #+n,k{mnl @ #+n, hln{> +5,4@5 ] ] g{ #+n,@knm#l @ g{ #+{,knm{l @ #+{, hÃln{= +5,4@5

515 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv Lq nhhslqj zlwk wkh jhqhudo vfkhph lqwurgxfhghduolhu/ wkh vhfrqgri wkh srvwxodwhv wkdw zh zloo h{soruh ghvfulehv wkh qdwxuh ri wkh revhuydeohv ri txdqwxp phfkdqlfdo v|vwhpv1

Hyhu| revhuydeoh D ri d txdqwxp phfkdqlfdo v|vwhp lv dvvrfldwhgzlwk d olqhdu Khuplwldq rshudwru D zkrvh hljhqvwdwhv irup d frpsohwh ruwkrqrupdo edvlv iru wkh txdqwxp phfkdqlfdo vwdwh vsdfh1

Zh duh ohg/ wkhuhiruh/ wr lqyhvwljdwh wkh qdwxuh ri olqhdu rshudwruv gh qhg rq olqhdu yhfwru vsdfhv1 51514 Rshudwruv dqg Wkhlu Surshuwlhv Dq rshudwru D dvvrfldwhgzlwk d olqhdu yhfwru vsdfh V dfwv rq wkh hohphqwv m"l lq V dqgpdsv wkhp rqwr +srvvleo|, rwkhu hohphqwv m"Dl ri wkh vdph vsdfh1 Zh h{suhvv wklv pdsslqj ri rqh yhfwru rqwr dqrwkhu lq wkh irup

Dm"l @ m"Dl= +5199,

Dq rshudwru D lv olqhdu li lw vdwlv hv wkh iroorzlqj olqhdulw| frqglwlrq

D+m"l . m#l,@m"Dl . m#Dl @ Dm"l . Dm#l> +519:, iru duelwudu| vwdwhv m"l> m#l/ dqgduelwudu| vfdoduv dqg 1 Lq zkdw iroorzv zh dvvxph/ xqohvv rwkhuzlvh vwdwhg/ wkdw doo rshudwruv xqghu frqvlghudwlrq duh olqhdu1 Rqh ri wkh xvhixo surshuwlhv ri olqhdu rshudwruv lv wkdw wkhlu dfwlrq rq duelwudu| vwdwhv lv ghwhuplqhg rqfh wkhlu dfwlrq rq wkh hohphqwv ri dq| RQE lv vshfl hg1 Wr vhh wklv/ ohw imllj eh dq duelwudu| RQE/ dqgohw wkh dfwlrq

m!ll @ Dmll +519;, ri wkh olqhdu rshudwru D rq wkhvh vwdwhv eh nqrzq1 Zkhq dq duelwudu| vwdwh m#l lv dfwhg xsrq e| D zh fdq xvh wkh h{sdqvlrq ri m#l lq wklv edvlv wr vhh wkdw [ [ [ Dm#l @ D #lmll @ #l Dmll @ #lm!ll +519<, l l l zklfk xqltxho| ghwhuplqhv wkh uhvxowlqj yhfwru1 Zh ghvfuleh ehorz vrph ri wkh frpprq surshuwlhv dvvrfldwhgzlwk olqhdu rshudwruv1 Wkh vxp dqg glhuhqfh ri rshudwruv duh gh qhg wkurxjk yhfwru dgglwlrq

+D . E,m#l @ Dm#l . Em#l @ m#Dl . m#El +51:3, 75 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

+D E,m#l @ Dm#lEm#l @ m#Dlm#El= +51:4, Wkh surgxfw ri rshudwruv lv gh qhgwkurxjk wkh frpelqhgdfwlrq ri hdfk1 Li F @ DE/ wkhq

Fm#l @ DEm#l @ Dm#El= +51:5, Lq jhqhudo wkh rshudwru surgxfw lv qrw frppxwdwlyh/ vlqfh uhyhuvlqj wkh rughu fdq jlyh d glhuhqw uhvxow/ l1h1/ wkh yhfwru

EDm#l @ Em#Dl +51:6, qhhgqrw kdyh dq| uhodwlrq wr wkh yhfwru Dm#El1 Lw lv xvhixo/ wkhuhiruh/ wr gh qh wkh frppxwdwru ri wzr rshudwruv

^D> E`@DE ED @ Ê>D` +51:7, zklfk lv dovr dq rshudwru1 Li ^D> E`@3/wkhqDE @ ED> dqgwkh wzr rshudwruv frppxwh1 Iurp wkh gh qlwlrq ri wkh frppxwdwru lw lv vwudljkwiruzdug wr suryh wkh iroorzlqj xvhixo uhodwlrqv ^D> D`@3 ^D> E . F`@^D> E`.^D> F` ^D . E>F`@^D> F`.Ê> F` ^D> EF`@E^D> F`.^D> E`F ^DE> F`@DÊ> F`.^D> FÈ ^DÊ> F`` . ^F^D> E``.Ê^F> D`` @ 3=

Wkh qxoo rshudwru/ pdsv hdfk yhfwru lq wkh vsdfh rqwr wkh qxoo yhfwru/ l1h1/ 3m#l @31 Wkh lghqwlw| rshudwru/ pdsv hdfk yhfwru lq wkh vsdfh rqwr lwvhoi/ l1h1/ 4m#l @ m#l1 Wkh lqyhuvh ri dq rshudwru D/lilwh{lvwv/lvghqrwhgDÃ4 dqgreh|v wkh surshuw|

DDÃ4 @ DÃ4D @ 4= +51:8,

D qrq}hur yhfwru m"l lv vdlgwr eh dq hljhqyhfwru ri dq rshudwru D zlwk hljhqydoxh d +zkhuh jhqhudoo|/ d 5 F,lilwvdwlv hvwkhhljhqydoxh htxdwlrq

Dm"l @ dm"l= +51:9,

Wkh vhw ri hljhqydoxhv idj iru zklfk vroxwlrqv wr wklv htxdwlrq h{lvw lv uhihuuhgwr dv wkh vshfwuxp ri D/ dqgghqrwhgvshfwuxp +D,1 Lw lv dovr srvvleoh wr gh qh rshudwruv wkdw duh/ wkhpvhoyhv/ ixqfwlrqv ri rwkhu rshudwruv1 Wklv fdq eh grqh lq d qxpehu ri zd|v1 Iru h{dpsoh/ iurp wkh surgxfw uxoh jlyhq deryh/ lw lv fohdu wkdw lq jhqhudo wkh q0irogsurgxfw ri dq rshudwru D zlwk lwvhoi lv zhoo0gh qhg1 Wkxv/ zh pd| dozd|v vshdn ri srvlwlyh lqwhjhu srzhuv Dq ri dq rshudwru1 Li wkh lqyhuvh DÃ4 ri dq Ãq rshudwru lv dovr gh qhg/ wkhqS zh fdq gh qh qhjdwlyh srzhuv wkurxjk wkh uhodwlrq D @ DÃ4 q1Wkhq/lii+{,@ i { lv dq| srzhu vhulhv h{sdqgdeoh ixqfwlrq zlwk d vxlwdeoh q q q S udglxv ri frqyhujhqfh/ zh fdq gh qh wkh rshudwru ydoxhg ixqfwlrq I +D,@ q iqDq ri wkh rshudwru D1 Zh zloo ljqruh iru wkh prphqw d glvfxvvlrq ri wkh frqglwlrqv xqghu zklfk vxfk vhulhv frqyhujh/ ehfdxvh/ dv zh zloo vhh/ wkhuh duh rwkhu zd|v ri gh qlqj rshudwru ydoxhgixqfwlrqv wkdw duh riwhq pruh xvhixo wkdw doorz wklv txhvwlrq wr eh dyrlghg1 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 76

Zh qrz frqvlghu d ydulhw| ri glhuhqw rshudwruv1 Zh ehjlq e| qrwlqj wkdw li zh pxowlso| dq| vwdwh m#l lq wkh vsdfh e| d vfdodu zh jhqhudwh d qhz yhfwru m#l @ m#l1Wkxv/ zh fdq gh qh d yhu| vlpsoh w|sh ri rshudwru zklfk fduulhv rxw wklv rshudwlrq1 Wr dyrlg dq| fxpehuvrph qrwdwlrq zh vlpso| zloo ghqrwh e| wkdw rshudwru zklfk pxowlsolhv d yhfwru e| wkh vfdodu 1 Wklv doorzv xv/ h1j1/ wr irup rshudwruv ri wkh irup . D> zkhuh D lv dq duelwudu| rshudwru dqg d vfdodu/ zkrvh dfwlrq lv wkh reylrxv rqh/ l1h1/ + . D, m#l @ m#l . Dm#l1 Dv vshfldo fdvhv ri rshudwruv iruphgiurp vfdoduv/ zh kdyh @4fruuhvsrqglqj wr wkh lghqwlw| rshudwru dqg @3fruuhvsrqglqj wr wkh qxoo rshudwru1 Pxowlsolfdwlrq e| d vfdodu lv dq rshudwlrq wkdw dozd|v frppxwhv zlwk dq| rwkhu olqhdu rshudwru/ l1h1/ vfdoduv dozd|v frppxwh zlwk hyhu|wklqj1 Wkxv/ zh fdq zulwh ^> D`@3=

51515 Pxowlsolfdwlyh Rshudwruv Lq wkh vsdfh ri d txdqwxp sduwlfoh prylqj lq rqh glphqvlrq ohw xv lqwurgxfh dq rshudwru [ e| gh qlqj lwv dfwlrq rq wkh rqh0glphqvlrqdo srvlwlrq vwdwhv im{lj dv iroorzv=

[m{l @ {m{l= +51::,

Wkxv/ [ mxvw pxowlsolhv wkh edvlv yhfwru m{l e| lwv odeho/ l1h1/ e| wkh srlqw zkhuh wkh dvvrfldwhgghowd ixqfwlrq lv fhqwhuhg1 +Qrwh wkdw wklv lpsolhv wkdw wkh edvlv vwdwhv m{l duh doo hljhqvwdwhv ri wkh rshudwru [> dqgduh lq idfw odehohge| wkhlu dvvrfldwhghljhqydoxhv,1 Zlwk wklv gh qlwlrq/ zh qg wkdw wkh dfwlrq ri [ rq dq duelwudu| vwdwh m#l lv udwkhu vlpso| h{suhvvhglq wkh srvlwlrq uhsuhvhqwdwlrq/ l1h1/ ] ] ] [m#l @ [ g{ #+{,m{l @ g{ #+{,[m{l @ g{ #+{,{m{l= +51:;,

Uhduudqjlqj d olwwoh/ zh vhh wkdw ] [m#l @ g{ ^{#+{,`m{l= +51:<,

Wklv vkrzv wkdw wkh zdyh ixqfwlrq uhsuhvhqwlqj [m#l lv mxvw {#+{,1Zkhqzhgrq*wplqg ehlqj d olwwoh lpsuhflvh zh zloo vd| wkdw lq wkh {0uhsuhvhqwdwlrq/ [#+{,@{#+{,1 ru [ pxowlsolhv wkh zdyh ixqfwlrq e| {1 Zh kdyh wr eh fduhixo/ wkrxjk= lq uhdolw| wkh rshudwru [ grhv qrw dfwxdoo| dfw rq wkh zdyh ixqfwlrq +zklfk lv mxvw d vfdodu0ydoxhg ixqfwlrq,/ lw dfwv rq wkh nhwv lq wkh h{sdqvlrq iru wkh vwdwh/ jlylqj ulvh wr wklv dssduhqw hhfw1 Wklv lghd lv hdvlo| h{whqghg wr ixqfwlrqv ri {1 Iru dq| ixqfwlrq i+{, zh fdq gh qh dq rshudwru I zklfk kdv wkh dfwlrq

I m{l @ i+{,m{l +51;3, ri pxowlso|lqj hdfk edvlv yhfwru m{l lq wkh srvlwlrq uhsuhvhqwdwlrq e| wkh ixqfwlrq i hydo0 xdwhgdw wkh srlqw { odeholqj wkh edvlv yhfwru1 Zkhq I dfwv rq duelwudu| yhfwruv lw ohdgv wr wkh uhvxow ] ] I m#l @ g{ #+{, I m{l @ g{ #+{,i+{,m{l +51;4, vr wkdw wkh zdyh ixqfwlrq uhsuhvhqwlqj I m#l lv mxvw i+{,#+{,1Lqwkh{ uhsuhvhqwdwlrq/ I pxowlsolhv wkh zdyh ixqfwlrq e| i+{,1 Wkh srwhqwldo hqhuj| ixqfwlrq Y +{, lv dvvrfldwhg zlwk dq rshudwru ri wklv w|sh1 Lq idfw/ doo zh duh grlqj khuh lv surylglqj dqrwkhu zd| ri gh qlqj d ixqfwlrq ri dq rshudwru/ l1h1/ zh fdq irupdoo| ylhz I dv d ixqfwlrq ri wkh rshudwru [> l1h1/ I @ I +[,= Lqduhsuhvhqwdwlrqlqzklfk[ mxvw pxowlsolhv e| { wkh dfwlrq ri I +[, lv wr pxowlso| e| i+{,= Wklv gh qlwlrq lv hdvlo| yhul hg wr djuhh zlwk rxu iruphu 77 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv rqh uhjduglqj vhulhv h{sdqvlrqv lq wkrvh vlwxdwlrqv zkhuh wkh vhulhv frqyhujhv/ dqg h{whqgv lw wr doorz iru pruh jhqhudo nlqgv ri rshudwru ixqfwlrqv1 Wkhvh lghdv dovr h{whqg wr kljkhu glphqvlrqv1 Zh fdq gh qh/ h1j1/ wkh Fduwhvldq rshudwru frpsrqhqwv [> \> dqg ]/riwkhsrvlwlrq rshudwru U/ wkurxjk wkhlu dfwlrq rq wkh edvlv yhfwruv imulj ri wkh srvlwlrq uhsuhvhqwdwlrq lq wkuhh glphqvlrqv=

[mul @ [m{> |> }l @ {m{> |> }l @ {mul> +51;5,

\ mul @ \ m{> |> }l @ |m{> |> }l @ |mul> +51;6,

]mul @ ]m{> |> }l @ }m{> |> }l @ }mul= +51;7,

Wkxv/ [> \> dqg ] pxowlso| wkh edvlv yhfwruv ri wkh srvlwlrq uhsuhvhqwdwlrq e| wkh fduwhvldq frpsrqhqwv ri wkh srlqwv zkhuh wkh| duh fhqwhuhg1 Wkh srvlwlrq rshudwru U lv d yhfwru rshudwru/ l1h1/ d froohfwlrq ri wkh wkuhh rshudwruv [> \> dqg ] zklfk wudqvirup olnh wkh frpsrqhqwv ri d yhfwru lq U61 Wkh hhfw ri wkh rshudwru U rq wkh srvlwlrq vwdwhv

Umul @ umul +51;8, lv wr pxowlso| wkhp e| wkh srvlwlrq yhfwru zlwk zklfk wkh| duh odehohg1 Lq wklv uhsuhvhq0 wdwlrq/ wkhq/ ] ] ] Um#l @ g6u#+u, Umul @ g6u#+u,umul @ g6u ^u#+u,` mul= +51;9,

Wkxv/ wkh hhfw ri U lv wr pxowlso| wkh zdyh ixqfwlrq #+u, e| u1 Ilqdoo|/ zh fdq h{whqg wklv wr ixqfwlrqv ri u/ dv lq rqh0glphqvlrq1 Iru hdfk ixqfwlrq Y +u, zh fdq gh qh dq rshudwru Y @ Y +U, vxfk wkdw Y mul @ Y +u,mul +51;:, dqgvr ] ] Y m#l @ g6u#+u, Y mul @ g6uY+u,#+u, mul= +51;;,

Wkxv wkh rshudwru Y dfwv lq wkh u uhsuhvhqwdwlrq wr pxowlso| wkh zdyh ixqfwlrq e| wkh ixqfwlrq Y +u,1 Wkhvh nlqgv ri pxowlsolfdwlyh rshudwruv fdq eh gh qhg iru dq| uhsuhvhqwdwlrq1 Li imlj lv dq RQE iru wkh vsdfh/ zh fdq gh qh dq rshudwru D vxfk wkdw

Dml @ ml iru doo edvlv yhfwruv ri wklv uhsuhvhqwdwlrq1 Wkhq/ iru dq| ixqfwlrq j+, zh fdq gh qh dq rshudwru J @ J+D, vxfk wkdw Jml @ j+,ml> +51;<, wkhq ] ] Jm#l @ g #+, Jml @ g j+,#+, ml> +51<3, vr wkdw J dfwv lq wkh uhsuhvhqwdwlrq wr pxowlso| wkh zdyh ixqfwlrq lq wkdw uhsuhvhqwdwlrq e| wkh ixqfwlrq j+,1 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 78

51516 Glhuhqwldo Rshudwruv Dqrwkhu fodvv ri rshudwruv fdq dovr eh gh qhg wkurxjk wkh srvlwlrq uhsuhvhqwdwlrq1 Ohw xv gh qh rshudwruv G{>G|> dqg G} lq vxfk d zd| wkdw li ] m#l @ g6u#+u, mul> +51<4, wkhq ] C# G m#l @ g6u mul= +51<5, { C{

Wkxv/ G{ uhsodfhv wkh zdyh ixqfwlrq lq wkh srvlwlrq uhsuhvhqwdwlrq e| lwv sduwldo ghulydwlyh zlwk uhvshfw wr {1 Vlplodu dfwlrqv duh lpsolflwo| gh qhg iru G| dqg G}1 Vxfk rshudwruv duh glhuhqwldo rshudwruv lq wklv uhsuhvhqwdwlrq1 Wkhvh wkuhh rshudwruv irup wkh frpsrqhqwv ri wkh yhfwru rshudwru G zklfk wdnhv wkh judglhqw lq wkh u uhsuhvhqwdwlrq1 Wkdw lv wr vd|/ ] G m#l @ g6u u # mul= +51<6,

D pruh xvhixo yduldwlrq ri wklv rshudwru lv rewdlqhge| pxowlso|lqj lw e| wkh vtxduh urrw ri 41 Zh wkxv lqwurgxfh wkh rshudwru N @ lG zklfk zh zloo uhihu wr dv wkh zdyhyhfwru rshudwru/ dqggh qh wkurxjk wkh h{suhvvlrq ] N m#l @ g6u ^lu #+u,` mul= +51<7,

Lw lv lqvwuxfwlyh wr frqvlghu wkh dfwlrq ri wklv rshudwru lq wkh n uhsuhvhqwdwlrq1 E| dv0 vxpswlrq/ dq duelwudu| vwdwh fdq eh h{sdqghg lq wkh n uhsuhvhqwdwlrq lq wkh irup ] m#l @ g6u#+n, mnl> +51<8, zkhuh ] g6n #+u,@ hlnÄu#+n,= +51<9, +5,6@5 Wkxv/ ] g6n lu #+u,@ ^lu hlnÄu`#+n,= +51<:, +5,6@5 Wkh judglhqw rshudwru/ zklfk dfwv rqo| rq wkh srvlwlrq yduldeohv/ mxvw sxoov grzq wkh zdyhyhfwru iurp wkh h{srqhqwldo/ l1h1/ ] g6n lu #+u,@ ^nhlnÄu`#+n,= +51<;, +5,6@5

Wkxv/ zh ghgxfh wkdw ] ] ] k l g6n N m#l @ g6u lu #+u, mul @ g6u nhlnÄu #+n, mul= +51<<, +5,6@5

Lqwhufkdqjlqj wkh rughu ri lqwhjudwlrq/ wklv ehfrphv ] ] g6u N m#l @ g6n n#+n, hlnÄumul= +51433, +5,6@5 79 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Lq wkh odvw lqwhjudo zh uhfrjql}h wkh gh qlwlrq ri wkh edvlv vwdwhv mnl1Wkxvzhrewdlqwkh vlpsoh uhvxow ] N m#l @ g6n n#+n,mnl= +51434,

Wkxv/ N dfwv lq wkh n uhsuhvhqwdwlrq wr pxowlso| wkh zdyh ixqfwlrq lq wkdw uhsuhvhqwdwlrq e| n1VlqfhN uhdoo| dfwv rqo| rq wkh nhwv mnl/ zh ghgxfh wkh dfwlrq

N mnl @ nmnl= +51435,

Wkxv/ wkh rshudwru N sod|v wkh vdph uroh lq wkh n uhsuhvhqwdwlrq wkdw wkh rshudwru U sod|v lq wkh u uhsuhvhqwdwlrq/ l1h1/ lw vlpso| pxowlsolhv wkh edvlv yhfwruv e| wkh ydoxh ri wkh sdudphwhu n wkdw odehov wkhp1 +Lw lv xvhixo wr wklqn ri wklv rshudwlrq dv rqh lq zklfk wkh rshudwru sxoov rxw wkh odeho1, Vlploduo| wkh nlqhwlf hqhuj| rshudwru

|5N5 |5 K @ @ N N 3 5p 5p lv d pxowlsolfdwlyh rshudwru lq wkh n uhsuhvhqwdwlrq wkdw pxowlsolhv wkh zdyh ixqfwlrq e| 5 5 | n @5p/ exw lv d glhuhqwldo rshudwru lq wkh u uhsuhvhqwdwlrq zkrvh dfwlrq lv wdnh #+u, rqwr wkh ixqfwlrq |5@5p u5#1 Wkxv/ zkhwkhu dq rshudwru lv d pxowlsolfdwlyh rshudwru ru d glhuhqwldo rshudwru lv yhu| pxfk d uhsuhvhqwdwlrq0ghshqghqw vwdwhphqw1 Lw lv ohiw dv dq h{huflvh wr vkrz wkdw lq wkh n uhsuhvhqwdwlrq wkh srvlwlrq rshudwru U dfwxdoo| dfwv dv d glhuhqwldo rshudwru/ l1h1/ wkdw ] 6 Um#l @ g n ^lun#+n,`mnl> +51436,

zkhuh un phdqv wr wdnh wkh judglhqw zlwk uhvshfw wr wkh yduldeohv n{>n|/dqgn}/dqg wkdw lq wkh srvlwlrq uhsuhvhqwdwlrq wkh nlqhwlf hqhuj| rshudwru K3 lv d glhuhqwldo rshudwru sursruwlrqdo wr wkh Odsodfldq +dv lq Vfkuùglqjhu*v htxdwlrq,1

51517 Nhw0Eud Rshudwruv Dyhu|xvhixofodvvrirshudwrufdqehgh qhgxvlqjdq|wzryhfwruvlqwkhvsdfh1Lim!l dqg m"l duh yhfwruv lq V wkhq zh fdq gh qh dq rshudwru

D @ m!lk"m +51437, zkrvh dfwlrq rq dq| vwdwh m#l lv dv iroorzv=

Dm#l @ m!lk"m#l +51438, zklfk lv mxvw wkh yhfwru m!l pxowlsolhge| wkh qxpehu k"m#l1 D olqhdu vxp ri rshudwruv ri wklv irup lv lwvhoi dq rshudwru1 Wklv nhw0eud irup lv sduwlfxoduo| xvhixo iru h{suhvvlqj zkdw duh uhihuuhgwr dv surmhfwlrq rshudwruv1

51518 Surmhfwlrq Rshudwruv= Wkh frpsohwhqhvv uhodwlrq Dq rshudwru S lv vdlgwr eh d surmhfwlrq rshudwru/ruvlpso|dsurmhfwru/lilwvdwlv hv wkh lghpsrwhqf| frqglwlrq S 5 @ S= +51439, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 7:

Qrwh wkdw wklv lpsolhv wkdw S q @ S iru doo lqwhjhuv q 41 Dv dq h{dpsoh/ li m!l lv vtxduh qrupdol}hgwr xqlw|/ vr wkdw k!m!l @4/ wkhq wkh rshudwru

S! @ m!lk!m +5143:, lv d surmhfwru rqwr wkh gluhfwlrq ri wkh yhfwru m!l1Lwlvdsurmhfwruehfdxvh

5 S! @+m!lk!m,+m!lk!m,@m!lk!m!lk!m @ m!lk!m @ S!> +5143;, zkhuh wkh fhqwudo lqqhu surgxfw froodsvhg wr xqlw| ehfdxvh ri wkh qrupdol}dwlrq ri wkh vwdwh m!l1Wkxv/ wkh dfwlrq ri S! rq dq duelwudu| vwdwh m#l lv wr wdnh dzd| wkrvh sduwv ri m#l qrw o|lqj dorqj m!l> dqgwr ohdyh wkh sduw o|lqj dorqj wkh gluhfwlrq ri m!l dorqh1 Dv d vlpsoh h{whqvlrq ri wklv lghd zh qrwh wkdw/ d vhw ri vwdwhv imllj irupv dq ruwkrqrupdo vhw ri yhfwruv vr wkdw klmml @ lm/ wkhq wkh rshudwru [q S @ mllklm +5143<, l@4 lv dovr d surmhfwlrq rshudwru ehfdxvh # $ 3 4 [q [q [q [q [q [q [q 5 C D S @ mllklm mmlkmm @ mllklmmlkmm @ mlllm kmm @ mllklm @ S= l@4 m@4 l@4 m@4 l@4 m@4 l@4 +51443,

Frpphqw= Surmhfwlrq rshudwruv dozd|v surmhfw rqwr vrphwklqj1 Lq wklv odwwhu fdvh/ wkh rshudwru S surmhfwv rqwr wkh vxevsdfh vsdqqhge| wkh yhfwruv lq wkh ruwkrqrupdo vhw1 Uhfdoo wkh gh qlwlrq ri d vxevsdfh=

D vhw ri yhfwruv V3 V zklfk lv d vxevhw ri d yhfwru vsdfh V lv d vxevsdfh ri V li lw lv forvhgxqghu wkh vdph rshudwlrqv wkdw duh gh qhglq wkh sduhqw vsdfh1

Dq| vxevhw ri yhfwruv vsdqv vrph vxevsdfh/ qdpho| wkh vxevsdfh ri doo yhfwruv wkdw fdq eh surgxfhg e| wkhp e| iruplqj doo srvvleoh olqhdu frpelqdwlrqv ri yhfwruv lq wkh vxevhw1 Lq wkh h{dpsoh deryh/ li wkh ruwkrqrupdo vwdwhv imllj zhuh frpsohwh/vrwkdwwkh|dfwxdoo| iruphgdq RQE iru wkh vsdfh/ wkhq wkh vxevsdfh wkdw wkh| surmhfw xsrq zrxogeh wkh hqwluh vsdfh1 Vlqfh/ iru vxfk d edvlv/ dq h{sdqvlrq ri wkh irup [ m#l @ mllklm#l +51444, l h{lvwv iru dq| vwdwh m#l lq wkh vsdfh/ wkh dfwlrq ri wkh rshudwru S rq vxfk d vwdwh [ S m#l @ mllklm#l @ m#l +51445, l lv wr mxvw uhsurgxfh wkh vwdwh lw dfwhg rq1 Zh ghgxfh wkdw li wkh vwdwhv imllj irup dq ruwkrqrupdo edvlv/ wkhq [ mllklm @ 4= +51446, l Wklv uhodwlrq/ zklfk lv ri ixqgdphqwdo lpsruwdqfh lv uhihuuhg wr dv d ghfrpsrvlwlrq ri xqlw| lq wkh edvlv imllj> ru dv d vwdwhphqw ri wkh frpsohwhqhvv uhodwlrq iru wkh vwdwhv imllj= 7; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Wkhvh lghdv duh dovr h{whqvleoh zlwk vrph fduh wr frqwlqxrxvo|0lqgh{hg vwdwhv1 Li wkh vwdwhv imlj irup dq RQE iru wkh vsdfh V wkhq wkh rshudwru

@ mlkm +51447, lv qrw d surmhfwlrq rshudwru/ vlqfh wkh vwdwh ml lv qrw vtxduh0qrupdol}hgwr xqlw|1 Lqghhg/ lw lv ri lq qlwh qrup/ vlqfh kml @ +,@+3,1 Krzhyhu/ wkh lqwhjudo ri wklv rshudwru ryhu dq| uhjlrq ri wkh srvvleoh ydoxhv wdnhq rq e| wkh sdudphwhu lv d surmhfwru1 Wkdw lv/ li zh gh qh ] e ] e Sde @ g @ g mlkm +51448, d d wkhq # $# $ ] e ] e ] e ] e 5 3 3 3 3 3 3 Sde @ g mlkm g m lk m @ g g mlkm lk m= +51449, d d d d

Lq wklv odvw h{suhvvlrq/ wkh qrupdol}dwlrq frqglwlrq rq wkh vwdwhv ml |lhogd ghowd ixqfwlrq/ zklfk phdqv wkdw

] e ] e ] e 5 3 3 3 Sde @ g g ml+ ,k m @ g mlkm @ Sde +5144:, d d d +Qrwh/ wkdw lq dq| h{suhvvlrq lq zklfk rqh lv lqwhjudwlqj ryhu d ghowd ixqfwlrq/ rqh vlpso| uhpryhv wkh ghowd ixqfwlrq dqgwkh lqwhjudo vljq dqguhsodfhv wkh lqwhjudwlrq yduldeoh zkhuhyhu lw rffxuv zlwk wkh ydoxh zklfk pdnhv wkh dujxphqw ri wkh ghowd ixqfwlrq ydqlvk1,1

m#l @ mlkm#l @ #+,ml dorqj wkh gluhfwlrq ri wkh vwdwh ml pxowlsolhge| wkh h{sdqvlrq frh!flhqw #+,1Wkxv/ zh zloo frqwlqxh wr uhihu wr m#l dv wkh sduw ri m#l o|lqj dorqj wkh vwdwh ml1 Dw dq| udwh/ lw lv hdv| wr vhh wkdw vwulsv dzd| dq| sduw ri m#l qrw o|lqj dorqj wkdw gluhfwlrq lq vwdwh vsdfh1 Dv lq wkh glvfuhwh fdvh/ li zh qrz frqvlghu wkh surmhfwru zklfk lqfoxghv doo wkh vwdwhv lq wkh edvlv/ zh surmhfw rqwr wkh hqwluh vsdfh1 Wkxv/ vlqfh zh fdq dozd|v zulwh dq duelwudu| vwdwh lq wkh irup ] m#l @ g mlkm#l +5144;, wkh dfwlrq ri wkh rshudwru ] S @ g mlkm> +5144<, zklfk kdv qr uhvwulfwlrqv rq wkh ydoxhv ri / lv wr uhsurgxfh zkdwhyhu vwdwh lw dfwv xsrq/ l1h1/ ] S m#l @ g mlkm#l @ m#l +51453,

Wklv ehlqj wuxh iru doo m#l/zhghgxfhwkhfrpsohwhqhvvuhodwlrq ] g mlkm @ 4 +51454, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 7< iru frqwlqxrxvo|0lqgh{hg vwdwhv1 Wkhvh ghfrpsrvlwlrqv ri wkh lghqwlw| rshudwru idflolwdwh wkh ghyhorsphqw ri h{sdqvlrqv iru yhfwruv dqg lqqhu surgxfwv/ dv ghprqvwudwhg ehorz= [ [ m"l @ 4m"l @ mllklm"l @ "l mll +51455, l l ] ] m"l @ 4m"l @ g mlkm"l @ g "+, ml +51456, [ [ Æ k#m"l @ k#m +4m"l,@ k#mllklm"l @ #l "l +51457, l l ] ] k#m"l @ k#m +4m"l,@ g k#mlkm"l @ g #Æ+,"+,= +51458,

Lq wklv zd| uhsuhvhqwdwlrq lqghshqghqw h{suhvvlrqv +rq wkh ohiw, duh frqyhuwhg lqwr uhsuh0 vhqwdwlrq ghshqghqw h{suhvvlrqv +rq wkh uljkw, e| lqvhuwlqj dq dssursuldwh frpsohwh vhw ri vwdwhv1 H{suhvvlrqv vxfk dv k#m"l duh uhsuhvhqwdwlrq lqghshqghqw1 H{suhvvlrqv vxfk dv [ [ Æ k#mllklm"l @ #l "l +51459, l l duh uhsuhvhqwdwlrq ghshqghqw/ ehfdxvh wkh| ghshqg xsrq d sduwlfxodu fkrlfh ri uhsuhvhqwd0 wlrq/ l1h1/ ri edvlv1 +Qrwh wkdw h{suhvvlrqv vxfk dv m#+u,l ru m#a+n,l duh dfwxdoo| loo0gh qhg/ kdyh qr r!fldo phdqlqj/ dqgduh wr eh dyrlghg1 Zkloh zh suredeo| fdq jxhvv zkdw lv lqwhqghg/ wkh| duh dpeljxrxv dqg dq dexvh ri vwdqgdug xvdjh1 Wkh ylhz khuh lv wkdw wkh ixqfwlrqv #+u, dqg #+n, jlyh vshfl f uhsuhvhqwdwlrqv ri d vlqjoh xqghuo|lqj vwdwh yhfwru m#l> dqgqrw wkh rwkhu zd| durxqg1,

51519 Pdwul{ Hohphqwv Wkh pdwul{ hohphqw ri dq rshudwru D ehwzhhq +ru frqqhfwlqj, wkh vwdwhv m"l dqg m#l lv wkh vfdodu txdqwlw|

k#m +Dm"l,@k#m"Dl> +5145:, zkhuh m"Dl @ Dm"l1 5151: Dfwlrq ri Rshudwruv rq Eudv ri VÆ Zh kdyh gh qhg wkh dfwlrq ri rshudwruv lq wkh vsdfh ri nhwv m"l5V1Zhqrzh{whqg wkh gh qlwlrq wr doorz wkhp wr rshudwh lq wkh vsdfh VÆ ri eudv k"m/e|uhtxlulqjwkdwdq| pdwul{ hohphqw eh xqfkdqjhgli wkh rshudwru dfwv wr wkh ohiw/ udwkhu wkdq wr wkh uljkw1 Wkxv/ iru doo m"l> m#l5V zh uhtxluh wkdw

k#m +Dm"l,@+k#mD, m"l @ k#mDm"l +5145;, zkhuh lq wkh odvw h{suhvvlrq zh kdyh uhpryhgwkh sduhqwkhvhv vlqfh qrz +e| frqvwuxfwlrq, D fdq dfw lq hlwkhu gluhfwlrq1 +Qrwh wkdw h{suhvvlrqv olnh Dk#m dqg m"lD duh xqgh qhg1 Rshudwruv fdq rqo| dfw rq yhfwruv zkhq wkh| duh qh{w wr wkh yhuwlfdo olqh dsshdulqj lq wkh qrwdwlrq1, Li wkh vwdwh m!l lv d xqlw yhfwru/ wkhq wkh pdwul{ hohphqw k!mDm!l lv uhihuuhgwr dv wkh h{shfwdwlrq ydoxh ri wkh rshudwru D wdnhq zlwk uhvshfw wr wkh vwdwh m!l1 Dq h{wuhpho| lpsruwdqw frqvhtxhqfh ri wklv gh qlwlrq ri wkh dfwlrq ri dq rshudwru rq eudv lv eurxjkw rxw e| wkh dqvzhu wr wkh iroorzlqj txhvwlrq= Li Dm"l @ m#l/ grhv lw iroorz wkdw k"mD @ k#mB Rqh pljkw eh whpswhgwr wklqn vr/ vlqfh wkh vwdwhv m"l dqg m#l5V duh 83 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv vxssrvhgwr eh lq 404 fruuhvsrqghqfh zlwk wkh vwdwhv k"m dqg k#m5V1 Qrqhwkhohvv/ wkh dqvzhu lv qr1 Wkhuh lv d uhodwlrqvkls vlplodu wr wklv wkdw zh fdq zulwh/ exw lw lqyroyhv zkdw lv uhihuuhgwr dv wkh Khuplwldq dgmrlqw ri wkh rshudwru D1 Wkh srlqw lv wkdw zh kdyh wr h{whqg wkh 404 fruuhvsrqghqfh wr lqfoxgh wkh uhodwlrqvkls wkdw h{lvwv ehwzhhq wkh rshudwruv wkdw dfw lq V dqgwkrvh wkdw dfw lq VÆ1 Zh h{soruh wklv ehorz1

5151; Khuplwldq Frqmxjdwlrq Zh kdyh suhylrxvo| hvwdeolvkhgd 404 fruuhvsrqghqfh ehwzhhq wkh nhw*v m"l ri V dqgwkh eud*v k"m ri VÆ +zklfk lq dq| uhsuhvhqwdwlrq fdq eh wkrxjkw ri dv wkh fruuhvsrqghqfh eh0 wzhhq froxpq yhfwruv dqgwkh frpsoh{0frqmxjdwhgurz yhfwruv,1 Fruuhvsrqglqj hohphqwv ri V dqg VÆ duh vdlgwr eh Khuplwldq Frqmxjdwhv/ Khuplwldq Dgmrlqwv/ruvlpso| Dgmrlqwv ri rqh dqrwkhu1 Wkxv V lv wkh olqhdu yhfwru vsdfh dgmrlqw wr VÆ1 Vlploduo|/ wkh eud k"m lv wkh dgmrlqw ri wkh nhw m"l1 Lw lv frqyhqlhqw wr xvh wkh qrwdwlrq ^ `. wr ghqrwh wkh dgmrlqw ri ^ `1Wkxvzhzulwh

^m"l`. @ k"m ^k"m`. @ m"l +5145<, zklfk vkrzv wkdw ^^ `.`. @^ `1 Zh fdq dovr uhihu/ h1j1/ wr wkh RQE ri eud*v iklmj dv ehlqj dgmrlqw wr wkh RQE ri nhw*v imllj1 Wklv jlyhv ulvh wr wkh iroorzlqj srlqw1 Jlyhq wkh h{sdqvlrq [ m"l @ el mll +51463, l zkhuh/ ri frxuvh/ el @ klm"l/ dqgwkh vlplodu h{sdqvlrq [ k"m @ fl klm +51464, l dvvrfldwhgzlwk wkh gxdo vsdfh/ krz duh wkh h{sdqvlrq frh!flhqwv el dqg fl uhodwhgwr rqh dqrwkhuB Wr qgrxw/ qrwh wkdw [ [ Æ Æ k"mml @ fl klmml @ fl lm @ fm @+kmm"l, @ em = +51465, l l

Æ Wkxv zh fdq zulwh fl @ el /dqgwkhuhiruhli [ [ m"l @ "l mll @ mllklm"l +51466, l l wkhq [ [ Æ k"m @ "l klm @ k"mllklm= +51467, l l Lq jhqhudo/ wklv lpsolhv wkdw li lv dq hohphqw ri wkh hogdvvrfldwhgzlwk wkh vsdfh V/ wkhq Æ lv wkh fruuhvsrqglqj hohphqw ri wkh hog lq VÆ +l1h1/ wkh vfdodu zklfk sod|v wkh vdph uroh lq VÆ wkdw sod|v lq V,1 Wkxv/ wkh uxoh iru wdnlqj wkh Khuplwldq frqmxjdwh ri dq| frpsoh{ qxpehu lv vlpso| wr wdnh lwv frpsoh{ frqmxjdwh1Zhzulwh

^`. @ Æ ^Æ`. @ = +51468,

Zh qrz h{whqgwklv lghdwr rshudwruv1 Li wkh rshudwru D pdsv wkh nhw m"l/vd|/rqwr wkh nhw m#l wkhq wkh dgmrlqw ri wkh rshudwru D pxvw kdyh wkh fruuhvsrqglqj hhfw lq wkh dgmrlqw vsdfh1 Wkxv/ wkh rshudwru D. +zklfk lv uhdg D dgmrlqw ru D gdjjhu, kdv wkh hhfw wkdw li Dm"l @ m#l +51469, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 84 wkhq k"mD. @ k#m= +5146:, Wklv lv wkh uhodwlrqvkls zh zhuh orrnlqj iru lq wkh txhvwlrq srvhgderyh1 Lw vkrzv wkdw zkhq zh ls wklqjv durxqg zh kdyh wr uhsodfh rshudwruv e| wkhlu dgmrlqwv1 Wkxv zh fdq zulwh ^Dm"l`. @ k"mD. +5146;, D ihz prphqwv ri vwxg| ri wkh dgmrlqw surfhvv doorzv wkh iroorzlqj uxohv wr eh ghyhorshg= Wr wdnh wkh dgmrlqw ri dq| surgxfw ri rshudwruv/ qxpehuv/ eud*v/ nhw*v hwf1/ +4, uhsodfh doo hohphqwv e| wkhlu dgmrlqwv +eud*v duh uhsodfhg e| nhw*v/ rshudwruv e| wkhlu dgmrlqwv/ qxpehuv e| wkhlu frqmxjdwhv,/ dqg+5, uhyhuvh wkh rughuri doo hohphqwv lq wkh ruljlqdo surgxfw1 Rqfh wklv rshudwlrq lv shuiruphg/ dq| qxpehuv fdq eh frppxwhg sdvw dq| rshudwruv ru yhfwruv wr vlpsoli| wkh h{suhvvlrq1 Dv dq h{dpsoh/ qrwh wkdw

^k!ml`. @ km!l @ k!mlÆ +5146<,

^k#mDm!l`. @ k!mD.m#l @ k#mDm!lÆ Wkh uvw ri lghqwlw| lq hdfk ri wkh odvw wzr htxdwlrqv iroorzv iurp wkh uxohv iru wdnlqj wkh dgmrlqw/ wkh vhfrqgiurp wkh idfw wkdw erwk txdqwlwlhv duh vfdoduv/ dqgwkh dgmrlqwv ri zklfk duh mxvw wkh frpsoh{ frqmxjdwhv1 Ilqdoo|/ dv d pruh frpsolfdwhgh{dpsoh zh qrwh wkdw wkh rshudwru D @ k!mEmlmlkm +51473, kdvdvlwvdgmrlqw D. @ mlkmkmE.m!lÆ= +51474,

D vkruw olvw ri surshuwlhv ri wkh Khuplwldq dgmrlqw duh jlyhq ehorz=

^D.`. @ D +51475, ^D`. @ ÆD. +51476, ^D . E`. @ D. . E. +51477, ^DE`. @ E.D. +51478,

Wklv odvw uxoh/ zklfk glvsod|v wkh uhyhuvdo ri rughu ri wkh dgmrlqw ri d surgxfw/ lv hdvlo| suryhg1 Li DEm#l @ Dm#El @ m!l/ wkhq wkh dgmrlqw lv

...... k!m @ k#EmD @^m#El` D @^Em#l` D @ k#mE D +51479, iurp zklfk zh vhh wkdw li DEm#l @ m!l/wkhqk#mE.D. @ k!m> zklfk suryhv wkh uhvxow1 Zh duh qrz lq d srvlwlrq wr gh qh vrph dgglwlrqdo whupv/ rqh ri zklfk dsshduv lq wkh vwdwhphqw ri wkh vhfrqgsrvwxodwh1

5151< Khuplwldq/ Dqwl0Khuplwldq/ dqg Xqlwdu| Rshudwruv Dq rshudwru D lv Khuplwldq ru vhoi dgmrlqw li lw lv htxdo wr lwv Khuplwldq dgmrlqw/ l1h1/ li

D @ D.= +5147:,

Lq whupv ri pdwul{ hohphqwv/ wkh surshuw|

k#mDm!l @ k!mD.m#lÆ> +5147;, 85 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv zklfk lv wuxh iru dq| rshudwru/ uhgxfhv iru Khuplwldq rshudwruv wr wkh uhodwlrq

k#mDm!l @ k!mDm#lÆ= +5147<,

Dv d vshfldo fdvh wklv lpsolhv wkdw k!mDm!l @ k!mDm!lÆ/ zklfk lpsolhv wkdw h{shfwdwlrq ydoxhv ri d Khuplwldq rshudwru duh vwulfwo| uhdo1 Dq rshudwru D lv dqwl0Khuplwldq li lw lv htxdo wr wkh qhjdwlyh ri lwv dgmrlqw/ l1h1/ li

D @ D.= +51483,

Wkh pdwul{ hohphqwv ri dq dqwl0Khuplwldq rshudwru reh| wkh hdvlo|0ghulydeoh frqglwlrq k#mDm!l @ k!mDm#lÆ>zklfk lpsolhv wkdw k!mDm!l @ k!mDm!lÆ1 Wkxv/ h{shfwdwlrq ydoxhv ri dqwl0Khuplwldq rshudwruv duh vwulfwo| lpdjlqdu|1 Qrwh wkdw li D lv dq| rshudwru/ lw pd| eh zulwwhq lq wkh irup 4 4 D @ D . D. . D D. +51484, 5 5 @ DK . DD

4 . 4 . zkhuh DK @ 5 +D . D , lv Khuplwldq +wdnh lwv dgmrlqw dqg vhh$, dqg DD @ 5 +D D , lv dqwl0Khuplwldq +olnhzlvh$,1 Wkxv dq duelwudu| rshudwru fdq eh xqltxho| ghfrpsrvhg lqwr d vxp ri Khuplwldq dqgdqwl0Khuplwldq rshudwruv1 Dq rshudwru X lv xqlwdu| li lwv dgmrlqw lv htxdo wr lw lqyhuvh1 Wkxv/ iru d xqlwdu| rshudwru

X . @ X Ã4> +51485, ru htxlydohqwo|/ XX. @ X .X @ 4= +51486, Zh zloo vhh wkdw xqlwdu| rshudwruv +ru wkh wudqvirupdwlrqv wkh| lqgxfh, sod| wkh vdph uroh lq txdqwxp phfkdqlfdo Kloehuw vsdfhv wkdw ruwkrjrqdo wudqvirupdwlrqv sod| lq Fduwhvldq yhfwru vsdfhv vxfk dv U61

515143 Pdwul{ Uhsuhvhqwdwlrq ri Rshudwruv Ohw imqlj eh dq RQE iru wkh vsdfh V dqgohw D eh dq rshudwru dfwlqj lq wkh vsdfh1 Iurp wkh wulyldo lghqwlw| D @ 4D4 +51487, zh rewdlq d uhsuhvhqwdwlrq iru D e| vxevwlwxwlqj d ghfrpsrvlwlrq ri xqlw| lq wkh imqlj edvlv1 Wkxv/ zh rewdlq # $ # $ [ [ D @ mqlkqm D mq3lkq3m +51488, q q3 Zlwk wkh glhuhqw gxpp| lqglfhv zh fdq qrz uhpryh wkh sduhqwkhvhv wr rewdlq [ D @ mqlkqmDmq3lkq3m +51489, q>q3 zklfk zh zulwh lq wkh irup [ 3 D @ mql Dqq3 kq m> +5148:, q>q3 zkhuh 3 Dqq3 @ kqmDmq l +5148;, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 86 lv wkh pdwul{ hohphqw ri D frqqhfwlqj wkh edvlv vwdwhv mql dqg mq3l1Wkxv/zhrewdlqd ghfrpsrvlwlrq ri wkh rshudwru D lq wkh nhw0eud irup/ zklfk pdnhv lwv dfwlrq rq dq| vwdwh vhoi0hylghqw1 Wkh rshudwru D/ wkhuhiruh/ lv frpsohwho| ghwhuplqhg e| lwv pdwul{ hohphqwv lq dq| RQE1 Wkxv/ vxssrvh wkdw m!l @ Dm#l +5148<, iru vrph vwdwhv m#l dqg m!l1 Wkh h{sdqvlrq frh!flhqwv iru wkh vwdwhv m#l dqg m!l duh fohduo| uhodwhg1 Qrwh wkdw li [ [ m!l @ !q mqlm#l @ #q mql +51493, q q wkhq [ 3 3 !q @ kqm!l @ kqmD#l @ kqmDmq lkq m#l +51494, q3 zklfk fdq eh zulwwhq [ !q @ Dqq3 #q3 = +51495, q3 Exw wklv lv suhflvho| wkh irup ri d pdwul{ pxowlsolfdwlrq 3 4 3 4 D D 3 4 ! 44 45 # 4 E D D F 4 E ! F E 54 55 F E # F C 5 D @ E 1 1 1 F C 5 D +51496, 1 C 1 1 11 D 1 1 1 ridpdwul{kdylqjhohphqwvDqq3 zlwkdfroxpqyhfwrukdylqjhohphqwv#q/ uhvxowlqj lq d froxpq yhfwru zlwk hohphqwv !q1 Wkxv zh vhh wkdw lq wkh urz0yhfwru0froxpq yhfwru uhsuhvhqwdwlrq lqgxfhg e| dq| glvfuhwh RQE/ dq rshudwru lv qdwxudoo| uhsuhvhqwhg e| d pdwul{ kdylqj hqwulhv zklfk duh mxvw wkh pdwul{ hohphqwv ri wkdw rshudwru frqqhfwlqj wkh glhuhqw phpehuv ri wkh edvlv1 Qrwh wkdw lq surgxflqj wkh pdwul{ ri hohphqwv Dqq3 @ kqmDmq3l wkh eud fruuhvsrqgv wr wkh urz lqgh{/ zkloh wkh nhw fruuhvsrqgv wr wkh froxpq lqgh{1 Qrwh dovr wkdw wklv h{sdqvlrq ri wkh rshudwru [ 3 D @ mql Dqq3 kq m @ m4l D44k4m . m4l D45k5m . . m5l D54k4m . === +51497, q>q3 lq nhw0eud irup/ kdv wkh pdwul{ lqwhusuhwdwlrq 3 4 3 4 3 4 3 4 D44 D45 D44 3 3 D45 33 E F E F E F E F C D54 D55 D @ C 33 D.C 33 D. .C D54 3 D. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 +51498, zkhuh zh duh vlpso| oolqj xs hdfk vorw ri wkh pdwul{ rqh hohphqw dw d wlph1 Lw lv zruwkzkloh orrnlqj dw d ihz dgglwlrqdo h{dpsohv1 Frqvlghu wkh pdwul{ hohphqw k#mDm!l ehwzhhq duelwudu| vwdwhv m!l dqg m#l1 Lqvhuwlqj rxu h{sdqvlrq iru D wklv ehfrphv [ 3 k#mDm!l @ k#mql Dqq3 kq m!l > +51499, q>q3

3 Æ lq zklfk zh uhfrjql}h !q3 @ kq m!l dqg #q @ k#mql1 Wkxv/ zh rewdlq wkh uhvxow [ Æ k#mDm!l @ #q Dqq3 !q3 > +5149:, q>q3 87 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv zklfk kdv wkh pdwul{ lqwhusuhwdwlrq dvvrfldwhgzlwk wkh iroorzlqj rshudwlrq 3 4 D D 3 4 44 45 ! E D D F 4 Æ Æ E 54 55 F E ! F k#mDm!l @ #4 #5 E 1 1 1 F C 5 D = +5149;, C 1 1 11 D 1 1

Dv dqrwkhu h{dpsoh/ frqvlghu wkh rshudwru surgxfw ri [ 3 D @ mql Dqq3 kq m +5149<, q>q3 dqg [ 3 E @ mql Eqq3 kq m= +514:3, q>q3 Wkh surgxfw rshudwru F @ DE kdv d vlplodu h{sdqvlrq/ l1h1/ [ 3 F @ mql Fqq3 kq m +514:4, q>q3 zkhuh [ 3 3 33 33 3 Fqq3 @ kqmFmq l @ kqmDEmq l @ kqmDmq lkq mEmq l +514:5, q33 Wkxv [ Fqq3 @ Dqq33 Eq33q3 = q33 zklfk lv htxlydohqw wr wkh pdwul{ pxowlsolfdwlrq 3 4 3 4 3 4 F44 F45 D44 D45 E44 E45 E F E F E F E F54 F55 F E D54 D55 F E E54 E55 F E 1 1 F @ E 1 1 F E 1 1 F = +514:6, C 1 1 D C 1 1 D C 1 1 D

Dv d qdo h{dpsoh/ frqvlghu wkh pdwul{ uhsuhvhqwlqj wkh dgmrlqw ri dq rshudwru1 Li [ 3 D @ mql Dqq3 kq m +514:7, q>q3 wkhq e| wkh wzr0sduw uxoh zh ghyhorshg iru wdnlqj wkh dgmrlqw/ lw iroorzv wkdw [ . 3 Æ D @ mq l Dqq3 kqm= +514:8, q>q3

Vlqfh q dqg q3 duh vlpso| vxppdwlrq lqglfhv zh fdq vzlwfk wkhp wr qg wkdw [ [ . Æ 3 . 3 D @ mql Dq3q kq m @ mql Dqq3 kq m> +514:9, q>q3 q>q3 iurp zklfk zh ghgxfh wkdw . Æ Dqq3 @ Dq3q= +514::, . Wr lqwhusuhw wklv surshuo| d olwwoh fduh pxvw eh wdnhq zlwk wklv qrwdwlrq= wkh v|pero Dqq3 3 . Æ phdqv wkh q> q pdwul{ hohphqw ri wkh rshudwru D > zkloh wkh v|pero Dq3q phdqv wkh Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 88 frpsoh{ frqmxjdwh ri wkh q3>q pdwul{ hohphqw ri wkh rshudwru D1 Wkxv/ lq dq| RQE wkh pdwul{ uhsuhvhqwlqj D. lv wkh frpsoh{0frqmxjdwh wudqvsrvh ri wkh pdwul{ uhsuhvhqwlqj D1 D Khuplwldq rshudwru lv htxdo wr lwv dgmrlqw/ vr wkdw wkh pdwul{ hohphqwv uhsuhvhqwlqj vxfk dq rshudwru reh| wkh uhodwlrq

3 3 Æ Æ kqmDmq l @ Dqq3 @ kq mDmql @ Dq3q= +514:;, Wkxv/ iru d Khuplwldq rshudwru Æ Dqq3 @ Dq3q zklfk lpsolhv/ h1j1/ wkdw wkh gldjrqdo hohphqwv ri wkh pdwul{ uhsuhvhqwlqj d Khuplwldq rshudwru duh uhdo1 Pruh jhqhudoo|/ wklv vkrzv wkdw dq| pdwul{ D uhsuhvhqwlqj d Khuplwldq rshudwru lv htxdo wr lwv frpsoh{0frqmxjdwh wudqvsrvh/ l1h1/ D @ DW Æ1 Dq| pdwul{ reh|lqj wklv uhodwlrqvkls lv d Khuplwldq pdwul{1 Wkh v|pphwu| surshuwlhv ri dq dqwl0Khuplwldq rshudwruv dqgpdwulfhv duh ohiw dv dq h{huflvh1 Ohw xv jlyh vrph h{dpsohv/ vxssrvh wkdw lq d 60glphqvlrqdo yhfwruv vsdfh wkh rshudwruv D dqg E duh uhsuhvhqwhglq vrph ruwkrqrupdo edvlv e| wkh iroorzlqj pdwulfhv 3 4 3 4 35l : 6l 7l 7l 47 D @ C 5l 67D E @ C 3 95.9l D = +514:<, :.6l 7; ;; ;

Qrwh wkdw zh xvh qrq0lwdolfl}hgerogidfh v|perov wr uhsuhvhqw wkh pdwulfhv lq rughuwr glvwlqjxlvk wkhp iurp wkh rshudwruv wkhpvhoyhv1 Wkh pdwulfhv D. dqg E. uhsuhvhqwlqj wkh dgmrlqwv ri wkh rshudwruv D dqg E duh 3 4 35l 6l .: D. @ DW Æ @ C 5l 67D @ D 6l .: 7 ; dqg 3 4 7l 3; E. @ EW Æ @ C 7l 9;D > 47 5 9l ; iurp zklfk zh vhh wkdw wkh rshudwru D lv Khuplwldq/ dqglv uhsuhvhqwhge| d Khuplwldq pdwul{/ zkloh wkh rshudwru E lv qrw Khuplwldq1 Wkh odwwhu rshudwru fdq eh zulwwhq dv d vxp ri Khuplwldq dqgdqwl0Khuplwldq sduwv/ krzhyhu/ dv fdq wkh pdwulfhv uhsuhvhqwlqj lw/ l1h1/ zh fdq zulwh E @ EK . ED zkhuh 3 4 35l 44 4 . C D EK @ E . E @ 5l 98.6l 5 44 8 6l ; dqg 3 4 7l 5l 6 4 E @ E E. @ C 5l 3 6.6l D = D 5 66.6l 3

Lw lv dq lqwhuhvwlqj idfw wkdw qhlwkhu wkh wudqvsrvh ru wkh frpsoh{ frqmxjdwh ri dq rshudwru duh/ e| wkhpvhoyhv/ zhoo gh qhg frqfhswv> l1h1/ jlyhq dq rshudwru D> wkhuh lv qr rshudwru wkdw fdq eh xqltxho| lghqwl hg zlwk wkh wudqvsrvh ri D1 Dowkrxjk rqh fdq irup wkh wudqvsrvh DW ri wkh pdwul{ D uhsuhvhqwlqj D lq dq| edvlv/ wkh rshudwru dvvrfldwhgzlwk 89 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv wkh wudqvsrvhgpdwul{ zloo qrw jhqhudoo| fruuhvsrqgwr wkh rshudwru dvvrfldwhgzlwk wkh wudqvsrvh ri wkh pdwul{ uhsuhvhqwlqj D lq dq| rwkhu edvlv1 Wkxv/ wkh dfw ri wudqvsrvlwlrq lv d uhsuhvhqwdwlrq ghshqghqw rshudwlrq1 Vlploduo|/ frpsoh{ frqmxjdwlrq fdq eh shuiruphg rq pdwulfhv/ ru rq pdwul{ hohphqwv/ exw lw lv qrw dq rshudwlrq wkdw lv xqltxho| gh qhg iru wkh rshudwruv wkhpvhoyhv1 Lw lv vrphzkdw vxusulvlqj/ wkhuhiruh/ wkdw wkh Khuplwldq dgmrlqw/ zklfk lq d vhqvh frpelqhv wkhvh wzr uhsuhvhqwdwlrq ghshqghqw rshudwlrqv/ |lhogv dq rshudwru wkdw lv lqghshqghqw ri uhsuhvhqwdwlrq1 Wklv djdlq hpskdvl}hv rqh ri wkh edvlf wkhphv/ zklfk lv wkdw eudv/ nhwv/ dqgrshudwruv duh qrw urz yhfwruv/ froxpq yhfwruv/ dqg pdwulfhv1 Wkh iruphu pd| eh uhsuhvhqwhge| wkh odwwhu/ exw wkh uhsuhvhqwdwlrq dqgwkdw zklfk lv uhsuhvhqwhgduh wzr frqfhswxdoo| glhuhqw wklqjv1 Pdwul{ uhsuhvhqwdwlrqv ri wklv irup zhuh ghyhorshgh{whqvlyho| e| Khlvhqehuj dqgjdyh ulvh wr wkh whup pdwul{ phfkdqlfv/ lq dqdorj| wr wkh zdyh phfkdqlfv ghyhorshg e| Vfkuùglqjhu/ zklfk irfxvhv rq d zdyh ixqfwlrq uhsuhvhqwdwlrq iru wkh xqghuo|lqj vsdfh1 Fohduo|/ krzhyhu/ zkhwkhu rqh kdv d zdyh phfkdqlfdo ru pdwul{ phfkdqlfdo uhsuhvhqwdwlrq ghshqgv vlpso| xsrq wkh fkrlfh ri edvlv +l1h1/ glvfuhwh ru frqwlqxrxv, lq zklfk rqh lv zrunlqj1 H{whqvlrq wr Frqwlqxrxv Uhsuhvhqwdwlrqv 0Ohwimlj eh d frqwlqxrxv RQE iru wkh vsdfh V dqgohw D eh dq rshudwru dfwlqj lq wkh vsdfh1 Iurp wkh wulyldo lghqwlw|

D @ 4D4 +514;3, zh rewdlq d uhsuhvhqwdwlrq iru D e| vxevwlwxwlqj d ghfrpsrvlwlrq ri xqlw| lq wkh imlj edvlv1 Wkxv/ zh rewdlq ] ] D @ g mlkm D g3 m3lk3m = +514;4,

Zlwk wkh glhuhqw gxpp| lqglfhv vdiho| lq sodfh zh fdq qrz uhpryh wkh sduhqwkhvhv wr rewdlq ] ] D @ g g3 mlkmD m3lk3m +514;5, zklfk zh zulwh lq wkh irup ] ] D @ g g3 ml D+> 3, k3m +514;6, zkhuh wkh nhuqho D+> 3,@kmD m3l +514;7, ri wklv lqwhjudo uhodwlrq lv mxvw wkh pdwul{ hohphqw ri D frqqhfwlqj wkh edvlv vwdwhv ml dqg m3l1 Wkxv/ zh rewdlq d ghfrpsrvlwlrq ri wkh rshudwru D lq wkh nhw0eud irup/ zklfk pdnhv lwv dfwlrq rq dq| vwdwh vhoi0hylghqw1 Wkh rshudwru D lv/ wkhuhiruh/ frpsohwho| ghwhuplqhg e| lwv pdwul{ hohphqwv lq dq| frqwlqxrxv RQE1 Wkxv/ li

m!l @ Dm#l +514;8, iru vrph vwdwhv ] m!l @ g !+, ml +514;9, dqg ] m#l @ g #+, ml +514;:, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 8: wkhq ] !+,@km!l @ kmDm#l @ g3 kmDm3lk3m#l +514;;, ru ] !+,@ g3 D+> 3, #+3,= +514;<,

Wklv lv ri wkh irup ri d frqwlqxrxv pdwul{ pxowlsolfdwlrq/ zlwk lqwhjudwlrq uhsodflqj wkh vxppdwlrq surfhvv1 Lw lv zruwkzkloh orrnlqj dw d ihz dgglwlrqdo h{dpsohv1 Frqvlghu wkh pdwul{ hohphqw k#mDm!l ehwzhhq duelwudu| vwdwhv m!l dqg m#l1 Lqvhuwlqj rxu h{sdqvlrq iru D wklv ehfrphv ] ] k#mDm!l @ g g3 k# ml D+> 3, k3m!l > +514<3, lq zklfk zh uhfrjql}h !+3,@k3m!l dqg #Æ+,@k#ml1Wkxv/zhrewdlqwkhuhvxow ] ] k#mDm!l @ g g3 #Æ+, D+> 3, !+3, > +514<4, zklfk lv wkh frqwlqxrxv dqdorj ri wkh pdwul{ h{suhvvlrq zh zurwh hduolhu1 Dv d qdo h{dpsoh/ frqvlghu wkh rshudwru surgxfw ri wkh rshudwruv ] ] D @ g g3 ml D+> 3, k3m +514<5, dqg ] ] E @ g g3 ml E+> 3, k3m = +514<6,

Wkh surgxfw F @ DE kdv dq h{sdqvlrq ] ] F @ g g3 ml F+> 3, k3m +514<7, lq zklfk ] F+> 3,@kmFm3l @ kmDEm3l @ g33 kmDm33lk33mEm3l= +514<8,

Wkxv/ zh qgwkdw ] F+> 3,@ g33 D+> 33, E+33>3, +514<9, zklfk lv wkh frqwlqxrxv dqdorj ri d pdwul{ pxowlsolfdwlrq1 Wkhdgmrlqwridqrshudwrukdvdnhuqhozklfklvwkhfrqwlqxrxvdqdorjriwkhfrpsoh{0 frqmxjdwh wudqvsrvh/ l1h1/ D.+> 3,@DÆ+3>,= +514<:, Iru wkh nhuqho uhsuhvhqwlqj d Khuplwldq rshudwru lq d frqwlqxrxv edvlv zh kdyh wkh vlpsohu uhodwlrqvkls DÆ+> 3,@D+3>,1 H{dpsohv= Dv dq h{dpsoh/ wkh rshudwru [ kdv dv lwv pdwul{ hohphqwv lq wkh srvlwlrq uhsuhvhqwdwlrq

ku 3m[mul @ ku 3m+{mul,@{ku 3mul @ {+u 3 u, +514<;,

Wklv doorzv xv wr frqvwuxfw wkh h{sdqvlrq iru wklv rshudwru ] ] ] [ @ g6u 3 g6u mu 3l{+u 3 u,kum @ g6u mul{kum +514<<, 8; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv zkhuh wkh grxeoh lqwhjudo kdv ehhq uhgxfhg wr d vlqjoh lqwhjudo ehfdxvh ri wkh ghowd ixqfwlrq1 Wkh rshudwru [ lv vdlgwr eh gldjrqdo lq wkh srvlwlrq uhsuhvhqwdwlrq/ ehfdxvh lw kdv qr qrq}hur hohphqwv frqqhfwlqj glhuhqw vwdwhv lq wklv uhsuhvhqwdwlrq1 Wklv frqfhsw ri gldjrqdolw| h{whqgv eh|rqg wkh srvlwlrq uhsuhvhqwdwlrq1 Lq sduwlfxodu/ dq rshudwru D lv vdlgwr eh gldjrqdo lq wkh imllj uhsuhvhqwdwlrq li

Dlm @ klmDmml @ Dllm +51533, vr wkdw [ [ [ D @ mll Dlm kmm @ mll Dllm kmm @ mll Dlklm> +51534, l>m l>m l zklfk rqo| kdv rqh vxppdwlrq lqgh{/ lq frqwudvw wr wkh jhqhudo irup zklfk uhtxluhv wzr1 Lq d uhsuhvhqwdwlrq lq zklfk dq rshudwru lv gldjrqdo/ wkhuhiruh/ lw lv uhsuhvhqwhg e| d gldjrqdo pdwul{ 3 4 D4 33 E F E 3 D5 3 F D @ E F C 33D6 D 1 1 1 1 1 1 1 11 Vlploduo|/ lq d frqwlqxrxv uhsuhvhqwdwlrq imlj> dq rshudwru D lv gldjrqdo li

D+> 3,@kmDm3l @ D+,+ 3,> +51535, vr wkdw ] ] ] ] D @ g g3 mlD+> 3,k3m @ g g3 mlD+,+ 3,k3m> +51536, ru ] D @ g mlD+,km= +51537,

Lw lv hdv| wr vkrz wkdw lq dq| edvlv lq zklfk dq rshudwru lv gldjrqdo/ lw lv zkdw zh uhihuuhg wr hduolhu dv d pxowlsolfdwlyh rshudwru1 Wkdw lv/ li ] J @ g ml j+, km> +51538, U lv gldjrqdo lq wkh imlj uhsuhvhqwdwlrq/ dqgli m#l @ g #+,ml/wkhq ] ] Jm#l @ g ml j+, km#l @ g ^j+, #+,`ml> +51539, zklfk vkrzv wkdw d gldjrqdo rshudwru J dfwv lq wkh imlj uhsuhvhqwdwlrq wr pxowlso| wkh zdyh ixqfwlrq e| j+,1 Zh olvw ehorz vrph dgglwlrqdo rshudwruv dqg pdwul{ hohphqwv lq wkh edvlv lq zklfk wkh| duh gldjrqdo1 Ghulydwlrq lv vwudljkwiruzdugdqgohiw dv dq h{huflvh1

41 Wkh srvlwlrq rshudwru ] U @ g6u mul u kumku 3mU mul @ u+u u 3, +5153:,

51 Wkh srwhqwldo hqhuj| rshudwru ] Y @ g6u mul Y +u, kumku 3mY mul @ Y +u, +u u 3,= +5153;, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 8<

61 Wkh zdyhyhfwru rshudwru ] N @ g6n mnln knm> kn 3mN mnl @ n+n n 3,> +5153<,

71 Wkh prphqwxp rshudwru ] S @ |N @ g6n mnl |n knmkn 3mS mnl @ |n+n n 3,= +51543,

81 Wkh nlqhwlf hqhuj| rshudwru ] |5N5 |5n5 |5n5 W @ @ g6n mnl knmkn 3mW mnl @ +n n 3,= +51544, 5p 5p 5p

Lw lv lpsruwdqw wr srlqw rxw wkdw/ dowkrxjk dq rshudwru pd| eh gldjrqdo lq rqh uhsuhvhq0 wdwlrq/ lw lv jhqhudoo| qrw gldjrqdo lq prvw rwkhuv1 Dv dq dgglwlrqdo h{dpsoh zh zrun rxw ehorz wkh pdwul{ hohphqwv ri wkh zdyhyhfwru rshudwru N lq wkh srvlwlrq uhsuhvhqwdwlrq1 Uhfdoo wkdw wkh zdyhyhfwru rshudwru N @ lG lv dU glhuhqwldo rshudwru lq wkh srvlwlrq uhsuhvhqwdwlrq1 Wklv phdqv wkdw iru dq| vwdwh m#l @ g6u#+u,mul> wkh vwdwh N m#l lv jlyhq e| wkh h{sdqvlrq ] k l N m#l @ g6u lu #+u, mul= +51545,

Rq wkh rwkhu kdqg/ zh nqrz wkdw zh fdq dozd|v zulwh ] ] N @ g6u g6u3 mul N +u>u 3, ku 3m +51546, zkhuh N +u>u 3,@ku m N m u 3l> vr wkdw ] ] ] ] N m#l @ g6u g6u3 mul N +u>u 3, ku 3m#l @ g6u g6u3 N +u> u 3, #+u 3, mul= +51547,

Frpsdulqj wkh odvw htxdwlrq wr +51545, zh ghgxfh wkdw iru dq| zdyhixqfwlrq #+u,> ] g6u3 N +u>u 3, #+u 3,@lu #+u,=

Frpsdulqj wklv wr wkh edvlf surshuw| dvvrfldwhgzlwk wkh judglhqw ri wkh ghowd ixqfwlrq/ qdpho| ] g6u3 u +u u 3, i+u 3,@i 3+u,> zklfk krogv iru dq| ixqfwlrq i+u,> zh ghgxfh wkdw N +u>u 3,@lu +u u 3,1Wkxvwkh pdwul{ hohphqwv ri wkh zdyhyhfwru rshudwru lq wkh srvlwlrq uhsuhvhqwdwlrq wdnh wkh irup

ku m N m u 3l @ lu +u u 3,=

Dowkrxjk wkhvh pdwul{ hohphqwv dsshdu wr eh }hur hyhu|zkhuh/ wkh zdyhyhfwru rshudwru lv qrw gldjrqdo lq wkh srvlwlrq uhsuhvhqwdwlrq1 Wkh uhdvrq iru wklv lv wkdw wkh judglhqw ri wkh ghowd ixqfwlrq +ru wkh ghulydwlyhv ri wkh ghowd ixqfwlrq lq jhqhudo,/ uhsuhvhqw d olplwlqj surfhvv lqyroylqj wkh glhuhqfh ehwzhhq wzr ydoxhv ri d ixqfwlrq lq qlwhvlpdoo| glvsodfhg iurp wkh gldjrqdo +h1j1/ u @ u u,1Wkhh{dpsohderyhlvxvhixolqwkdwlwvkrzvwkdw 93 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv hyhq glhuhqwldo rshudwruv fdq ohjlwlpdwho| eh ylhzhgdv kdylqj pdwul{ hohphqwv/ dowkrxjk wkhlu pdwul{ hohphqwv lqyroyh yhu| rgg glvwulexwlrqdo ixqfwlrqv/ vxfk dv wkh ghulydwlyhv ri ghowd ixqfwlrqv1 Lq d vlplodu idvklrq/ wkh iroorzlqj pdwul{ hohphqwv duh uhdglo| hvwdeolvkhg

3 3 kn m U m n l @ lun+n n ,> +51548, |5 k u m W m u 3l @ u5+u u 3, +51549, 5p zkhuh lq wkh odvw olqh lw lv wkh Odsodfldq ri wkh ghowd ixqfwlrq wkdw dsshduv lq wkh h{suhvvlrq iru wkh pdwul{ hohphqwv ri wkh nlqhwlf hqhuj| rshudwru1 515144 Fdqrqlfdo Frppxwdwlrq Uhodwlrqv Lw lv fohdu wkdw lq wkh vsdfh ri d vlqjoh txdqwxp phfkdqlfdo sduwlfoh/ wkhuh lv d yhu| forvh uhodwlrqvkls ehwzhhq wkh srvlwlrq rshudwru U dqgwkh zdyhyhfwru rshudwru N /ru htxlydohqwo|/ wkh prphqwxp rshudwru S @ |N 1 Wklv uhodwlrqvkls lv riwhq h{suhvvhglq whupv ri wkh frppxwdwlrq uhodwlrqv ehwzhhq wkh glhuhqw fduwhvldq frpsrqhqwv ri wkhvh rshudwruv1 Wkhvh uhodwlrqv/ zklfk duh uhihuuhgwr dv fdqrqlfdo frppxwdwlrq uhodwlrqv duh hdv| wr ghulyh1 Zh qrwh uvw wkdw wkh Fduwhvldq frpsrqhqwv ri wkh srvlwlrq rshudwru frppxwh zlwk rqh dqrwkhu/ l1h1/ wkhlu dfwlrq rq wkh edvlv lq zklfk wkh| duh gldjrqdo vkrzv/ h1j1/ wkdw [\ mul @ {|mul @ \[mul +5154:, vlqfh wklv lv wuxh iru hdfk hohphqw ri dq RQE zh ghgxfh dq rshudwru lghqwlw| [\ @ \[/ ru ^[> \ `@31 Wklv h{whqgv wr wkh rshudwru ] dv zhoo/ vr wkdw zh fdq zulwh/ txlwh jhqhudoo|/

^[l>[m`@3= +5154;, E| dq dqdorjrxv dujxphqw lw lv irxqgwkdw wkh Fduwhvldq frpsrqhqwv ri wkh zdyhyhfwru ru prphqwxp rshudwru frppxwh zlwk rqh dqrwkhu/ l1h1/

^Nl>Nm`@3@^Sl>Sm`= +5154<, Rq wkh rwkhu kdqg/ wkh Fduwhvldq frpsrqhqwv ri srvlwlrq gr qrw jhqhudoo| frppxwh zlwk wkh Fduwhvldq frpsrqhqwv ri zdyhyhfwru ru prphqwxp1 Wr vhh wklv lw lv xvhixo wr zrun lq d vshfl f uhsuhvhqwdwlrq +hlwkhu rqh zrxogvx!fh,1 Lq wkh srvlwlrq uhsuhvhqwdwlrq/ zh qrwh wkdw iru dq duelwudu| vwdwh m#l uhsuhvhqwhge| wkh zdyh ixqfwlrq #+u,/ C# C# [lNm#+u,@{l l @ l{l = +51553, C{m C{m Rq wkh rwkhu kdqg/ C C# Nm[l#+u,@l i{l#+u,j @ lilm#+u,.{l j +51554, C{m C{m zkhuh zh kdyh xvhgwkh vwdqgduguhodwlrq C{l@C{m @ lm 1 Wkxv wkh dfwlrq ri wkh frppx0 wdwru ^[l>Nm>`@[lNm Nm[l rq vxfk d vwdwh lv jlyhq lq wkh srvlwlrq uhsuhvhqwdwlrq e| wkh h{suhvvlrq C# C# ^[l>Nm`#+u,@l{l . lilm #+u,.{l j @ llm #+u,= +51555, C{m C{m Wklv ehlqj wuxh iru doo vwdwhv m#l zh ghgxfh wkh rshudwru lghqwlw|

^[l>Nm`@llm +51556, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 94 ru wkh htxlydohqw uhodwlrq h{suhvvhglq whupv ri wkh Fduwhvldq frpsrqhqwv ri wkh prphq0 wxp rshudwru

^[l>Sm`@l|lm1 +51557, Wkxv/ wkh frpsrqhqwv ri srvlwlrq dqg prphqwxp dorqj wkh vdph gluhfwlrq ri vsdfh gr qrw frppxwh zlwk rqh dqrwkhu1 Pruh jhqhudoo|/ rqh fdq ghulyh wkh uhodwlrqvkls

^U d>a N ae`@lda ae +51558, iru duelwudu| frpsrqhqwv ri wkh srvlwlrq dqgzdyhyhfwru rshudwru1 Wkhvh edvlf frppxwd0 wlrq uhodwlrqv fdq eh xvhgwr ghyhors pruh frpsolfdwhgfrppxwdwruv lqyroylqj ixqfwlrqv ri wkh srvlwlrq dqgzdyhyhfwru +ru prphqwxp, rshudwruv1

515145 Pdwul{ Hohphqwv ri Xqlwdu| Rshudwruv +Fkdqjlqj Uhsuhvhqwdwlrq, Li X lv d xqlwdu| rshudwru wkhq lw reh|v wkh xqlwdulw| frqglwlrq

X .X @ XX. @ 4= +51559,

Zh fdq h{suhvv wklv uhodwlrqvkls lq whupv ri wkh pdwul{ hohphqwv ri X lq dq| RQE im!llj iru wkh vsdfh/ lq wkh irup [ . XlnXnm @ lm= +5155:, n . Lq wklv h{suhvvlrq/ Xnm @ k!nmXm!ml/ zkloh wkh pdwul{ hohphqwv ri X vdwlvi| wkh uhodwlrq . Æ Xln @ Xnl1Wklvohdgvwrwkhuhvxow [ Æ XnlXnm @ lm> +5155;, n zklfk/ zh dvvhuw/ orrnv vrphwklqj olnh wkh ruwkrqrupdolw| uhodwlrq iru d vhw ri yhfwruv1 Wr pdnh wklv d olwwoh pruh fohdu/ ohw xv gh qh d vhw ri yhfwruv [ mxll @ Xnlm!nl +5155<, n dqg [ mxml @ Xnmm!nl +51563, n zkrvh h{sdqvlrq frh!flhqwv lq wkh im!llj uhsuhvhqwdwlrq duh wkh froxpqv ri wkh pdwul{ uhsuhvhqwlqj wkh xqlwdu| rshudwru X1 Wkh lqqhu surgxfw ri wkhvh yhfwruv lv [ Æ kxlmxml @ XnlXnm @ lm= +51564, n

Wkxv/ wkh vhw ri yhfwruv imxllj irup dq ruwkrqrupdo vhw1 Vlqfh wkh| duh dovr htxdo lq qxpehu wr wkh froxpqv ri wkh pdwul{ uhsuhvhqwlqj X> dqgwkhuhiruh wr wkh qxpehu ri edvlv yhfwruv lq wkh ruljlqdo edvlv/ wkhvh yhfwruv irup dqrwkhu RQE iru wkh vdph vsdfh1 D xqlwdu| rshudwru X/ wkhuhiruh/ doorzv xv wr frqvwuxfw d qhz ruwkrqrupdo edvlv iurp wkh ruljlqdo rqh1 Lq idfw/ X lv suhflvho| wkdw rshudwru zklfk pdsv wkh ruljlqdo edvlv yhfwruv rqwr wkh qhz rqhv1 Wr vhh wklv/ qrwh wkdw e| frqvwuxfwlrq/ [ [ mxll @ Xnlm!nl @ k!nmXm!llm!nl= +51565, n n 95 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Fkdqjlqj wkh rughu ri wkh pdwul{ hohphqw +d qxpehu, dqg wkh yhfwru lq wkh odvw h{sdqvlrq/ zh rewdlq [ mxll @ m!nlk!nmXm!ll @ 4Xm!ll> +51566, n lq zklfk zh kdyh lghqwl hg d ghfrpsrvlwlrq ri wkh xqlw rshudwru lq wkh ruljlqdo edvlv1 Wkxv/ zh qgwkdw

mxll @ Xm!ll= +51567, Wkxv/ dq| xqlwdu| rshudwru X pdsv dq duelwudu| RQE rqwr dqrwkhu RQE1 Wkh lqyhuvh ri wklv lv dovr wuxh1 Jlyhq dq| wzr RQE*v iru wkh vsdfh/ wkhuh h{lvwv d xqlwdu| rshudwru zklfk frqqhfwv wkhp/ dqgzklfk fdq eh xvhgwr fkdqjh uhsuhvhqwdwlrqv iurp rqh edvlv wr wkh rwkhu1 Ohw im!llj dqg im#llj eh wzr duelwudu| RQE*v iru d vsdfh1 Ohw X eh wkh rshudwru wkdw pdsv wkh lwk hohphqw ri wkh vhw im!llj rqwr wkh fruuhvsrqglqj hohphqw ri wkh vhw im#llj/ l1h1/ Xm!ll @ m#ll iru l @4> 5> +51568, Zh zloo vkrz wkdw wklv rshudwru lv/ lq idfw/ xqlwdu|1 Wr vhh wklv/ qrwh wkdw wkh pdwul{ hohphqwv ri wklv rshudwru lq wkh im!llj uhsuhvhqwdwlrq duh jlyhq e| wkh h{suhvvlrq

Xlm @ k!lmXm!ml @ k!lm#ml/ +51569, vr wkdw zh fdq h{sdqg X lq wkh irup [ [ [ X @ m!llXlm k!mm @ m!llk!lm#mlk!mm @ m#mlk!mm= +5156:, l>m l>m m

Qrwh wkdw lq wkh odvw h{suhvvlrq zh kdyh djdlq lghqwl hg d ghfrpsrvlwlrq ri wkh xqlw rshudwru/ doorzlqj iru wkh vlpsohu irup1 Wkh dgmrlqw ri wklv uhodwlrq lv [ . X @ m!llk#lm= +5156;, l Wkxv/ wkh surgxfw ri X dqglwv dgmrlqw jlyhv [ [ [ [ [ . X X @ m!llk#lm#mlk!mm @ m!lllm k!mm @ m!llk!lm @ 4> +5156<, m l m l l wkh lghqwlw| rshudwru1 Wdnlqj wkh surgxfw lq uhyhuvh rughu/ rq wkh rwkhu kdqg/ |lhogv d vlplodu uhvxow [ [ [ [ [ . XX @ m#mlk!mm!llk#lm @ m#mlmlk#lm @ m#mlk#mm @ 4= +51573, m l m l m

Qrwhwkdwzhkdyhxvhgwkhruwkrqrupdolw|uhodwlrqdvvrfldwhgzlwkhdfkRQElqvlp0 soli|lqj wkhvh h{suhvvlrqv1 Khqfh/ wkh rshudwruv X dqg X . duh xqlwdu|1 Wkh rshudwru X . wdnhv nhwv lq im!llj wr nhwv lq im#llj/ zkloh X wdnhvnhwvlqim#llj wr nhwv lq im!llj1Wkdw lv/ xvlqj wkh h{sdqvlrq iru X ./ [ . X m#ll @ m!mlk#mm#ll @ m!ll> +51574, m

. Lq wkh im#llj edvlv/ X kdv pdwul{ hohphqwv

. . Xlm @ k#lmX m#ml @ k#lm!ml= +51575, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 96

Wkxv/ zh vhh wkdw

Xlm @ k!lm#ml +51576, . Xlm @ k#lm!ml= +51577, Wkh sudfwlfdo xvh ri wkhvh pdwul{ hohphqwv frphv zkhq zh zlvk wr wudqvirup iurp rqh uhsuhvhqwdwlrq wr dqrwkhu1 Zh uvw frqvlghu wkh wudqvirupdwlrq ri yhfwruv1 d, Wudqvirupdwlrq ri Nhwv 0Ohwm"l eh dq duelwudu| nhw lq wkh vsdfh1 Lw fdq eh h{sdqghg lq hlwkhu ri wkh wzr edvhv frqvlghuhg deryh/ l1h1/ zh fdq zulwh [ m"l @ "l m!ll +51578, l zkhuh "l @ k!lm"l/ru [ 3 m"l @ "l m#ll/ +51579, l 3 zkhuh "l @ k#lm"l1 Wkh txhvwlrq lv krz duh wkh h{sdqvlrq frh!flhqwv lq wkhvh wzr edvhv uhodwhgwr rqh dqrwkhu1 Wr qgrxw zh xvh dq dssursuldwh ghfrpsrvlwlrq ri wkh lghqwlw| rshudwru/ l1h1/ zh zulwh [ "l @ k!lm"l @ k!lm4m"l @ k!lm#mlk#mm"l +5157:, m

Exwzhkdyhvhhqderyh/wkdwwkhtxdqwlwlhvk!lm#ml @ Xlm duh mxvw wkh pdwul{ hohphqwv 3 ri wkh xqlwdu| rshudwru frqqhfwlqj wkhvh wzr edvhv/ zkloh k#mm"l @ "m lv wkh h{sdqvlrq frh!flhqw lq wkh rwkhu edvlv1 Wkxv zh kdyh wkh uhodwlrq [ 3 "l @ Xlm"m> +5157;, m zklfk lv ri wkh irup ri d pdwul{ pxowlsolfdwlrq 3 4 3 4 X44 X45 3 "4 E F "4 E F E X54 X55 F 3 C "5 D @ E 1 1 F "5 = +5157<, 1 C 1 1 D 1 1 1

E| d vlplodu dssurdfk lw fdq eh vkrzq wkdw wkh uhyhuvh wudqvirupdwlrq lv hhfwhge| wkh pdwul{ uhsuhvhqwlqj X .1Wkxv/zhkdyhwkhuhodwlrq [ 3 . "l @ Xlm "m= +51583, m e, Wudqvirupdwlrq ri Pdwulfhv 0LiD lv dq rshudwru lw kdv pdwul{ hohphqwv lq wkh wzr edvhv frqvlghuhg deryh ri wkh irup

Dlm @ k!lmDm!ml +51584, dqg 3 Dlm @ k#lmDm#ml= +51585, Wr qgwkh uhodwlrqvkls ehwzhhq wkh pdwulfhv uhsuhvhqwlqj wklv rshudwru lq wkhvh wzr edvhv zh zulwh

Dlm @ k!lm4D4m!ml +51586, 97 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

dqglqvhuw ghfrpsrvlwlrqv ri xqlw| lq wkh im#mlj1 Wklv |lhogv [ Dlm @ k!lm#nlk#nmDm#n3 lk#n3 m!ml +51587, n>n3 zklfk zh lghqwli| iurp deryh dv [ 3 . Dlm @ Xln Dnn3 Xn3m +51588, n>n3 zklfk lv ri wkh irup ri d pdwul{ pxowlsolfdwlrq D @ XD3X./zkhuhD lv wkh pdwul{ zlwk hohphqwv Dlm1 Wkh uhyhuvh wudqvirupdwlrq lv irxqglq wkh vdph zd|/ dqg|lhogvwkh uhvxow D3 @ X.DX1 Ehiruh frqvlghulqj dq h{dpsoh/ lw vkrxogeh qrwhgwkdw xqlwdu| rshudwruv suhvhuyh wkh qrup ri dq| yhfwru wkdw wkh| dfw xsrq1 Wklv lv lqwxlwlyho| uhdvrqdeoh/ vlqfh wkh| kdyh wkh vlpsoh hhfw ri fkdqjlqj wkh frruglqdwh v|vwhp/ lq wkh vdph zd| wkdw ruwkrjrqdo wudqvirupdwlrqv gr lq uhdo yhfwru vsdfhv/ exw lv dovr txlwh hdv| wr suryh1 Li m"l lv dq duelwudu| yhfwru zklfk lv wdnhq e| d xqlwdu| rshudwru X rqwr wkh yhfwru m!l @ Xm"l wkhq wkh vtxduhgqrup ri wkh wudqviruphgyhfwru lv jlyhq e| wkh uhodwlrq

k!m!l @+k"mX .,+Xm"l,@k"mX .Xm"l @ k"m"l +51589, vlqfh X .X @ 41 H{dpsoh +H{whqvlrq wr Frqwlqxrxv Uhsuhvhqwdwlrqv, Ohw m#l eh dq duelwudu| yhfwru lq wkh vsdfh ri d txdqwxp sduwlfoh lq wkuhh glphqvlrqv/ l1h1/ wkh vsdfh vsdqqhge| wkh yhfwruv imulj ri wkh srvlwlrq uhsuhvhqwdwlrq dqge| wkh yhfwruv imnlj ri wkh zdyhyhfwru uhsuhvhqwdwlrq1 Zh fdq h{sdqgwkh nhw m#l lq hlwkhu ri wkhvh wzr edvhv/ l1h1/ ] m#l @ g6u#+u, mul +5158:, zkhuh #+u,@kum#l dqg ] m#l @ g6n#+n, mnl +5158;, zkhuh #+n,@knm#l1 Krz duh wkh h{sdqvlrq frh!flhqwv #+u, uhodwhgwr wkh h{sdqvlrq frh!flhqwv #+n,1 Zh fdq qg rxw lq wkh vdph zd| dv zh mxvw glg iru wkh glvfuhwh fdvh/ l1h1/ zh zulwh ] #+u,@kum#l @ kum4m#l @ g6n kumnlknm#l +5158<, zklfk zh zulwh dv ] #+u,@ g6nX+u>n,#+n, +51593, zkhuh wkh +frqwlqxrxv, pdwul{ hohphqwv ri wkh xqlwdu| rshudwru frqqhfwlqj wkhvh wzr edvhv duh hlnÄu X+u>n,@kumnl @ = +51594, +5,6@5 Wkxv/ zh qgwkdw ] hlnÄu #+u,@ g6n #+n, +51595, +5,6@5 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 98 zklfk/ ri frxuvh/ zh douhdg| nqhz1 Vlploduo|/ zh qg wkdw ] ] hÃlnÄu #+n,@ g6uX.+n>u,#+u,@ g6u #+u,= +51596, +5,6@5

Wkxv/ wkh Irxulhu wudqvirup lv mxvw dq h{dpsoh ri d xqlwdu| wudqvirupdwlrq iurp rqh frqwlqxrxv edvlv wr dqrwkhu1 Lw lv dovr srvvleoh wr xvh wkh xqlwdu| wudqvirupdwlrq uhsuhvhqwhge| wkh Irxulhu wudqvirup wr ghulyh wkh pdwul{ hohphqwv ri vrph ri wkh rshudwruv douhdg| hqfrxqwhuhg1 Dv dq h{dpsoh/ frqvlghu wkh srvlwlrq rshudwru U> zkrvh pdwul{ hohphqwv lq wkh srvlwlrq uhsuhvhqwdwlrq duh jlyhq e| wkh h{suhvvlrq ku m U m u 3l @ U+u> u 3,@u+u u 3,= Wkh pdwul{ hohphqwv lq wkh zdyhyhfwru uhsuhvhqwdwlrq fdq eh rewdlqhgiurp wklv e| d xqlwdu| wudqvirupdwlrq/ l1h1/ ] ] U+n>n 3,@ g6u g6u 3 X .+n>u, U+u>u 3, X+u 3>n, ] ] 4 3 3 @ g6u g6u 3 hÃlnÄu u+u u 3, hln Äu +5,6 ] ] 6 4 3 g u 3 @ g6uuhÃl+nÃn ,Äu @ lu hl+nÃn ,Äu +51597, +5,6 n +5,6 zklfk jlyhv wkh uhvxow vwdwhghduolhu zlwkrxw surri/ qdpho|/ wkdw

3 3 3 kn m U m n l @ U+n> n ,@lun+n n ,= +51598,

515146 Uhsuhvhqwdwlrq Lqghshqghqw Surshuwlhv ri Rshudwruv Wkhuh duh d qxpehu ri surshuwlhv dvvrfldwhgzlwk rshudwruv zklfk duh lqghshqghqw ri wkh uhsuhvhqwdwlrq xvhgwr h{suhvv wkhp1 Wkhvh surshuwlhv lqfoxgh=

Wkh wudfh ri dq rshudwru D/ ghqrwhg e| Wu+D,/ lv wkh vxp ri wkh gldjrqdo hohphqwv ri dq| pdwul{ uhsuhvhqwlqj wkh rshudwru/ l1h1/ [ [ Wu+D,@Wu+D,@ Dll @ klmDmll +51599, l l iru dq| ruwkrqrupdo edvlv ri vwdwhv imllj1 Lq d frqwlqxrxv edvlv/ e| gh qlwlrq/ ] Wu+D,@ g D+> ,= +5159:,

Wkh wudfh ri dq rshudwru +ru pdwul{, kdv pdq| lqwhuhvwlqj surshuwlhv1 Lw lv hdvlo| yhul hg/ h1j1/ wkdw lq dq| qlwh0glphqvlrqdo vsdfh wkh wudfh ri d surgxfw ri pdwulfhv +ru rshudwruv, lv lqyduldqw xqghu f|folf shupxwdwlrq ri wkh hohphqwv lq wkh surgxfw1 Wkdw lv/

Wu+DEFG,@Wu+EFGD,@Wu+FGDE,@Wu+GDEF,= +5159;,

Dv dq lpsruwdqw frqvhtxhqfh ri wklv idfw/ lw iroorzv wkdw

Wu+XDX.,@Wu+XX.D,@Wu+D, +5159<, zklfk vkrzv wkdw wkh wudfh ri D lv lqyduldqw xqghu d xqlwdu| wudqvirupdwlrq/ dqg khqfh lv lqghshqghqw ri wkh vshfl f uhsuhvhqwdwlrq xvhg wr hydoxdwh lw1 99 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Wkh ghwhuplqdqw ri dq rshudwru D/ ghqrwhg ghw+D,/ lv wkh ghwhuplqdqw ri dq| pdwul{ uhsuhvhqwlqj wkh rshudwru/ l1h1/ D44 D45 D54 D55 ghw+D,@ 1 1 +515:3, 1 1 1 1

Edvlf idploldulw| zlwk jhqhudo surshuwlhv ri wkh ghwhuplqdqw ri d pdwul{ zloo eh dvvxphg1 Iru h{dpsoh/ wkh ghwhuplqdqw ri d 5 5 pdwul{ lv de @ dg ef> +515:4, fg zkloh wkh ghwhuplqdqw ri d gldjrqdo pdwul{ lv mxvw wkh surgxfw d44 33 3 d55 3 \ @ d +515:5, 33d66 3 l 1 1 1 l 1 1 3 11 ri wkh gldjrqdo hohphqwv1 Wkxv/ h1j1/ wkh lghqwlw| rshudwru kdv d ghwhuplqdqw ri xqlw|/ ghw+4,@41 Lq dgglwlrq/ lw wxuqv rxw wkdw wkh ghwhuplqdqw ri d surgxfw lv htxdo wr wkh surgxfw ri wkh ghwhuplqdqwv/ l1h1/

ghw+DEF,@ghw+D, ghw+E, ghw+F,= +515:6,

Wklv odvw uhvxow lpsolhv wkdw

ghw+XDX.,@ghw+X, ghw+D, ghw+X .,@ghw+X .X, ghw+D,@ghw+D,> +515:7, lq zklfk zh kdyh xvhgwkh uhvxow lq erwk gluhfwlrqv wr uhfrpelqh wkh surgxfw ri wkh ghwhuplqdqw lqwr wkh ghwhuplqdqw ri wkh surgxfw ghw+XX.,@ghw+4,@41Wkxv/wkh ghwhuplqdqw ri dq rshudwru lv dovr lqyduldqw zlwk uhvshfw wr d fkdqjh ri uhsuhvhqwdwlrq1 Ilqdoo|/ |rx pd| uhfdoo wkdw d qhfhvvdu| dqgvx!flhqw frqglwlrq iru wkh lqyhuvh ri d pdwul{ wr h{lvw lv wkdw lwv ghwhuplqdqw qrw ydqlvk1 Wklv frqglwlrq h{whqgv wr dq| rshudwru uhsuhvhqwhge| vxfk d pdwul{/ l1h1 Li ghw+D,@3/wkhqD lv qrq0lqyhuwleoh ru vlqjxodu1 Li ghw+D, 9@3> wkhq wkhuh h{lvwv dq lqyhuvh rshudwru DÃ4 vxfk wkdw DDÃ4 @ DÃ4D @ 41

515147 Hljhqydoxhv dqg Hljhqyhfwruv D qrq}hur yhfwru m"l lv vdlgwr eh dq hljhqyhfwru ri wkh rshudwru D zlwk hljhqydoxh d +zkhuh jhqhudoo|/ d 5 F,lilwvdwlv hvwkhhljhqydoxh htxdwlrq

Dm"l @ dm"l= +515:8,

Wkh vhw ri hljhqydoxhv idj iru zklfk vroxwlrqv wr wklv htxdwlrq h{lvw lv uhihuuhgwr dv wkh vshfwuxp ri wkh rshudwru D/ dqgzh zulwh vshfwuxp +D,@idj1Wkhvshfwuxpri dq duelwudu| rshudwru fdq eh uhdo/ frpsoh{/ frqwlqxrxv/ glvfuhwh/ pl{hg/ erxqghg/ ru xqerxqghg1 Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 9:

D qxpehu ri ihdwxuhv iroorz iurp wkh hljhqydoxh htxdwlrq1 Iru h{dpsoh/ lw iroorzv wkdw li m"l lv dq hljhqyhfwru ri D wkhq vr lv dq| pxowlsoh m"l ri m"l1 Wklv iroorzv iurp wkh idfw wkdw D lv d olqhdu rshudwru vr wkdw

D+m"l,@Dm"l @ dm"l @ d+m"l,= +515:9,

Wkxv/ rqo| wkh gluhfwlrq lq Kloehuw vsdfh ri d jlyhq hljhqyhfwru lv xqltxh1 E| wdnlqj wkh dgmrlqw ri wkh hljhqydoxh htxdwlrq

Dm"l @ dm"l> +515::, zh vhh wkdw li m"l lv dq hljhqnhw zlwk hljhqydoxh d wkhq

k"mD. @ k"mdÆ> +515:;, zklfk lpsolhv wkdw k"m lv dq hljhqeud ri D. zlwk hljhqydoxh dÆ1 Dq hljhqydoxh d ri dq rshudwru lv ghjhqhudwh li wkhuh h{lvwv pruh wkdq rqh olqhduo| lqghshqghqw hljhqyhfwru fruuhvsrqglqj wr wkdw hljhqydoxh1 Wkh ghjhqhudf| qd ri dq hljhqydoxh d lv htxdo wr wkh pd{lpxp qxpehu ri olqhduo| lqghshqghqw hljhqyhfwruv dvvrfl0 dwhgzlwk lw1 Zh dovr vd| wkdw dq hljhqydoxh zlwk ghjhqhudf| qd lv qd0irogghjhqhudwh1 Dq hljhqydoxh zlwk rqo| rqh olqhduo| lqghshqghqw hljhqyhfwru lv vdlg wr eh qrqghjhqhudwh1 Lw vkrxogeh fohdu/ wkdw dq| vhw ri olqhduo| lqghshqghqw yhfwruv irup d edvlv iru d vxevsdfh ri wkh ruljlqdo vsdfh +qdpho|/ wkh vxevsdfh iruphgiurp doo srvvleoh olqhdu frpelqdwlrqv ri wkrvh yhfwruv,1 Lw iroorzv/ dovr/ wkdw dq| vhw ri q olqhduo| lqghshqghqw yhfwruv m"ll/hdfk ri zklfk lv dq hljhqyhfwru ri dq rshudwru D dvvrfldwhgzlwk wkh vdph q0irogghjhqhudwh hljhqydoxh d/ irupv d edvlv iru dq hqwluh vxevsdfh Vd/ hdfk yhfwru ri zklfk lv dq hljhqyhfwru ri D zlwk wkdw vdph hljhqydoxh1 Djdlq/ wklv iroorzv iurp wkh dvvxpswlrq wkdw zh duh ghdolqj zlwk olqhdu rshudwruv/ vlqfh li

Dm"ll @ dm"ll> +515:<, iru l @4> 5> q/ wkhq wkh dfwlrq ri D rq dq| olqhdu frpelqdwlrq [q m#l @ lm"ll +515;3, l@4 ri wkhvh yhfwruv lv [q [q [q Dm#l @ lDm"ll @ ldm"ll @ d lm"ll @ dm#l= +515;4, l@4 l@4 l@4

Wkxv/ dq| yhfwru m#l lq Vd lv dovr dq hljhqyhfwru zlwk wkh vdph hljhqydoxh1 Zlwklq wklv vxevsdfh zh pd| irup olqhdu frpelqdwlrqv ri wkh olqhdu lqghshqghqw yhfwruv m"ll xvlqj wkh Judp0Vfkplgw surfhvv wr frqvwuxfw dq ruwkrqrupdo edvlv ri hljhqyhfwruv iru wklv hljhqvxevsdfh1 Iurp wkh gh qlwlrqv jlyhq deryh lw lv uhdglo| yhul hg wkdw wkh edvlv vwdwhv ri wkh srvlwlrq uhsuhvhqwdwlrq duh hljhqvwdwhv ri wkh srvlwlrq rshudwru/ dqgduh dfwxdoo| odehohge| wkh dvvrfldwhghljhqydoxhv = Wkh srvlwlrq vwdwhv duh dovr hljhqvwdwhv ri wkh srwhqwldo hqhuj| rshudwru1 Vlploduo|/ wkh edvlv yhfwruv ri wkh zdyhyhfwru uhsuhvhqwdwlrq duh hljhqvwdwhv ri wkh zdyhyhfwru rshudwru dqgduh vlploduo| odehohge| wkhlu dvvrfldwhghljhqydoxhv1 Wkh| duh dovr hljhqvwdwhv ri wkh prphqwxp rshudwru dqgri wkh nlqhwlf hqhuj| rshudwru1 9; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

515148 Hljhqsurshuwlhv ri Khuplwldq Rshudwruv Wkh vhfrqgsrvwxodwh +zklfk zh lqwurgxfhgvrph wlph djr qrz, dvvrfldwhv revhuydeohv D zlwk Khuplwldq rshudwruv D1 Wkh uhdvrq iru wklv odujho| vwhpv iurp wkh vshfldo surshuwlhv dvvrfldwhgzlwk vxfk rshudwruv1 Wkhvh surshuwlhv lqfoxghwkh iroorzlqj= Uhdolw| ri wkh Hljhqydoxhv 0LiD lv d Khuplwldq rshudwru/ vr wkdw D @ D./dqgm"l lv rqh ri lwv hljhqyhfwruv/ vr wkdw Dm"l @ dm"l/wkhq

k"mDm"l @ dk"m"l= +515;5,

Qrz iru d Khuplwldq rshudwru wkh dgmrlqw ri wklv htxdwlrq lv

k"mDm"l @ dÆk"m"l= +515;6,

Frpsdulqj wkh odvw wzr uhodwlrqv zh ghgxfh wkdw

dÆ @ d= +515;7,

Wkxv/ zh frqfoxgh wkdw wkh hljhqydoxhv ri Khuplwldq rshudwruv duh uhdo1 Iruphuo| zh vkrzhgwkdw h{shfwdwlrq ydoxhv ri Khuplwldq rshudwruv duh uhdo1 Wkh wzr vwdwhphqwv duh reylrxvo| forvho| uhodwhg1 Wkh uhtxluhphqw wkdw phdvxudeoh txdqwlwlhv eh uhdo ydoxhg prwlydwhv wkh lghqwl fdwlrq ri revhuydeohv zlwk Khuplwldq rshudwruv1 Qrwh wkdw/ ehfdxvh ri wkh uhdolw| ri wkh hljhqydoxhv/ wkh dgmrlqw ri wkh hljhqydoxh htxdwlrq iru d Khuplwldq rshudwru kdv wkh vlpsoh irup k"mD @ k"md= +515;8,

Ruwkrjrqdolw| ri Hljhqyhfwruv 0 Lw lv vwudljkwiruzdugwr vkrz wkdw hljhqyhfwruv ri d Khu0 plwldq rshudwru fruuhvsrqglqj wr glhuhqw hljhqydoxhv duh qhfhvvdulo| ruwkrjrqdo1 Ohw m"l dqg m"3l/ eh hljhqyhfwruv ri d Khuplwldq rshudwru D fruuhvsrqglqj wr hljhqydoxhv d dqg d3/ uhvshfwlyho|1 Wkxv/ zh fdq zulwh

Dm"l @ dm"l> +515;9, dqg Dm"3l @ d3m"3l= +515;:, Wdnlqj wkh lqqhu surgxfw ri wkh uvw ri wkhvh zlwk m"3l zh qgwkdw

k"3mDm"l @ dk"3m"l= +515;;,

Exw wkh dgmrlqw ri wkh vhfrqg h{suhvvlrq vkrzv wkdw

k"3mD @ k"3md3> +515;<, zkhuh zh kdyh xvhgwkh uhdolw| ri wkh hljhqydoxhv ghgxfhgderyh1 Wdnlqj wkh lqqhu surgxfw ri wklv htxdwlrq rq wkh uljkw zlwk m"l/zh qgwkdw

k"3mDm"l @ d3k"3m"l= +515<3,

Htxdwlqj wkhvh wzr h{suhvvlrqv iru wkh pdwul{ hohphqw k"3mDm"l zh qgwkdw

dk"3m"l @ d3k"3m"l> +515<4, ru +d d3,k"3m"l @3= +515<5, Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv 9<

Wkhuh duh wzr zd|v lq zklfk wklv surgxfw fdq ydqlvk1 Hlwkhu d @ d3/lqzklfkfdvhzh kdyhq*w irxqgrxw dq|wklqj/ ru d 9@ d3/ lq zklfk fdvh zh ghgxfh

k"3m"l @3 +515<6, vkrzlqj wkdw wkh hljhqvwdwhv ri d Khuplwldq rshudwru fruuhvsrqglqj wr wzr glhuhqw hljhq0 ydoxhv duh dozd|v ruwkrjrqdo1 515149 Rewdlqlqj Hljhqyhfwruv dqg Hljhqydoxhv Wkh phwkrgv wkdw rqh wdnhv wr dfwxdoo| vroyh wkh hljhqydoxh sureohp ghshqg wr vrph h{whqw xsrq wkh vl}h ri wkh vsdfh wkdw rqh lv zrunlqj lq1 Iru qlwh glphqvlrqdo vsdfhv wkh sureohp lv uhgxfhg wr d vwdqgdug rqh ri olqhdu dojheud1 Zh vhhn qrqwulyldo vroxwlrqv wr wkh hljhqydoxh htxdwlrq Dm"l @ dm"l> +515<7, zklfk phdqv/ jhqhudoo| vshdnlqj/ wkh wzr vwhs surfhvv ri qglqj wkh hljhqydoxhv d iru zklfk dffhswdeoh vroxwlrqv h{lvw/ dqgwkhq qglqj wkh dvvrfldwhghljhqyhfwruv1 Qrwh uvw wkdw wkh qxoo yhfwru lv dozd|v d vroxwlrq wr wkh hljhqydoxh htxdwlrq iru dq| ydoxh ri d/ exw lv ri qr lqwhuhvw vlqfh lw grhv qrw uhsuhvhqw d wuxh g|qdplfdo vwdwh ri wkh v|vwhp/ dqg vr lv uhihuuhgwr dv d wulyldo vroxwlrq1 Wkxv/ zh vhhn qrqwulyldo hljhqyhfwruv ri qrq}hur ohqjwk1 Wr wklv hqgzh uhzulwh wkh hljhqydoxh htxdwlrq lq wkh irup

+D d,m"l @3 +515<8, zkhuh d @ d4 lv d vfdodu pxowlsolfdwlyh rshudwru wkdw pxowlsolhv doo yhfwruv e| wkh vfdodu d1Zhqrzuhzulwhwklvdvhfrqgwlph/lqwkhirup

Em"l @3 +515<9, zkhuh wkh rshudwru E @ E+d,@D d +515<:, lv dq rshudwru ixqfwlrq ri wkh sdudphwhu d1Qrz/li wkhlqyhuvhriE h{lvwhg/ wkh vroxwlrq wr wklv htxdwlrq frxogeh rewdlqhge| pxowlso|lqj erwk vlghve| EÃ4=

m"l @ EÃ4Em"l @ EÃ43@3 +515<;,

Wklv vkrzv wkdw li EÃ4 h{lvwv/ wkh rqo| vroxwlrq lv wkh wulyldo rqh1 Lw iroorzv wkdw iru wkrvh ydoxhv ri d iru zklfk qrqwulyldo vroxwlrqv h{lvw/ wkh rshudwru E+d, fdqqrw srvvhvv dq lqyhuvh1 Vlqfh wkh lqyhuvh ri E zloo h{lvw xqohvv wkh ghwhuplqdqw ri E ydqlvkhv/ zh frqfoxgh wkdw wkh hljhqydoxhv ri D duh wkrvh ydoxhv zklfk pdnh ghw+E,@31Wkxv/zh lghqwli| wkh hljhqydoxhv ri D zlwk wkh urrwv ri wkh fkdudfwhulvwlf ru vhfxodu htxdwlrq

ghw+D d,@3= +515<<,

Vlqfh wkh ghwhuplqdqw ri dq rshudwru lv uhsuhvhqwdwlrq lqghshqghqw/ vr zloo eh wkh hljhq0 ydoxhv1 Wkxv/ wkh vshfwuxp ri dq rshudwru lv uhsuhvhqwdwlrq lqghshqghqw/ l1h1/ lqyduldqw xqghu xqlwdu| wudqvirupdwlrqv/ dqg dq| uhsuhvhqwdwlrq fdq eh xvhg wr hydoxdwh wkh gh0 whuplqdqw1 Lq d yhfwru vsdfh ri glphqvlrq Q/ wkh rshudwru D zloo eh uhsuhvhqwhge| dq Q Q pdwul{ D/ dqgwkh fkdudfwhulvwlf htxdwlrq zloo lqyroyh d sro|qrpldo ri ghjuhh Q D44 dD45 D4Q D54 D55 d D5Q ghw+Dd4,@ 1 1 1 @ f .f d. .f dQ @3 +51633, 1 1 11 1 3 4 Q 1 1 1 1 DQ 4 DQ 5 DQQ d :3 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv lq wkh yduldeoh d> uhihuuhgwr dv wkh fkdudfwhulvwlf sro|qrpldo1 Wkh ixqgdphqwdo wkhr0 uhp ri dojheud jxdudqwhhv wkdw dq| vxfk sro|qrpldo zloo kdyh/ lqfoxglqj srvvleoh pxowlsolf0 lwlhv/ h{dfwo| Q urrwv +zklfk zloo jhqhudoo| eh frpsoh{ qxpehuv,1 Wkxv/ wkh fkdudfwhulvwlf sro|qrpldo fdq jhqhudoo| eh idfwruhglqwr wkh irup

q4 q5 qp ghw+D d4,@+d d4, +d d5, +d dp, @3 +51634, zkhuh d4>d5> dp uhsuhvhqw wkh p Q glvwlqfw urrwv/ zklfk duh dvvxphg wr kdyh pxo0 wlsolflwlhv q4>q5> qp/vxfkwkdw [p ql @ Q= +51635, l@4

Rqfh wkh hljhqydoxhv duh irxqge| vroylqj wkh fkdudfwhulvwlf htxdwlrq/ wkh hljhqyhfwruv duh jhqhudoo| irxqgrqh dw d wlph e| vxevwlwxwlqj wkh hljhqydoxhv edfn lqwr wkh hljhqydoxh htxdwlrq dqgvroylqj wkh olqhdu v|vwhp ri htxdwlrqv rewdlqhge| h{suhvvlqj wkh hljhqyhfwruv dqgrshudwruv lq dq| frqyhqlhqw uhsuhvhqwdwlrq1 Wkxv/ li wkh vwdwhv imllj uhsuhvhqw dq RQE iru wkh vsdfh/ wkh hljhqydoxh htxdwlrq +D d4,m"l fdq eh zulwwhq [ +Dlm dlm,"m @3 l @4> 5> Q> +51636, m lq whupv ri wkh pdwul{ hohphqwv ri wkh rshudwru D lq wklv uhsuhvhqwdwlrq dqgwkh frh!flhqwv

"l ri wkh h{sdqvlrq iru wkh hljhqyhfwru [ m"l @ "lmll +51637, l ri lqwhuhvw1 Qrwh wkdw zkloh wklv surfhgxuh jlyhv d vhw ri Q olqhdu htxdwlrq lq Q xqnqrzqv zklfk zh fdq vroyh xvlqj vwdqgdug whfkqltxhv ri olqhdu dojheud/ wkh Q htxdwlrqv duh qrw olqhduo| lqghshqghqw1 Wklv iroorzv iurp wkh idfw wkdw wkh pdwul{ ri frh!flhqwv kdv d ydqlvklqj ghwhuplqdqw1 Lqghhg/ zh fkrvh wkh hljhqydoxh d wr pdnh wklv ghwhuplqdqw ydqlvk1 Wkh srlqw khuh lv wkdw wkh vroxwlrq wr wkhvh olqhdu htxdwlrqv lv qrw xqltxh1 Jhqhudoo|/ zh zloo qgwkdw wkh htxdwlrqv doorz xv wr ghwhuplqh doo ri wkh frh!flhqwv lq whupv ri rqh xqghwhuplqhg frh!flhqw/ zklfk fdq wkhq wdnh dq| ydoxh1 Exw wklv lv htxlydohqw wr wkh revhuydwlrq wkdw d vfdodu pxowlsoh ri dq| hljhqyhfwru lv dovr dq hljhqyhfwru1 Lqghhg/ lw lv wklv iuhhgrp wkdw doorzv xv wr surgxfh d vhw ri vtxduh0qrupdol}hg hljhqyhfwruv1 Zh qrz dvvhuw vrph edvlf surshuwlhv ri olqhdu dojheud zlwkrxw surri1 Iluvw zh gh qh wkh frqfhsw ri d qrupdo rshudwru1 Dq rshudwru lv vdlgwr eh qrupdo li lw frppxwhv zlwk lwv dgmrlqw1 Wkxv/ li ^D> D.`@3 +51638, wkhq D lv qrupdo1 Lw iroorzv wkdw Khuplwldq rshudwruv dqgxqlwdu| rshudwruv duh qrupdo1 Lw fdq eh vkrzq/ wkdw lq d qlwh glphqvlrqdo yhfwru vsdfh wkh qxpehu ri olqhduo| lqghshqghqw hljhqyhfwruv ri d qrupdo rshudwru lv dozd|v htxdo wr wkh glphqvlrq ri wkh vsdfh/ dqg wkdw wkh pxowlsolflw| ri dq| urrw lq wkh fkdudfwhulvwlf htxdwlrq lv htxdo wr wkh ghjhqhudf| ri wkdw hljhqydoxh1 Wkxv/ wkh hljhqyhfwruv ri d qrupdo rshudwru irup d edvlv iru wkh vsdfh1 Frpelqlqj wklv zlwk wkh ruwkrjrqdolw| ri wkh hljhqyhfwruv ri Khuplwldq rshudwruv/ zh ghgxfh dq lpsruwdqw idfw= Lq dq| qlwh glphqvlrqdo vsdfh d Khuplwldq rshudwru srvvhvvhv dq ruwkrqrupdo edvlv ri hljhqyhfwruv1 Lq dq ruwkrqrupdo edvlv frpsrvhgri lwv rzq hljhqvwdwhv/ dq rshudwru lv fohduo| gldjrqdo/ vlqfh kdlmDmdml @ dllm > khqfh [ D @ mdlldlkdlm= +51639, l Srvwxodwh LL= Revhuydeohv ri Txdqwxp Phfkdqlfdo V|vwhpv :4

Lw iroorzv wkdw wkh wudfh ri dq| qrupdo rshudwru lv mxvw wkh vxp [ Wu +D,@ dl +5163:, l ri lwv hljhqydoxhv +hdfk hljhqydoxh ehlqj lqfoxghg lq wkh vxp ql wlphv,/ zkloh wkh ghwhu0 plqdqw ri dq| qrupdo rshudwru lv mxvw wkh surgxfw \ ghw +D,@ dl +5163;, l +lqfoxglqj ghjhqhudflhv, ri lwv hljhqydoxhv1 H{whqghg H{dpsoh= Frqvlghu d wkuhh glphqvlrqdo yhfwru vsdfh1 Ohw wkh vwdwhv im4l> m5l> m6lj irup dq ruwkrqrupdo edvlv iru wklv vsdfh1 Ohw D eh dq rshudwru rq wkh vsdfh kdylqjs wkh iroorzlqj irxu qrq}hur pdwul{ hohphqwv k4mDm5l @ k5mDm4l @ k5mDm6l @ k6mDm5l @ 5/zlwk doo rwkhu pdwul{ hohphqwv ehlqj htxdo wr }hur1 Lq wklv uhsuhvhqwdwlrq/ dq duelwudu| vwdwh S6 m#l @ l@4 #lmll fdq eh uhsuhvhqwhge| d froxpq yhfwru zlwk frpsrqhqwv #l dqgwkh rshudwru D fdq eh uhsuhvhqwhge| wkh pdwul{ 3 s 4 s3 53s C D D @ 53s 5 = 3 53

Zkhq D dfwv rq m#l lw surgxfhv d vwdwh m!l zklfk fdq eh uhsuhvhqwhge| wkh froxpq yhfwru 3 4 3 s 4 3 4 3 s 4 3 53 5# !4 s s #4 s s5 C ! D @ C D C # D @ C D 1 5 53s 5 5 5#s4 . 5#6 !6 3 53 #6 5#5 Wr qgwkh hljhqydoxhv ri D zh qgwkh urrwv ri wkh fkdudfwhulvwlf htxdwlrq s sd 53s ghw+D d,@ 5 sd 5 @3= 3 5 d

Hydoxdwlqj wkh ghwhuplqdqw |lhogv wkh frqglwlrq d+d5 7, @ 3/ zklfk kdv wkuhh glvwlqfw urrwv1 d4 @ 5/ d5 @3>d6 @.51 Wkxv/ zh fdq zulwh vshfwuxp+D,@i5> 3> 5j1Ohw xv ghqrwh wkh hljhqyhfwruv ri D e| imd4l> md5l> md6lj= Wr qg md4l zh vxevwlwxwh lqwr wkh hljhqydoxh htxdwlrq +D d4,md4l @+D .5,md4l @3> zklfk lq wklv uhsuhvhqwdwlrq fdq eh zulwwhq 3 s 4 3 4 3 s 4 s5 53s d44 s 5d44 . 5d45s C D C D C D 55s 5 d45 @3@ 5d44s.5d45 . 5d46 = 3 55 d46 5d45 .5d46

Iurps wkh uvws dqgodvw frpsrqhqwv ri wkh froxpq yhfwru rq wkh uljkw zh qgwkdw d45 @ 5d44 @ 5d461 ^Qrwh wkdw zh fdq dozd|v ljqruh wkh htxdwlrq iurp rqh ri wkh frpsrqhqwv= lw zloo mxvw jlyh xv uhgxqgdqws lqirupdwlrq ehfdxvh ghw+D d4,@3`1 Wkxv wkh froxpq yhfwru zlwk frpsrqhqwv i4> 5> 4j lv dq hljhqyhfwru zlwk hljhqydoxh d4 @ 5/ exw lv qrw qrupdol}hg1 Wr qrupdol}h/ zh glylgh e| wkh ohqjwk wr rewdlq wkh froxpq yhfwru 3 4 4@s5 C D md4l$ 4@ 5 4@5 :5 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

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Dmdl @ dmdl= +51655,

Wkh dfwlrq ri D rq wkh vwdwh Emdl lv wkhq hdvlo| ghwhuplqhg/ l1h1/

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d6 :9 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

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S+d,@k#mSdm#l> +51669, zkhuh [qd Sd @ md> lkd> m +5166:, @4

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S +d,@4 S+d3,@3 iru d3 9@ d +51683,

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Srvwxodwh LLL+e, +Froodsvh ri wkh Vwdwh Yhfwru, 0 Lpphgldwho| diwhu d phd0 vxuhphqw ri dq revhuydeoh D shuiruphgrqdv|vwhplqwkhvwdwhm#l wkdw |lhogv wkh ydoxh d/ wkh vwdwh ri wkh v|vwhp lv wkh qrupdol}hgsurmhfwlrq ri m#l rqwr wkh Srvwxodwh LLL= Wkh Phdvxuhphqw ri Txdqwxp Phfkdqlfdo V|vwhpv ;4

hljhqvxevsdfh Vd dvvrfldwhgzlwk wkh hljhqydoxh phdvxuhg/l1h1/ lw lv mxvw wkdw sduw ri m#l o|lqj zlwklq wkh vxevsdfh1 Zh vfkhpdwlfdoo| lqglfdwh wklv fkdqjh lq wkh vwdwh yhfwru dv iroorzv= D S m#l m#l $ d = +51684, mmS m#lmm d d

Wkxv/ vlpso| vshdnlqj/ qdwxuh wkurzv dzd| wkrvh sduwv ri wkh vwdwh yhfwru zklfk duh qrw frqvlvwhqw zlwk wkh dfwxdo ydoxh rewdlqhg1 Lw vkrxog eh qrwhg wkdw wklv rqo| lqglfdwhv rqh srvvleoh eudqfk ri wkh hyroxwlrq ri wkh v|vwhp gxulqj wkh frxuvh ri wkh phdvxuhphqw surfhvv/ qdpho|/ wkdw rqh zklfk rffxuv zkhq wkh sduwlfxodu hljhqydoxh d lv rewdlqhg1 Dv zh kdyh vhhq/ lw lv qrw srvvleoh wr suhglfw zklfk ri wkhvh eudqfkhv zloo dfwxdoo| eh iroorzhg e| dq| vlqjoh txdqwxp phfkdqlfdo v|vwhp1 Wkxv/ wklv fkdqjh lq wkh vwdwh yhfwru gxulqj phdvxuhphqw lv lqkhuhqwo| qrq0ghwhuplqlvwlf= Wkh ylhzsrlqw xvxdoo| wdnhq lv wkdw wkh froodsvh ri wkh vwdwh yhfwru wr wkh dvvrfldwhg hljhqvxevsdfh rffxuv dv wkh uhvxow ri dq xqvshfl hglqwhudfwlrq ri wkh v|vwhp zlwk wkh +fodvvlfdo, phdvxuhphqw ghylfh xvhg wr phdvxuh wkh revhuydeoh1 Zh zloo dyrlg wkh pdq| lqwhuhvwlqj txhvwlrqv dqgdssduhqw sdudgr{hv zklfk dulvh lq wkh dwwhpsw wr vlpxowdqh0 rxvo| wuhdw wkh v|vwhp0soxv0phdvxulqj0ghylfh dv d forvhg txdqwxp v|vwhp/ dv zhoo dv dq| glvfxvvlrq ri zkhwkhu wkh froodsvh ri wkh vwdwh yhfwru lv d sk|vlfdo surfhvv ru d vwdwlvwlfdo rqh1 Zh zloo krzhyhu pdnh d ihz sudfwlfdo revhuydwlrqv1

Iluvw/ lw lv hdv| wr vkrz wkdw wkh qrupdol}dwlrq idfwru mmSdm#lmm zklfk dsshduv lq wkh ghqrplqdwru ri wkh uhgxfhgvwdwh yhfwru lv vlpso| uhodwhgwr wkh suredelolw| wkdw v|vwhp zrxogkdyh wdnhq wkdw sduwlfxodu urxwh gxulqj wkh phdvxuhphqw surfhvv/ l1h1/ s s s mmSdm#lmm @ k#mSdSdm#l @ k#mSdm#l @ S +d,> +51685,

5 zkhuhzhkdyhxvhgwkhfkdudfwhulvwlfihdwxuhSd @ Sd dvvrfldwhgzlwk surmhfwlrq rshu0 dwruv dorqj zlwk wkh idfw wkdw d surmhfwlrq rshudwru lv Khuplwldq1 Wkxv/ zh fdq h{suhvv wkh uhgxfwlrq surfhvv lq wkh irup

D S m#l m#l $ s d = +51686, d k#mSdm#l

Ylhzhgiurp wkh rwkhu gluhfwlrq/ wklv vkrzv wkdw wkh suredelolw| S+d, lv mxvw wkh vtxduhg qrup ri wkdw sduw ri m#l o|lqj zlwklq wkh hljhqvxevsdfh1

Vhfrqgo|/ zh vkrxog qrwh wkdw li d lv qrqghjhqhudwh/ wkhq Sd @ mdlkdm/dqg S m#l mdlkdm#l d @ @ hl!mdl> +51687, mmSdm#lmm mkdm#lm zkhuh doo wkdw lv ohiw ri wkh ruljlqdo vwdwh lv wkh skdvh lqirupdwlrq frqwdlqhglq wkh idfwru kdm#l hl! @ +51688, mkdm#lm zklfk lv d frpsoh{ qxpehu ri xqlw prgxoxv hl! @4/ vlqfh zh kdyh glylghg wkh frpsoh{ qxpehu kdm#l @ mkdm#lm hl! e| lwv pdjqlwxgh1 Lw vkrxog eh qrwhg wkdw doo h{shfwdwlrq ;5 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv ydoxhv ri wkh vwdwhv hl!mdl dqg mdl duh lghqwlfdo/ dqg khqfh wkh| duh qrw sk|vlfdoo| glvwlqfw vwdwhv1 Wkxv/ lq wkh qrqghjhqhudwh fdvh wkh v|vwhp lv mxvw ohiw lq wkh hljhqvwdwh dvvrfldwhg zlwk wkdw hljhqydoxh1 H{whqvlrq wr Frqwlqxrxv Vshfwud 0LiwkhvshfwuxpriD lv frqwlqxrxv/ d phdvxuhphqw ri D zlwk lq qlwh suhflvlrq zrxogohdyh wkh v|vwhp lq dq dvvrfldwhghljhqyhfwru/ vxfk wkdw

D m#l m#l $ = +51689, mk#m m#lm zklfk lv vlplodu wr wkh suhylrxv uhvxow h{fhsw wkh surmhfwru kdv ehhq uhsodfhge| wkh surmhfwru ghqvlw|1 Lq wkh fdvh ri d qrqghjhqhudwh vshfwuxp wklv zrxog lpso| wkh uhgxfwlrq

D mlkm#l m#l $ @ hl!ml= +5168:, mkm#lm

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3 3 5L 5@L Reylrxvo| zh fdq gurs wkh sduw sursruwlrqdo wr }hur dqg zulwh ] 3 3 3 S @ g m lk m= +5168<,

3 5L Exw wklv lv d surmhfwlrq rshudwru/ ehlqj dq lqwhjudo ryhu d surmhfwru ghqvlw|1 Wkxv/ phd0 vxulqj d folfn/ whoov xv wkdw wkh v|vwhp kdv d ydoxh lq wkh dvvrfldwhglqwhuydo/ dqg Srvwxodwh LLL= Wkh Phdvxuhphqw ri Txdqwxp Phfkdqlfdo V|vwhpv ;6 surmhfwv wkh vwdwh ri wkh v|vwhp rqwr d qrupdol}deoh vwdwh yhfwru kdylqj lwv frpsrqhqwv lq wklv lqwhuydo/ l1h1/ S @4 S m#l m#l $ = +51693, mmSm#lmm 5 L Wkxv/ dq| uhdo phdvxuhphqw +l1h1/ rqh pdgh zlwk qlwh suhflvlrq, ohdyhv wkh v|vwhp lq d qrupdol}hgvwdwh yhfwru1 Rq wkh rwkhu kdqg/li wkh suhflvlrq > dowkrxjk qlwh/ lv yhu| vpdoo frpsduhgwr wkh yduldwlrq ri #+, ryhu wklv lqwhuydo/ zh fdq vwloo surshuo| vshdn ri wkh suredelolw| ghqvlw| +,@m#+,m5 ri rewdlqlqj d sduwlfxodu hljhqydoxh/ zlwkrxw kdylqj wr h{solflwo| phqwlrq wkh dfwxdo suhflvlrq ri wkh ghylfh1 Wkxv/ lq zkdw iroorzv zh zloo uhihu wr phdvxuhphqwv ri srvlwlrq/ prphqwxp/ nlqhwlf hqhuj|/ hwf1/ zlwkrxw zruu|lqj wrr pxfk derxw wkh dfwxdo phdvxulqj dssdudwxv1 Zh qrz frqvlghu vrph frqvhtxhqfhv ri wkh wzr sduwv ri wkh phdvxuhphqw srvwxodwh1 51614 Vxp ri Suredelolwlhv Iluvw/ zh qrwh wkdw wkh suredelolw| ri rewdlqlqj vrph ydoxh lv jxdudqwhhgwkurxjk wkh pdwkhpdwlfdo vwuxfwxuh ri wkh wkhru|/ vlqfh # $ [ [ [qd [ [qd S +d,@ k#md> lkd> m#l @ k#m md> lkd> m m#l @ k#m#l @4> +51694, d d @4 d @4 zkhuh zh kdyh lghqwl hg wkh ghfrpsrvlwlrq ri xqlw| lpsolflw lq wkh idfw wkdw wkh rshudwru D lv dq revhuydeoh1 Lw lv wklv frpsohwhqhvv surshuw| ri frxuvh/ zklfk prwlydwhgrxu pdwkhpdwlfdo gh qlwlrq ri wkh whup revhuydeoh lq wkh uvw sodfh1 51615 Phdq Ydoxhv Jlyhq wkdw wkh suhglfwlrqv ri wkh srvwxodwhv duh vwdwlvwlfdo lq qdwxuh/ wkhuh duh d qxpehu ri vwdwlvwlfv ri wkh phdvxuhphqw surfhvv zklfk duh xvhixo wr hydoxdwh1 Frqvlghu/ h1j1/ dq hqvhpeoh ri Q lghqwlfdoo| suhsduhg v|vwhpv +zlwk Q 4,> doo lq wkh vdph txdqwxp phfkdqlfdo vwdwh m#l1 Li wkh vdph revhuydeoh D lv phdvxuhgrq hdfk phpehu ri wklv hq0 vhpeoh/ wkh uhvxow zloo eh d froohfwlrq ri ydoxhv idj +doo ehlqj hljhqydoxhv ri D,1 Hdfk ydoxh d zloo rffxu zlwk d iuhtxhqf| id @ QS+d,> uhodwhgwr wkh dvvrfldwhgsuredelolw| dsshdulqj lq wkh srvwxodwhv1 Zh fdq wkhq frpsxwh wkh phdq ydoxh kDl dv wkh dulwkphwlf dyhudjh dvvrfldwhgzlwk wklv vhulhv ri phdvxuhphqwv1 Wklv phdq ydoxh/ lw vkrxogeh hpskdvl}hg/ pd| qrw dfwxdoo| frlqflgh zlwk dq| ri wkh phdvxuhphqwv dfwxdoo| shuiruphg/ exw lw grhv jlyhv vrph lqirupdwlrq derxw wkh xqghuo|lqj glvwulexwlrq1 Wr frpsxwh wkh phdq ydoxh zh surfhhgdv iroorzv=

4 [ [ [ [qd kDl @ i d @ dS +d,@ d k#md> lkd> m#l= +51695, Q d d d d @4 Qrwh/ krzhyhu/ wkdw li zh pryh wkh hljhqydoxh d ehwzhhq wkh hohphqwv ri wkh surmhfwruv/ zh fdq lghqwli| wkh uhvxowlqj h{suhvvlrq dv dq h{sdqvlrq iru wkh rshudwru D lq wkh edvlv frqvlvwlqj ri lwv rzq hljhqvwdwhv/ l1h1/ # $ [ [qd [ [qd k#md> ldkd> m#l @ k#m md> l d kd> m m#l @ k#mDm#l= +51696, d @4 d @4

Wkxv zh rewdlq wkh yhu| vlpsoh uhvxow wkdw wkh phdq ydoxh kDl @ k#mDm#l +51697, ;7 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv lv htxlydohqw wr wkh h{shfwdwlrq ydoxh ri wkh rshudwru zlwk uhvshfw wr wkh jlyhq vwdwh1 Wkh xvhixo wklqj derxw wklv uhvxow lv wkdw zh fdq frpsxwh lw lq dq| edvlv wkdw zh olnh= zh gr qrw dfwxdoo| kdyh wr nqrz wkh hljhqvwdwhv dqg hljhqydoxhv ri dq rshudwru wr frpsxwh lwv h{shfwdwlrq ydoxh1 Iru h{dpsoh/ iru dq| RQE ri vwdwhv imqlj iru zklfk zh nqrz wkh h{sdqvlrq frh!flhqwv #q @ kqm#l iru wkh vwdwh m#l dqgwkh pdwul{ hohphqwv Dqq3 @ kqmDmq3l zh fdq frpsxwh wkh h{suhvvlrq [ Æ kDl @ k#mDm#l @ #qDqq3 #q3 > +51698, q>q3 rewdlqhg e| lqvhuwlqj ghfrpsrvlwlrqv li xqlw| rq hdfk vlgh ri wkh rshudwru D1 Iru uhihuhqfh/ zh olvw ehorz vrph ri wkh pruh frpprq h{shfwdwlrq ydoxhv dvvrfldwhgzlwk d vlqjoh sduwlfoh1 Lq wkh srvlwlrq uhsuhvhqwdwlrq/ zh kdyh ] ] ] kUl @ k#mUm#l @ g6u k#mulukum#l @ g6u#Æ+u,u#+u,@ g6uu+u,> +51699, ] ] ] kY l @ k#mY m#l @ g6u k#mulY +u,kum#l @ g6u#Æ+u,Y +u,#+u,@ g6uY+u,+u,>

] ] +5169:, kN l @ k#mN m#l @ g6u k#mulkumN m#l @ l g6u#Æ+u,u #+u,> +5169;, ] ] kS l @ k#mSm#l @ g6u k#mulkumS m#l @ l| g6u#Æ+u,u #+u,> +5169<, ] ] S 5 4 |5 5 kW l @ k#m m#l @ g6u k#mulkumS 5m#l @ g6u#Æ+u,u #+u,> +516:3, 5p 5p 5p zkloh lq wkh zdyhyhfwru uhsuhvhqwdwlrq ] ] 6 6 Æ kUl @ g n k#mnlknmUm#l @ l g n# +n,un #+n,> +516:4, ] ] ] ] kY l @ g6n g6n3 k#mnlknmY mn 3lkn 3m#l @ g6n g6n3 #Æ+n,Y +n n 3,#+n3, +516:5, ] ] ] kSl @ g6n k#mnl|nknm#l @ g6n#Æ+n,|n#+n,@ g6n |n+n,> +516:6, ] ] ] |5 |5 |5 kW l @ g6n k#mnln5knm#l @ g6n#Æ+n,n5#+n,@ g6nn5+n,= +516:7, 5p 5p 5p

51616 Vwdwlvwlfdo Xqfhuwdlqw| Wkh phdq ydoxh ri dq revhuydeoh whoov xv urxjko| zkhuh zh fdq h{shfw wkh pdmrulw| ri wkh ydoxhv rewdlqhglq dq hqvhpeoh ri phdvxuhphqwv wr eh foxvwhuhg1 Lw whoov xv qrwklqj derxw krzeljriduhjlrqdurxqgwkhphdqydoxhzhpljkwh{shfwwrrewdlqwkhvhydoxhv1Lwlv xvhixo wr kdyh d phdvxuh ri wklv vwdwlvwlfdo vsuhdg/ zklfk uh hfwv wkh lqwulqvlf txdqwxp phfkdqlfdo xqfhuwdlqw| dvvrfldwhgzlwk wkh phdvxuhphqw surfhvv1 Rqh xvhixo phdvxuh ri wklv glvshuvlrq lv wkh urrw0phdq0vtxduh xqfhuwdlqw| ri d vhulhv ri phdvxuhphqwv/ gh qhg wkurxjk wkh uhodwlrq s D @ k+D kDl,5l> +516:8, zklfk zh fdq zulwh lq dq htxlydohqw dqgvrphwlphv pruh xvhixo irup e| h{sdqglqj wkh txdgudwlf k+D kDl,5l @ kD5 5DkDl . kDl5l= +516:9, Srvwxodwh LLL= Wkh Phdvxuhphqw ri Txdqwxp Phfkdqlfdo V|vwhpv ;8

Vlqfh kDl lv d frqvwdqw/ wklv uhgxfhv wr

k+D kDl,5l @ kD5l5kDl5 . kDl5 @ kD5lkDl5> +516::, vr wkdw s D @ kD5lkDl5= +516:;,

Qrwh wkdw li wkh v|vwhp lv lq d qrupdol}hghljhqvwdwh mdl ri D wkhq

kDql @ kdmDqmdl @ dqkdmdl @ dq> +516:<, lq zklfk fdvh s D @ d5 d5 @3> +516;3, vr wkdw wkhuh lv qr xqfhuwdlqw| zkhq wkh v|vwhp lv lq dq hljhqvwdwh ri wkh rshudwru/ dv zh kdyh uhshdwhgo| dvvhuwhg1 Wkxv/ lq d vwdwlvwlfdo vhqvh/ wkh xqfhuwdlqw| lq dq revhuydeoh dvvrfldwhgzlwk d jlyhq txdqwxp vwdwh m#l lv d phdvxuh ri wkh h{whqw wr zklfk wkh vwdwh fdq eh vdlgwr dfwxdoo| srvvhvv d ydoxh ri wkh dvvrfldwhgrevhuydeoh1 Lw lv lqwhuhvwlqj/ lq wklv frqwh{w/ wr dvn derxw wkh vlpxowdqhrxv srvvlelolw| ri uhgxflqj wkh xqfhuwdlqw| dvvrfldwhg zlwk wzr glhuhqw revhuydeohv1 Zh nqrz/ iru h{dpsoh/ wkdw li E lv dq revhuydeoh zklfk frppxwhv zlwk D/ wkhq lw lv srvvleoh wr qgvlpxowdqhrxv hljhqvwdwhv md> el ri erwk revhuydeohv1 Iru vxfk d vwdwh wkh xqfhuwdlqw| lq erwk revhuydeohv zloo ydqlvk1 Li/ krzhyhu/ E lv dq rshudwru zklfk grhv qrw frppxwh zlwk D/ wkhq wkhuh qhhgeh qr vl0 pxowdqhrxv hljhqvwdwhv +dowkrxjk d ihz pd| h{lvw/ wkhuh zloo qrw jhqhudoo| h{lvw d frpprq edvlv ri hljhqvwdwhv,1 Xqghu wkhvh flufxpvwdqfhv/ lw lv qrw dozd|v srvvleoh wr uhgxfh wkh vlpxowdqhrxv vwdwlvwlfdo xqfhuwdlqw| dvvrfldwhgzlwk wkh phdvxuhphqw ri erwk revhuydeohv rq d jlyhq txdqwxp vwdwh1 Wkhuh wxuqv rxw wr eh d suhflvh vwdwhphqw zklfk fdq eh pdgh derxw wkh vr0fdoohg xqfhuwdlqw| surgxfw s s DE @ k+D kDl,5l k+E kEl,5l +516;4, dvvrfldwhgzlwk dq| jlyhq vwdwh ri wkh v|vwhp1 Wklv surgxfw lv fohduo| d phdvxuh ri wkh mrlqw xqfhuwdlqw| dvvrfldwhgzlwk wkhvh wzr revhuydeohv1 Lq sduwlfxodu/ zh suryh ehorz wkh zhoo0nqrzq xqfhuwdlqw| sulqflsoh/ wkh vwdwhphqw ri zklfk iroorzv=

51617 Wkh Xqfhuwdlqw| Sulqflsoh Iru dq| txdqwxp vwdwh m#l/ wkh mrlqw xqfhuwdlqw| lq wkh ydoxhv ri wzr revhuydeohv D dqg E dv phdvxuhgwkurxjk wkh xqfhuwdlqw| surgxfw DE lv erxqghg iurp ehorz wkurxjk wkh uhodwlrq 4 DE mk^D> E`lm> +516;5, 5 lw ehlqj xqghuvwrrg wkdw doo h{shfwdwlrq ydoxhv duh wr eh wdnhq zlwk uhvshfw wr wkh vdph txdqwxp vwdwh m#l1 Wr suryh wkh xqfhuwdlqw| sulqflsoh/ zh qhhg uvw wr suryh d vlpsoh exw xvhixo wkhruhp nqrzq dv Vfkzdu}*v lqhtxdolw|/ zklfk vwdwhv wkdw li m{l dqg m|l duh dq| wzr vwdwhv lq wkh vsdfh/ wkhq k{m{lk|m|lk{m|lk|m{l ru/ htxlydohqwo|/ mm{mm5mm|mm5 mk{m|lm5> ru pruh vlpso|/ mm{mmmm|mm mk{m|lm= +516;6, ;9 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Wr suryh wklv uhodwlrq/ vhw ml @ m{l. m|l/ iru vrph frqvwdqw wr eh fkrvhq odwhu1 Lw wkhq iroorzv wkdw wkh ohqjwk ri wklv yhfwru lv srvlwlyh vr wkdw

kml @^k{m . Æk|m`^m{l . m|l` 3= +516;7,

H{sdqglqj/ zh qg wkdw

k{m{l . k|m{l . Æk|m{l . Æk|m|l3= +516;8,

Wklv vwdwhphqw lv wuxh iru duelwudu| / vr zh fdq vhw

k{m|l k|m{l @ Æ @ > +516;9, k|m|l k|m|l zklfk ohdgv wr wkh uhvxow k{m|lk|m{l k|m{lk{m|l k{m|l k|m{l k{m{l . k|m|l3= +516;:, k|m|l k|m|l k|m|l k|m|l

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k#mDa5m#lk#mEa5m#lk#mDaEam#lk#mEaDam#l @ mk#mDaEam#lm5= +516<9,

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51618 Suhsdudwlrq ri d Vwdwh Xvlqj d FVFR Lw iroorzv iurp wkh odvw glvfxvvlrq dqgwkh vwdwhphqw ri wkh vhfrqgsduw ri wkh phdvxuhphqw srvwxodwh/ wkdw li d v|vwhp lv lqlwldoo| lq dq xqnqrzq vwdwh/ lw vkrxogeh srvvleoh wr froodsvh lw lqwr d nqrzq vwdwh wkurxjk d vhulhv ri qhduo| lqvwdqwdqhrxv phdvxuhphqwv shuiruphgxvlqj wkh rshudwruv lq d frpsohwh vhw ri frppxwlqj revhuydeohv iD> E> Fj1 Vxssrvh/ iru h{dpsoh/ wkh v|vwhp lv lq wkh vwdwh m#l/ zklfk zh pd| ru pd| qrw nqrz1 Wklv vwdwh fdq eh h{sdqghglqwkh edvlv yhfwruv dvvrfldwhgzlwk wkh vlpxowdqhrxv hljhqvwdwhv ri wkh rshudwruv D> E> dqg F lq wkh xvxdo zd|/ l1h1/ [ 3 3 3 m#l @ #d3>e3>f3 md >e>fl= +51734, d3>e3>f3

Li zh qrz phdvxuh wkhvh wkuhh frpsdwleoh revhuydeohv lq d yhu| vkruw lqwhuydo ri wlph/ zh zloo vhh wkh vxevhtxhqw uhgxfwlrq ri wkh vwdwh yhfwru rqwr rqh ri wkhvh edvlv yhfwruv/ dv uhsuhvhqwhge| wkh gldjudp

D [ E [ F # m#l $ # md> e3>f3l $ # md> e> f3l $ d>e>f md> e> fl= d>e3>f3 d>e>f3 m# m d e3>f3 e f3 f d>e>f +51735, ;; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Wklv frqfhswxdo delolw| wr suhsduh d v|vwhp lq d zhoo fkdudfwhul}hgvwdwh lv h{wuhpho| lpsruwdqw iru wkh wkhru|/ iru lw doorzv xv wr whvw wkh vwdwlvwlfdo suhglfwlrqv ri wkh wkhru|/ zklfk uho| rq wkh lghd ri dq hqvhpeoh ri vlploduo| suhsduhg v|vwhpv xsrq zklfk wr shuirup d vxevhtxhqw phdvxuhphqw1 Wkxv/ diwhu shuiruplqj vxfk d frpsohwh vhulhv ri phdvxuh0 phqwv rq dq hqvhpeoh ri duelwudu| lqlwldo vwdwh yhfwruv/ zh fdq h{wudfw wkrvh zklfk hqgxs lq d sduwlfxodu txdqwxp vwdwh md> e> fl wr surgxfh d vxehqvhpeoh ri v|vwhpv xsrq zklfk wr shuirup ixuwkhu h{shulphqwv1 Srvwxodwh LY = Hyroxwlrq ;<

Zh duh qrz uhdg| wr qlvk xs wkh vhw ri srvwxodwhv wkdw zh kdyh ehhq ghyhorslqj wr ghvfuleh wkh irupdolvp ri txdqwxp phfkdqlfv1 Wkh odvw srvwxodwh ghvfulehv wkh zd| d txdqwxp phfkdqlfdo v|vwhp ehkdyhv lq ehwzhhq wkh wlphv gxulqj zklfk phdvxuhphqwv duh ehlqj pdgh1 Dv zh kdyh vhhq/ gxulqj d phdvxuhphqw surfhvv/ d txdqwxp phfkdqlfdo v|vwhp/ lq frqwdfw zlwk d fodvvlfdo phdvxulqj ghylfh/ hyroyhv qrq0ghwhuplqlvwlfdoo| dv wkh vwdwh yhfwru froodsvhv lqwr rqh ri wkh hljhqvxevsdfhv ri wkh sduwlfxodu revhuydeoh ehlqj phdvxuhg1 Lq ehwzhhq wkhvh phdvxuhphqw hyhqwv/ hyroxwlrq lv jryhuqhg e| wkh irxuwk srvwxodwh1

517 Srvwxodwh LY = Hyroxwlrq Ehwzhhq phdvxuhphqwv wkh vwdwh yhfwru m#+w,l ri d txdqwxp v|vwhp hyroyhv ghwhuplqlvwlfdoo| dffruglqj wr Vfkuùglqjhu*v htxdwlrq ri prwlrq

g l| m#+w,l @ Km#+w,l> +51736, gw lq zklfk wkh Kdplowrqldq rshudwru K lv wkh revhuydeoh dvvrfldwhgzlwk wkh wrwdo hqhuj| ri wkh v|vwhp dw wlph w1

Lq sudfwlfh/ wr xvh Vfkuùglqjhu*v htxdwlrq zh surmhfw lw rqwr wkh edvlv yhfwruv ri dq dssur0 suldwh uhsuhvhqwdwlrq1 Wkxv/ li wkh yhfwruv imqlj irup dq RQE iru wkh vsdfh ri lqwhuhvw/ wkhq zh fdq zulwh g kqml| m#+w,l @ kqmKm#+w,l= +51737, gw Volglqj wkh eud kqm sdvw wkh wlph ghulydwlyh dqg lqvhuwlqj d frpsohwh vhw ri vwdwhv wr wkh uljkw ri wkh Kdplowrqldq/ zh rewdlq

g [ l| kqm#+w,l @ kqmKmq3lkq3m#+w,l +51738, gw q3 lq zklfk zh uhfrjql}h frh!flhqwv iru wkh h{sdqvlrq [ [ m#+w,l @ mqlkqm#+w,l @ #q+w, mql= +51739, q q Wkxv/ lq wklv uhsuhvhqwdwlrq/ Vfkuùglqjhu*v htxdwlrq wdnhv wkh irup [ g#q l| @ K 3 # +w, +5173:, gw qq q3 p ri d vhw ri uvw0rughu frxsohg glhuhqwldo htxdwlrqv iru wkh wlph0ghshqghqw h{sdqvlrq frh!flhqwv iru wkh vwdwh m#+w,l lq wklv edvlv1 Lq d frqwlqxrxv uhsuhvhqwdwlrq ml> wkh vwdwh ri wkh v|vwhp lv uhsuhvhqwhge| wkh zdyh0 ixqfwlrq #+, dqgwkh Kdplowrqldq ehfrphv dq lqwhjur0glhuhqwldo rshudwru dfwlqj rq wklv ixqfwlrq1 Lq wkh prvw jhqhudo fdvh/ wkh pdwul{ hohphqwv ri K ehwzhhq wkh frqwlqxrxv edvlv vwdwhv ml duh gh qhg e| vrph nhuqho K+> 3,@kmKm3l= Surmhfwlrq ri wkh Vfkuùglqjhu htxdwlrq rqwr wkh edvlv vwdwhv ri wkdw uhsuhvhqwdwlrq wkhq ohdgv wr wkh h{suhvvlrq

g kml| m#+w,l @ kmKm#+w,l= +5173;, gw <3 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Dv ehiruh/ wdnlqj wkh ghulydwlyh zlwk uhvshfw wr wlph lv d olqhdu rshudwlrq/ vr zh fdq zulwh

g g C#+> w, kml| m#+w,l @ l| km#+w,l @ l| +5173<, gw gw Cw zkhuh wkh h{dfw glhuhqwldo iru wkh yhfwru m#+w,l +zklfk rqo| ghshqgv sdudphwulfdoo| rq wlph/ qrw rq ru dq|wklqj hovh, wxuqv lqwr d sduwldo ghulydwlyh zkhq lw dfwv rq wkh ixqfwlrq #+> w,@km#+w,l> zklfk lv irupdoo| d ixqfwlrq ri wzr yduldeohv= Pdnlqj wklv vxevwlwxwlrq dqglqvhuwlqj d frpsohwh vhw ri vwdwhv ehwzhhq K dqg #+w, rq wkh uljkw zh rewdlq dq lqwhjudo htxdwlrq ] C# l| @ g3 K+> 3,#+3>w,= +51743, Cw iru wkh zdyh ixqfwlrq #+> w,1 Xqghu fhuwdlq vshfldo vlwxdwlrqv +zklfk rffxu udwkhu riwhq, wkh pdwul{ hohphqwv ri K zloo lqyroyh ghulydwlyhv ri ghowd ixqfwlrqv/ dqg wkh lqwhjudo htxdwlrq zloo uhgxfh wr d glhuhqwldo htxdwlrq/ dv zh kdyh vhhq rffxu zlwk wkh hqhuj| hljhqydoxh htxdwlrq lq wkh srvlwlrq uhsuhvhqwdwlrq1

Wkxv/ iru d vlqjoh sduwlfoh lq 6G prylqj xqghu wkh lq xhqfh ri d srwhqwldo Y +u> w,> wkh Kdplowrqldq lv vlpso| wkh vxp ri wkh nlqhwlf dqgsrwhqwldo hqhuj| rshudwruv

S 5 K @ . Y +U> w,= +51744, 5p Xqghu wkhvh flufxpvwdqfhv/ wkh Vfkuùglqjhu htxdwlrq fdq eh zulwwhq lq wkh srvlwlrq uhs0 uhvhqwdwlrq e| surmhfwlqj lw rqwr wkh edvlv yhfwruv ri wkdw uhsuhvhqwdwlrq/ l1h1/

g S 5 l| m#+w,l @ Km#+w,l @ m#+w,l . Y m#+w,l> +51745, gw 5p

g 4 l| kum#+w,l @ kumS 5m#+w,l . kumY m#+w,l> +51746, gw 5p zklfk zh uhfrjql}h dv

C |5 l| #+u> w,@ u5#+u> w,.Y +u> w,#+w,> +51747, Cw 5p zklfk lv Vfkuùglqjhu*v htxdwlrq lq lwv ruljlqdo irup1 Dowhuqdwlyho|/ zh fdq fkrrvh wr zrun lq wkh prphqwxp ru zdyhyhfwru uhsuhvhqwdwlrq=

g 4 l| knm#+w,l @ knmS 5m#+w,l . knmY m#+w,l> +51748, gw 5p zklfk zh fdq zulwh lq wkh irup ] C |5n5 l| #+n> w,@ #+n> w,. g6n3 Y +n n 3>w,#+n 3>w,= +51749, Cw 5p

Xqohvv Y +n n 3>w, kdv vshfldo surshuwlhv zklfk hqdeoh d vlpsol fdwlrq/ Vfkuùglqjhu*v htxdwlrq iru d vlqjoh sduwlfoh lq wkh zdyhyhfwru uhsuhvhqwdwlrq lv dq lqwhjurglhuhqwldo htxdwlrq1 Srvwxodwh LY = Hyroxwlrq <4

51714 Frqvwuxfwlrq ri wkh Kdplowrqldq dqg Rwkhu Revhuydeohv Lq sulqflsoh/ wkh hyroxwlrq ri d txdqwxp phfkdqlfdo v|vwhp lv uhgxfhg wr wkh vroxwlrq ri d vhw ri frxsohg uvw rughuglhuhqwldo htxdwlrqv rqfh wkh Kdplowrqldq lv nqrzq1 Dv lq fodvvlfdo phfkdqlfv/ wkhuhiruh/ wkh uvw vwhs lq vroylqj wkh g|qdplfdo sureohp lv wkh frqvwuxfwlrq ri d vxlwdeoh Kdplowrqldq1 Lq pdq| fdvhv d Kdplowrqldq rshudwru fdq eh rewdlqhgiurp wkh Kdplowrqldq ixqfwlrq ri dq dvvrfldwhgfodvvlfdo v|vwhp1 Iru wkh sdu0 dgljpdwlf fdvh ri d vlqjoh vslqohvv sduwlfoh prylqj lq wkuhh glphqvlrqv/ iru h{dpsoh/ zh pryh iurp wkh fodvvlfdo ghvfulswlrq/ zklfk lv edvhg xsrq wkh g|qdplfdo yduldeohv u dqg s/ wr wkh txdqwxp phfkdqlfdo rqh/ e| uhsodflqj wkh g|qdplfdo yduldeohv e| wkh Khuplwldq rshudwruv U dqg S / zkrvh frpsrqhqwv reh| wkh fdqrqlfdo frppxwdwlrq uhodwlrqv

^[l>[m`@^Sl>Sm`@3 +5174:,

^[l>Sm`@l|lm= +5174;,

Lq d vlplodu idvklrq/ lw vhhpv uhdvrqdeoh wr dvvrfldwh zlwk dq| fodvvlfdo revhuydeoh D+u>s> w, dq rshudwru D+U> S>w , rewdlqhge| uhsodflqj wkh g|qdplfdo yduldeohv dsshdulqj lq wkh ixqfwlrq zlwk wkh fruuhvsrqglqj rshudwruv1 Xqiruwxqdwho|/ zkloh wklv surfhgxuh zrunv d jrrg ghdo ri wkh wlph/ wkhuh duh flufxpvwdqfhv zkhuh lw fdq jlyh dpeljxrxv uhvxowv dqg2ru vxhu wkh gudzedfn wkdw wkh rshudwru zklfk lv surgxfhg lv qrw Khuplwldq1 Wr looxvwudwh wkh edvlf gl!fxowlhv wkdw dulvh/ frqvlghu wkh iroorzlqj fodvvlfdo revhuydeoh

{s{ @ s{{ +5174<, zklfk zh fdq zulwh lq hlwkhu ri wkhvh wzr zd|v/ vlqfh fodvvlfdo yduldeohv dozd|v frppxwh zlwk rqh dqrwkhu1 Txdqwxp phfkdqlfdoo|/ krzhyhu/ wkh rshudwruv rewdlqhge| uhsodflqj wkh g|qdplfdo yduldeohv zlwk dvvrfldwhg rshudwruv

[S{ 9@ S{[ +51753, duh qrw htxdo ehfdxvh wkhvh rshudwruv gr qrw frppxwh1 D prphqwv uh hfwlrq zloo uhyhdo wkdw wkhuh duh dfwxdoo| dq lq qlwh qxpehu ri fodvvlfdoo| htxlydohqws h{suhvvlrqv wkdw hdfk q q q jhqhudwh d glhuhqw txdqwxp phfkdqlfdo rshudwru +frqvlghu/ h1j1/ { s @ {s{,Zklfk ri wkhvh rshudwruv vkrxogeh xvhgwr uhsuhvhqw wkh fodvvlfdo revhuydeohB Prvw ri wkhp duh sdwhqwo| xqxvdeoh ehfdxvh wkh| duh qrw Khuplwldq1 Lq wkh vlpsoh h{dpsoh deryh/ h1j1/ qhlwkhu [S{ ru S{[ lv d Khuplwldq rshudwru/

. . . ^[S{` @ S{ [ @ S{[= +51754, dqgvr fdqqrw uhsuhvhqw dq revhuydeoh1 Wr uhvroyh wklv sureohp zh xvh wkh lghd ri Khuplwldq v|pphwul}dwlrq ri dq rshudwru1 Uhfdoo wkdw dq| rshudwru D fdq eh zulwwhq lq wkh irup

D @ DK . DD> +51755,

4 . 4 . zkhuh wkh rshudwru DK @ 5 +D.D , lv Khuplwldq dqg DD @ 5 +D.D , lv dqwl0Khuplwldq1 Wkxv/ dq duelwudu| rshudwru fdq eh ghfrpsrvhglq d xqltxh zd| lqwr Khuplwldq dqgdqwl0 Khuplwldq sduwv lq d zd| dqdorjrxv wr wkh pdqqhu lq zklfk dq duelwudu| frpsoh{ qxpehu fdq eh ghfrpsrvhglqwr uhdo dqglpdjlqdu| sduwv1 Zh uhihu wr DK /wkhuhiruh/dvwkh Khuplwldq sduw ri D/ dqgvwlsxodwh wkdw wkh revhuydeoh fruuhvsrqglqj wr dq| fodvvlfdo <5 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv txdqwlw| eh dvvrfldwhgzlwk wkh Khuplwldq sduw ri wkh rshudwru rewdlqhgwkurxjk wkh uhsodfhphqw glvfxvvhg deryh/ l1h1/

D+U> S>w ,.D.+U> S>w , D+u>s> w, $ = +51756, 5 Lq wkh sduwlfxodu h{dpsoh frqvlghuhg/ wklv lpsolhv wkdw 4 {s @ s { $ +[S . S [, +51757, { { 5 { { vr wkdw erwk ri wkh rshudwruv lq txhvwlrq duh wuhdwhg rq dq htxdo irrwlqj1 Dq dgglwlrqdo h{dpsoh zrxogeh 4 u s $ +U S . S U,= +51758, 5

Xqiruwxqdwho|/ wklv surfhgxuh rqo| uhgxfhv wkh xqghuo|lqj sureohp/ lw grhv qrw holplqdwh lw1 Wklv fdq eh vhhq e| frqvlghulqj wkh iroorzlqj htxlydohqw fodvvlfdo h{suhvvlrqv

5 5 { s{ @ s{{ @ {s{{> +51759, wkh uvw wzr ri zklfk duh erwk dvvrfldwhgxqghu wkh deryh suhvfulswlrq zlwk wkh rshudwru 4 +[5S . S [5,> +5175:, 5 { { zkloh wkh wklugri zklfk lv dvvrfldwhgzlwk wkh rshudwru

[S{[= +5175;,

Wklv gl!fxow| lv/ lq d vhqvh/ rqo| pdwkhpdwlfdo/ dqg suhvxpdeo| dulvhv ehfdxvh zh duh dwwhpswlqj wr jr iurp d ohvv frpsohwh ghvfulswlrq +l1h1/ fodvvlfdo phfkdqlfv, wr d pruh frp0 sohwh ghvfulswlrq +l1h1/ txdqwxp phfkdqlfv, ri wkh sk|vlfdo xqlyhuvh1 Lw lv qrw xquhdvrqdeoh wr h{shfw wkdw lq dq| grpdlq zkhuh fodvvlfdo ehkdylru lv revhuyhg/ wkh glhuhqfhv ehwzhhq wkh suhglfwlrqv dvvrfldwhg zlwk dq| ri wkhvh rshudwruv zloo ehfrph xqlpsruwdqw1 Lq wkh txdqwxp grpdlq/ krzhyhu/ wklv xqghuvfruhv wkh idfw wkdw wkh dvvrfldwlrq ri d phdvxulqj ghylfh zlwk dq revhuydeoh fdq vrphwlphv lqyroyh vxewoh glvwlqfwlrqv1 Lw vkrxogdovr eh srlqwhgrxw wkdw wklv sureohp uhdoo| rqo| dulvhv lq rshudwruv lqyroylqj surgxfwv ri qrq0frppxwlqj revhuydeohv1 Lq wkh prvw frpprq vlwxdwlrq/ qdpho| wkdw ri d sduwlfohv prylqj lq uhvsrqvh wr d fodvvlfdo srwhqwldo ixqfwlrq Y +u> w,/ wkh sureohp qhyhu dulvhv ehfdxvh vxfk surgxfwv grq*w dsshdu lq wkh Kdplowrqldq1 51715 Vrph Ihdwxuhv ri Txdqwxp Phfkdqlfdo Hyroxwlrq Ghwhuplqlvp 0 Qrwh wkdw wkh glhuhqwldo htxdwlrq jryhuqlqj wkh hyroxwlrq ri wkh vwdwh yhfwru lv uvw rughu lq wlph1 Wklv phdqv wkdw wkh vroxwlrq ghshqgv rqo| rq wkh lqlwldo vwdwh ri wkh v|vwhp/ dqgqrw/ h1j1/ rq lwv lqlwldo udwh ri fkdqjh1 Wkxv/ dq| lqlwldo vwdwh m#+w3,l ri wkh v|vwhp dw wlph w3 zloo hyroyh lqwr d vlqjoh xqltxh yhfwru m#+w,l dw wlph wAw31 Zh qrwh wkdw wklv lpsolflwo| gh qhv d pdsslqj ri wkh vsdfh rqwr lwvhoi/ dqg wkxv lpsolhv wkh h{lvwhqfh ri dq rshudwru X/ ru d idplo| ri rshudwruv X+w> w3,/ wkdw pds dq duelwudu| vwdwh dw wlph w3 rqwr wkh vwdwh lqwr zklfk lw hyroyh lq wlph w1Wklvhyroxwlrq rshudwru lv gh qhg wkurxjk wkh uhodwlrq

m#+w,l @ X+w> w3,m#+w3,l= +5175<, Srvwxodwh LY = Hyroxwlrq <6

Olqhdulw| 0 Wkh olqhdulw| ri wkh htxdwlrqv ri prwlrq lpso| d vxshusrvlwlrq sulqflsoh iru wkh vroxwlrqv ri wkh Vfkuùglqjhu htxdwlrq1 Wkdw lv li m#4+w,l dqg m#5+w,l duh wzr srvvleoh vroxwlrqv wr wkh Vfkuùglqjhu htxdwlrq +zklfk kdyh/ h1j1/ hyroyhg iurp wzr glhuhqw lqlwldo vwdwh yhfwruv m#4+w3,l dqg m#5+w3,l,/ wkhq wkh wlph0ghshqghqw yhfwru

m#+w,l @ m#4+w,l . m#5+w,l +51763, lv dovr d vroxwlrq wr wkh Vfkuùglqjhu htxdwlrq iru dq| frpsoh{ frqvwdqwv dqg / vlqfh

g g g l| m#+w,l @ +l| m# +w,l,.+l| m# +w,l, +51764, gw gw 4 gw 5 @ Km#4+w,l . Km#4+w,l @ K+m#4+w,l . m#5+w,l,> +51765, vr wkdw g l| m#+w,l @ Km#+w,l= +51766, gw

Wklv lpsolhv/ dv d frqvhtxhqfh/ wkdw li zh qgrxw krz wkh edvlv yhfwruv ri dq| RQE hyroyh xqghu wkh Vfkuùglqjhu htxdwlrq/ zh fdq ghwhuplqh wkh hyroxwlrq ri dq| rwkhu yhfwru lq wkh v|vwhp1 Lw dovr lpsolhv wkdw wkh hyroxwlrq rshudwru X+w> w3, lqwurgxfhg deryh lv d olqhdu rshudwru1 Frqvhuydwlrq ri wkh Qrup 0 Lw lv dovr uhodwlyho| hdv| wr vkrz wkdw txdqwxp phfkdqlfdo hyroxwlrq suhvhuyhv wkh qrup ri wkh vwdwh yhfwru/ d frqglwlrq zklfk lv reylrxvo| lpsruwdqw li zh zlvk wkh wrwdo vxp ri suredelolwlhv wr eh frqvhuyhg1 Wkxv/ zh frqvlghu wkh udwh ri fkdqjh ri wkh +vtxduhg, ohqjwk ri d yhfwru m#+w,l hyroylqj xqghu wkh Vfkuùglqjhu htxdwlrq g g g k#+w,m#+w,l @ k#m m#l . k#m m#l> +51767, gw gw gw zkhuhzhkdyhvlpso|xvhgwkhfkdlquxohrqwkhuljkwkdqgvlgh1IurpwkhVfkuùglqjhu htxdwlrq lwvhoi zh ghgxfh wkdw g l m#l @ Km#l +51768, gw | wkh dgmrlqw ri zklfk jlyhv g l k#m @ k#mK= +51769, gw | Vxevwlwxwlqj wkhvh lq deryh zh qgwkdw

g l l k#m#l @ k#mKm#l . k#mKm#l @3> +5176:, gw | | vr wkdw k#+w,m#+w,l @ k#+w3,m#+w3,l lv frqvwdqw1 Wklv lpsolhv wkdw wkh hyroxwlrq rshudwru X @ X+w> w3, wkdw zh gh qhg hduolhu lv xqlwdu|/vlqfh

. k#+w,m#+w,l @ k#+w3,mX Xm#+w3,l @ k#+w3,m#+w3,l= +5176;,

. Vlqfh wklv pxvw eh wuxh iru duelwudu| vwdwhv m#+w3,l/ lw iroorzv wkdw X X @ 41Wklvihdwxuh lv dovr ghvfulehg e| vd|lqj wkdw wkh Vfkuùglqjhu htxdwlrq ohdgv wr d xqlwdu| hyroxwlrq1 <7 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

51716 Hyroxwlrq ri Phdq Ydoxhv Ohw xv qrz frqvlghu krz wkh phdq ydoxh dvvrfldwhg zlwk dq duelwudu| revhuydeoh D+w, hyroyhv lq wlph1 Lq jhqhudo/ rshudwruv fdq kdyh dq lqwulqvlf wlph ghshqghqfh1 Dv dq h{dpsoh frqvlghu wkh srwhqwldo dvvrfldwhg zlwk wkh dssolfdwlrq ri d vsdwldoo| xqlirup vlqxvrlgdoo|0ydu|lqj hohfwulf hog/ l1h1/ Yh{w+w,@hH U frv+$w,1 +5176<,

Wkh phdq ydoxh ri wklv rshudwru

kYh{w+w,l @ k#+w,mYh{w+w,m#+w,l +51773, hyroyhv lq wlph/ vlqfh erwk wkh vwdwh dqgwkh rshudwru lwvhoi lv fkdqjlqj1 Dw dq| lqvwdqw ri wlph/ wklv phdq ydoxh jlyhv d phdvxuh ri wkh lqwhudfwlrq ri wkh v|vwhp zlwk wkh h{whuqdo hog1 Lq jhqhudo/ wkh phdq ydoxh ri dq duelwudu| revhuydeoh

kD+w,l @ k#+w,mD+w,m#+w,l +51774, pd| kdyh wzr vrxufhv ri wlph ghshqghqfh= d sduw gxh wr wkh rshudwru lwvhoi/ dqg d sduw gxh wr wkh hyroxwlrq ri wkh v|vwhp1 Zh fdq/ krzhyhu/ xvh wkh fkdlq uxoh wr zulwh g g kD+w,l @ k#+w,mD+w,m#+w,l +51775, gw gw g CD g @ k#m Dm#l . k#m m#l . k#mD m#l = +51776, gw Cw gw Iurp rxu hduolhu pdqlsxodwlrqv wklv fdq eh zulwwhq g l CD l kD+w,l @ k#mKDm#l . k l k#mDKm#l= +51777, gw | Cw | CD l @ k l k#mDK KDm#l +51778, Cw | lq zklfk zh uhfrjql}h wkh frppxwdwru ri D dqg K1 Wkxv/ zh kdyh wkh htxdwlrq ri prwlrq g CD l kD+w,l @ k l k^D> K`l= +51779, gw Cw |

Wklv irup pd| eh idploldu wr wkh vwxghqw ri fodvvlfdo phfkdqlfv/ lq wkdw lw uhvhpeohv wkh htxdwlrq ri prwlrq gD CD @ iD> Kj = +5177:, gw Cw SE iru d fodvvlfdo revhuydeoh D+t> s> w,/ zkhuh wkh eudfnhwhgtxdqwlw| uhsuhvhqwv wkh Srlvvrq eudfnhw ri wkh wzr ixqfwlrqv D+t> s> w, dqg K+t> s> w,/ gh qhg wkurxjk wkh uhodwlrq

[ Ci Cj Cj Ci ii>jj @ = +5177;, Ct Cs Ct Cs l l l l l Dv d frqvhtxhqfh ri wklv htxdwlrq ri prwlrq/ zh qrwh wkdw dq| wlph0lqghshqghqw rshudwru wkdw frppxwhv zlwk wkh Kdplowrqldq kdv d phdq ydoxh wkdw uhpdlqv frqvwdqw lq wlph/ vlqfh wkh htxdwlrq ri prwlrq wkhq suhglfwv wkdw gkD+w,l@gw @31 Srvwxodwh LY = Hyroxwlrq <8

51717 Hkhuhqihvw*v Wkhruhp Dv dq lqwhuhvwlqj dssolfdwlrq ri wkh xvh ri wkh htxdwlrqv ri prwlrq iru wkh phdq ydoxh ri dq revhuydeoh/ frqvlghu wkh prwlrq ri d sduwlfoh xqghu wkh lq xhqfh ri d irufh I +u,@uY +u, ghulydeoh iurp d vfdodu srwhqwldo Y +u,1 Txdqwxp phfkdqlfdoo|/ wklv fruuhvsrqgv wr wkh xvxdo Kdplowrqldq S 5 K @ . Y +U,= +5177<, 5p

Wkh fodvvlfdo g|qdplfdo yduldeohv u+w, dqg s+w, dvvrfldwhgzlwk vxfk d Kdplowrqldq hyroyh dffruglqj wr Kdplowrq*v htxdwlrqv gu+w, s+w, @ > +51783, gw p gs+w, @ uY +u+w,, @ I +u+w,,= +51784, gw zklfk duh htxlydohqw wr Qhzwrq*v vhfrqgodz1 Ohw xv qrz frqvlghukrz wkh phdq ydoxhv kU+w,l dqg kS+w,l dvvrfldwhgzlwk wkh fruuhvsrqglqj txdqwxp phfkdqlfdo revhuydeohv fkdqjh lq wlph1 Iluvw/ zh h{dplqh wkh htxdwlrq ri prwlrq iru wkh srvlwlrq rshudwru U/ zklfk ehlqj lqghshqghqw ri wlph +CU@Cw @3, ohdgv wr wkh htxdwlrq ri prwlrq g l kU+w,l @ kÛ> K`l= +51785, gw | Wklv ohdgv xv wr hydoxdwh 4 Û> K`@ Û> S 5`.Û> Y `= +51786, 5p Vlqfh Y lv d ixqfwlrq ri U/ wkh vhfrqgfrppxwdwru ydqlvkhv1 Wkh { frpsrqhqw ri wkh uvw frppxwdwru lv

5 5 5 5 5 ^[> S `@^[> S{ `.^[> S| `.^[> S} `@^[> S{ `> +51787, zkhuh zh kdyh uhfrjql}hgwkdw wkh rqo| qrq0frppxwlqj sduw lqyroyhv srvlwlrq dqgpr0 phqwxp rshudwruv dorqj wkh vdph gluhfwlrq1 Xvlqj wkh vwdqgdug wulfn iru hydoxdwlqj wkh frppxwdwru ri d surgxfw zh qg wkdw

5 ^[> S{ `@S{^[> S{`.^[> S{`S{ @5l|S{> +51788, dqgvlploduo| iru wkh rwkhu wzr fduwhvldq frpsrqhqwv ri wkh frppxwdwru lq txhvwlrq1 Dv d yhfwru rshudwru uhodwlrq/ wkhuhiruh/ zh kdyh wkh uhvxow wkdw

^U> S 5`@5l|S> +51789, zklfk zh fdq sxw edfn lqwr wkh htxdwlrq ri prwlrq iru kU+w,l wr rewdlq g l 4 4 kU+w,l @ k5l|Sl @ kSl= +5178:, gw | 5p p

Wkxv/ zh qgwkdw - . gkUl S @ > +5178;, gw p zklfk/ eudfnhwv dvlgh/ orrnv olnh lwv fodvvlfdo frxqwhusduw1 Wkxv/ dv lq fodvvlfdo phfkdqlfv wkh phdq yhorflw| htxdov wkh phdq prphqwxp glylghg e| wkh pdvv1 <9 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

Lq d vlplodu idvklrq zh fdq frpsxwh wkh htxdwlrq ri prwlrq iru wkh phdq prphqwxp/

g l kS+w,l @ k^S>K `l= +5178<, gw | zklfk ohdgv xv wr hydoxdwh 4 ^S>K `@ ^S>S 5`.^S>Y `= +51793, 5p Qrz wkh nlqhwlf hqhuj| whup glvdsshduv/ exw wkh srwhqwldo hqhuj| whup grhv qrw/ vlqfh lw lv d ixqfwlrq ri wkh rshudwru U> zklfk grhv qrw frppxwh zlwk S1 Vlqfh zh kdyh qrw vshfl hgwkh h{dfw ixqfwlrqdo irup ri wkh srwhqwldo hqhuj| ixqfwlrq/ lw lv frqyhqlhqw wr zrunlqduhsuhvhqwdwlrqlqzklfkY lv gldjrqdo/ qdpho| wkh srvlwlrq uhsuhvhqwdwlrq1 Lq wkh srvlwlrq uhsuhvhqwdwlrq zh fdq zulwh

kum^S>Y `m#l @ l|^u Y +u,#+u, Y +u,u #+u,` +51794,

@ l|^#+u,u Y +u,.Y +u,u #+u, Y +u,u #+u,` @ l|^u Y +u,`#+u,> +51795, vr wkdw lq wkh srvlwlrq uhsuhvhqwdwlrq/ ^S>Y ` dfwv wr pxowlso| wkh zdyh ixqfwlrq e| wkh ixqfwlrq l|u Y +u,@l|I +u,/zkhuhI+u, lv wkh fodvvlfdo irufh ixqfwlrq1 Wkxv/ zh pdnh wkh lghqwl fdwlrq ^S>Y `@l|I +U,> +51796, zkhuh ] I+U,@ g6u mulI +u,kum lv d yhfwru revhuydeoh dvvrfldwhgzlwk wkh irufh rq wkh sduwlfoh/ l1h1/ wkh irufh rshudwru1 Xvlqjwklvuhvxowlqwkhhtxdwlrqriprwlrqzh qgwkdw

g l kS +w,l @ kl|I+U,l @ kI l= +51797, gw |

Wkxv/ wkh htxdwlrqv ri prwlrq iru wkh srvlwlrq dqgprphqwxp rshudwruv fdq eh zulwwhq

gkUl @ kSl> +51798, gw

g kS l @ kI l= +51799, gw zklfk orrnv olnh Qhzwrq*v htxdwlrqv/ dvlgh iurp wkh wdnlqj ri h{shfwdwlrq ydoxhv1 Wkhvh fodvvlfdoo| idploldu0orrnlqj h{suhvvlrqv duh uhihuuhgwr dv Hkuhqihvw*v htxdwlrqv ri pr0 wlrq iru wkh phdq ydoxhv1 Wkhlu lqwhusuhwdwlrq uhtxluhv d olwwoh fduh1 Lw pljkw eh h{shfwhg/ iru h{dpsoh/ wkdw wkhvh htxdwlrqv lpso| wkdw li wkh lqlwldo phdq ydoxhv zhuh htxdo wr wkrvh ri vrph k|srwkhwlfdo fodvvlfdo v|vwhp zlwk wkh vdph srwhqwldo/ vr wkdw kU+3,l @ u+3, dqg kS +3,l @ s+3,> wkhq dv erwk v|vwhpv hyroyhgwkh phdq ydoxhv kU+w,l dqg kS +w,l iru wkh txdqwxp sduwlfoh zrxogvlpso| iroorz wkh fruuhvsrqglqjfodvvlfdo wudmhfwru| u+w, dqg s+w,1 Wklv lv/ krzhyhu/ qrw jhqhudoo| wkh fdvh1 Wr vhh wklv/ zh qrwh wkdw wkh fodvvlfdo htxdwlrq ri prwlrq gs @ I @ I+u+w,,> +5179:, gw Srvwxodwh LY = Hyroxwlrq <: htxdwhv wkh ghulydwlyh ri s wr wkh irufh ixqfwlrq hydoxdwhg dw wkh sduwlfoh*v lqvwdqwdqhrxv srvlwlrq1 Li wkh phdq ydoxh kS l zhuh wr reh| wklv vdph htxdwlrq/ lw zrxogkdyh wr vdwlvi| wkh iroorzlqj uhodwlrq gkS l @ I+kUl,> +5179;, gw zklfk lqyroyhv wkh irufh ixqfwlrq I+u, hydoxdwhgdw wkh phdq ydoxh ri wkh sduwlfoh*v srvl0 wlrq1 Exw wklv lv qrw wkh htxdwlrq zh ghulyhg/ zklfk frqwdlqv kI +U,l rq wkh uljkw kdqg vlgh/ qrw I +kUl,1 Wkxv/ wkh rqo| vlwxdwlrq lq zklfk wkh txdqwxp phdq ydoxhv zloo iroorz fodvvlfdo wudmhfwrulhv lv zkhq/ iru doo lqvwdqwv ri wlph/

I +kUl,@kI+U,l= +5179<,

Lq jhqhudo/ ri frxuvh wkhvh duh qrw wkh vdph1 Lw lv vwudljkwiruzdugwr vkrz/ krzhyhu/ wkdw li wkh srwhqwldo ixqfwlrq fdq eh zulwwhq dv d sro|qrpldo ri ghjuhh wzr ru ohvv lq wkh srvlwlrq ri wkh sduwlfoh/ wkhq wklv frqglwlrq lv vdwlv hg1 Wklv phdqv wkdw wkh phdq ydoxh ri srvlwlrq dqgprphqwxp iru d sduwlfoh vxemhfw wr qr irufh/ d frqvwdqw irufh/ ru d olqhdu +h1j1/ Krrnh*v odz, irufh zloo dozd|v iroorz wkh fruuhvsrqglqj fodvvlfdo wudmhfwru|1

51718 Hyroxwlrq ri V|vwhpv zlwk Wlph Lqghshqghqw Kdplowrqldqv Zh qrz frqvlghu wkh hyroxwlrq ri txdqwxp phfkdqlfdo v|vwhpv lq zklfk wkh Kdplowrqldq rshudwru lv lqghshqghqw ri wlph/ vr wkdw CK@Cw @31 Fodvvlfdoo|/ lq vxfk d v|vwhp wkh wrwdo hqhuj| lv frqvhuyhg1 Txdqwxp phfkdqlfdoo|/ wklv lpsolhv wkdw wkh phdq ydoxh ri wkh hqhuj| zloo eh frqvhuyhgvlqfh/ xqghuwkhvh flufxpvwdqfhv g CK l kK+w,l @ k l k^K> K`l @3= +517:3, gw Cw | Lw lv lpsruwdqw wr uhdol}h/ ri frxuvh/ wkdw wkh hqhuj| ri d txdqwxp v|vwhp lv vwloo jhqhudoo| xqgh qhg xqohvv wkh v|vwhp lv dfwxdoo| lq dq hljhqvwdwh ri wkh hqhuj| rshudwru1 Wkxv/ wkh phdq ydoxh rqo| suhglfwv wkh vwdwlvwlfdo rxwfrph dvvrfldwhg zlwk pdq| phdvxuhphqwv ri hqhuj| shuiruphgrq dq hqvhpeoh ri lghqwlfdoo|0suhsduhgtxdqwxp phfkdqlfdo v|vwhpv1 Qrqhwkhohvv/ zkhq wkh Kdplowrqldq lv wlph lqghshqghqw/ wkh hyroxwlrq ri wkh v|vwhp lv prvw hdvlo| h{suhvvhglq whupv ri wkh RQE ri hqhuj| hljhqvwdwhv1 Lq wkh fdvh ri d glvfuhwh v|vwhp/ zh fdq h{suhvv wkh hqhuj| hljhqvwdwhv lq whupv ri d glvfuhwh lqgh{ q vr wkdw

Kmql @ Hqmql> +517:4, zlwk 3 kqmq l @ q>q3 = +517:5, Xqghu vxfk flufxpvwdqfhv/ wkh lqvwdqwdqhrxv g|qdplfdo vwdwh fdq eh h{sdqghg lq wkh irup [ [ m#+w,l @ mqlkqm#+w,l @ #q+w,mql= +517:6, q q

Iru d v|vwhp zlwk d frqwlqxrxv hqhuj| vshfwuxp/ wkh hqhuj| hljhqvwdwhv fdq eh lqgh{hg e| d frqwlqxrxv lqgh{/ /vxfkwkdw

Kmyl @ Hymyl> +517:7, zlwk kymy3l @ +y y3,> +517:8, <; Wkh Irupdolvp ri Txdqwxp Phfkdqlfv dqg ] ] m#+w,l @ gy mylkym#+w,l @ gy #+y> w,myl= +517:9,

Lq jhqhudo/ wkh Kdplowrqldq fdq kdyh erwk d glvfuhwh dqg d frqwlqxrxv sduw wr lwv vshfwuxp/ zlwk erwk glvfuhwh dqg frqwlqxrxvo| glvwulexwhg hljhqyhfwruv imql> mylj/zlwk

3 kqmq l @ q>q3 = +517::, kymy3l @ +y y3,> +517:;, kqmyl @3> +517:<, dqg [ ] [ ] m#+w,l @ mqlkqm#+w,l . gy mylkym#+w,l @ #q+w,mql . gy #+y> w,myl= +517;3, q q Lq zkdw iroorzv/ zh zloo/ iru vlpsolflw|/ zulwh h{suhvvlrqv lq wkh irup ri d glvfuhwh lqgh{/ exw fruuhvsrqglqj h{suhvvlrqv iru wkh jhqhudo fdvh vkrxogeh vwudljkwiruzdugwr jhqhudwh1 Li zh surmhfw wkh Vfkuùglqjhu htxdwlrq rqwr wkh edvlv yhfwruv ri wkh hqhuj| rshudwru/ zh rewdlq [ g#q+w, l| @ K 3 # +w,> +517;4, gw qq q3 q3 3 3 zkhuh/ e| dvvxpswlrq/ Kqq3 @ kqmKmq l @ Hqkqmq l @ Hqqq3 = Pdnlqj wklv vxevwlwxwlrq zh qgwkdw wkh htxdwlrqv ri prwlrq lq wkh hqhuj| uhsuhvhqwdwlrq g l| # @ H # @ |$ # > +517;5, gw q q q q q duh xqfrxsohg1 Lq wklv odvw h{suhvvlrq zh kdyh lqwurgxfhg wkh qrwdwlrq $q @ Hq@|1Wklv htxdwlrq lv uhdglo| pdqlsxodwhg lqwr wkh lqwhjudo

] # +w, ] w q g# q @ l$ gw> +517;6, # q #q+w3, q w3 zklfk jlyhv Ãl$q+wÃw3, #q+w,@#q+w3,h > +517;7, ru li zh zlvk wr pdnh w3 @3/ Ãl$qw #q+w,@#q+3,h > +517;8, vr wkdw [ [ Ãl$qw Ãl$q+wÃw3, m#+w,l @ #q+3, h mql @ #q+w3, h mql q q Wkxv/ lq wkh hqhuj| uhsuhvhqwdwlrq wkh frh!flhqwv zklfk ghwhuplqh wkh vwdwh yhfwru dw dq lqlwldo lqvwdqw ri wlph hdfk dftxluh d vlpsoh wlph0ghshqghqw skdvh idfwru wkdw ghshqgv xsrq wkh hqhuj| ri wkh dvvrfldwhgedvlv vwdwh1 Li dw w @3wkh v|vwhp lv lq d vlqjoh hqhuj| hljhqvwdwh/ vr wkdw m#+3,l @ mql> +517;9, wkhq #q3 +3, @ q>q3 = Wkh v|vwhp zloo wkhq vlpso| vwd| lq wkdw hljhqvwdwh/ exw zloo dftxluh dq rvfloodwlqj skdvh idfwru/ l1h1/

m#+w,l @ hÃl$qwmql= +517;:, Srvwxodwh LY = Hyroxwlrq <<

kE+w,l @ k#+w,mEm#+w,l @ h.l$qwkqmEmqlhÃl$qw @ kqmEmql @ kE+3,l> +517;;, zklfk lv lqghshqghqw ri wlph1 Iru wklv uhdvrq/ wkh hqhuj| hljhqvwdwhv duh uhihuuhg wr dv vwdwlrqdu| vwdwhv1 Lq jhqhudo/ ri frxuvh/ wkh v|vwhp zloo eh lq d olqhdu vxshusrvlwlrq ri hqhuj| hljhqvwdwhv ri glhuhqw hqhujlhv/ dqg revhuydeohv ri wkh v|vwhp zloo wkhuhiruh hyroyh lq wlph1 Wkxv/ dq lqlwldo vwdwh [ m#+3,l @ #q+3,mql> +517;<, q zloo hyroyh lqwr wkh vwdwh [ Ãl$qw m#+w,l @ #q+3,h mql= +517<3, q Rqfh zh nqrz wkh h{sdqvlrq lq wkh hqhuj| edvlv zh fdq ylhz wkh hyroxwlrq lq rwkhu uhsuhvhqwdwlrqv dv zhoo1 Wkxv/ h1j1/ wkh uhdo vsdfh zdyh ixqfwlrq #+u> w, iru vxfk d vwdwh zloo hyroyh lq d pdqqhu wkdw ghshqgv xsrq wkh surmhfwlrq ri wkh vwdwh yhfwru rqwr rqh ri wkh edvlv vwdwhv ri wkh srvlwlrq uhsuhvhqwdwlrq/ l1h1/ [ [ Ãl$qw Ãl$qw #+u> w,@kum#+w,l @ #q+3, h kumql @ #q+3, h !q+u,= +517<4, q q zkhuh !q+u,@kumql duh wkh hqhuj| hljhqixqfwlrq lq wkh srvlwlrq uhsuhvhqwdwlrq/ dqg ] ] 6 6 Æ #q+3, @ kqm#+3,l @ g u kqmulkum#+3,l @ g u!q+u,#+u> 3, fdq eh frpsxwhgiurp wkh lqlwldo uhdo vsdfh zdyhixqfwlrq1 Wkxv/ zh rewdlq d qdwxudo ghfrpsrvlwlrq ri #+u> w, lq wkh ruwkrqrupdo ixqfwlrqv !q+u, dvvrfldwhgzlwk wkh hqhuj| hljhqvwdwhv1 Lq vxfk d vxshusrvlwlrq vwdwh/ wkh phdq ydoxh ri dq revhuydeoh zloo dovr hyroyh/ dv zh kdyh vhhq1 Wklv hyroxwlrq/ zkrvh htxdwlrq ri prwlrq zh kdyh douhdg| h{soruhg fdq dovr eh h{suhvvhglq wkh hqhuj| uhsuhvhqwdwlrq [ [ Æ kD+w,l @ k#+w,mDm#+w,l @ #q+w,Dqq3 #q3 +w, +517<5, 3 [ [ q q Æ .l+$qÃ$q3 ,w @ #q+3,Dqq3 #q3 +3, h > +517<6, q q3 lq zklfk zh kdyh lqvhuwhgwkh h{sdqvlrq iru m#+w,l dqglwv dgmrlqw [ Æ .l$qw k#+w,m @ #q+3, h kqm= +517<7, q Iurp wklv h{suhvvlrq lw lv hdv| wr vhh wkdw wkh phdq ydoxh ri doo wlph0lqghshqghqw revhuy0 deohv zloo kdyh frpsrqhqwv zklfk rvfloodwh lq wlph dw wkh vr0fdoohg Erku iuhtxhqflhv ri wkh v|vwhp/ Hq Hq3 3 @ $ $ 3 @ > +517<8, qq q q | zklfk duh vlpso| uhodwhg wr wkh hqhuj| glhuhqfhv ehwzhhq wkh glhuhqw hljhqvwdwhv ri K1 433 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv

51719 Wkh Hyroxwlrq Rshudwru Zh qlvk xs rxu irupdo glvfxvvlrq ri wkh frqvhtxhqfhv ri wkh hyroxwlrq srvwxodwh e| glvfxvvlqj wkh h{solflw irup ri wkh xqlwdu| rshudwru X+w> w3, zklfk hyroyhv wkh vwdwh yhfwru iurp dq lqlwldo vwdwh dw wlph w3 wr wkh vwdwh ri wkh v|vwhp dw vrph wlph w odwhu1 Zh wuhdw vhsdudwho| wkh fdvh ri d wlph0ghshqghqw dqg d wlph0lqghshqghqw Kdplowrqldq1 Wlph0Lqghshqghqw Kdplowrqldq 0Iru wkh fdvh lq zklfk K lv lqghshqghqw ri wlph lw lv srvvleoh wr h{solflwo| frqvwuxfw wkh hyroxwlrq rshudwru e| frqvlghulqj wkh h{sdqvlrq ri wkh hyroylqj vwdwh yhfwru lq wkh hqhuj| edvlv1 Wkh hyroxwlrq rshudwru lv gh qhg wkurxjk wkh uhodwlrq m#+w,l @ X+w> w3,m#+w3,l> +517<9, zkloh wkh htxdwlrqv ri prwlrq lpso| wkh h{sdqvlrq [ ÃlHq+wÃw3,@| m#+w,l @ #q+w3,h mql> +517<:, q lq zklfk

#q+w3,@kqm#+w3,l= +517<;,

Pdnlqj wklv vxevwlwxwlrq lqwr wkh h{sdqvlrq dqg grlqj vrph mxglflrxv uh0duudqjlqj ri whupv/ zh qgwkdw [ [ ÃlHq+wÃw3,@| ÃlHq+wÃw3,@| m#+w,l @ kqm#+w3,lh mql @ mqlh kqm#+w3,l> +517<<, q q lq zklfk zh fdq/ e| frpsdulvrq zlwk wkh gh qlwlrq ri wkh hyroxwlrq rshudwru/ pdnh wkh lghqwl fdwlrq [ ÃlHq+wÃw3,@| X+w> w3,@ mqlh kqm= +51833, q

Wkxv/ wkh hyroxwlrq rshudwru lv gldjrqdo lq wkh hqhuj| uhsuhvhqwdwlrq/ dqg lwv gldjrqdo hohphqwv duh d vlpsoh ixqfwlrq ri wkh dvvrfldwhghljhqydoxhv ri wkh hqhuj| rshudwru1 Zh fdq wkhuhiruh zulwh wkh hyroxwlrq rshudwru dv wkh fruuhvsrqglqj ixqfwlrq ri wkh hqhuj| rshudwru lwvhoi/ l1h1/ ÃlK+wÃw3,@| X+w> w3,@h = +51834,

Lq wklv irup lv fohdu wkdw zkhq K lv lqghshqghqw ri wlph wkh hyroxwlrq rshudwru X+w> w3,@ X+w w3, rqo| ghshqgv rq wkh ohqjwk ri wkh wlph lqwhuydo ryhu zklfk wkh v|vwhp lv ehlqj hyroyhg1 Wkxv/ zh fdq zulwh X+w,@hÃlKw@| dv wkh rshudwru zklfk hyroyhv wkh v|vwhp iru d wlph w= Zh dovr qrwh wkdw lq wklv irup wkh hyroxwlrq rshudwru lv h{solflwo| xqlwdu|/ vlqfh iru dq| Khuplwldq rshudwru D/ wkh dgmrlqw ri wkh rshudwru X @ hlD lv wkh rshudwru X . @ hÃlD/dqgvr XX. @ X .X @ hl+DÃD, @ 4= +51835,

Wklv irup uhvhpeohv dqrwkhu rshudwru ri wklv w|sh/ qdpho| wkh vsdwldo wudqvodwlrq rs0 hudwru W +O ,@hÃlOÄN @ hÃlOÄS@ | > +51836, zklfk lv d pxowlsolfdwlyh rshudwru lq wkh prphqwxp uhsuhvhqwdwlrq/ exw zklfk lq wkh srvlwlrq uhsuhvhqwdwlrq kdv wkh hhfw ri vkliwlqj wkh zdyh ixqfwlrq wkurxjk d glvsodfhphqw O / l1h1/ W +O ,#+u,@#+u O ,= +51837, Srvwxodwh LY = Hyroxwlrq 434

Wkxv/ li ruljlqdoo| wkh shdn ri wkh zdyh ixqfwlrq #+u, zdv orfdwhgdw wkh ruljlq +dw u @3,> lw zloo qrz eh vkliwhgwr wkh srlqw zkhuh wkh dujxphqw ri #+u O , lv }hur/ l1h1/ wr wkh srlqw u @ O 1 Wr vhh krz wklv frphv derxw/ frqvlghu wkh fruuhvsrqglqj rqh0glphqvlrqdo yhuvlrqriwklvrshudwru

5 6 +lON{, +lON{, W +O,@hÃlON{ @ 4 .+lON ,. . . +51838, { 5$ 6$ Li zh ohw wklv dfw rq d vwdwh ghvfulehg e| wkh zdyh ixqfwlrq #+{,/ dqgxvh wkh uhvxow dssursuldwh wr wkh srvlwlrq uhsuhvhqwdwlrq wkdw lN{ @ g@g{/ zh qgwkdw g O5 g5 O6 g6 W +O,#+{,@^4 O . . `#+{, +51839, g{ 5$ g{5 6$ g{6 g# O5 g5# O6 g6#+{, @ #+{, O . . `@#+{ O,> +5183:, g{ 5$ g{5 6$ g{6 zkhuh zh vhh wkdw wkh h{sdqvlrq ri wkh h{srqhqwldo rshudwru dxwrpdwlfdoo| jhqhudwhv wkh Wd|oru vhulhv iru wkh ixqfwlrq #+{ O, h{sdqghg derxw wkh srlqw {1Lqwhupvriwkhedvlv yhfwruv ri wkh srvlwlrq dqgprphqwxp uhsuhvhqwdwlrq lw lv vwudljkwiruzdugwr vkrz wkdw W +O ,mul @ mu . O l dqgwkdw W +O ,mnl @ hÃlOÄnmnl1 Wkxv/ e| dqdorj|/ wkh hyroxwlrq rshudwru X+w, lv vrphwlphv uhihuuhgwr dv wkh wlph0 wudqvodwlrq rshudwru1 Wudqvodwlrq rshudwruv iru rwkhu revhuydeohv duh dovr hdvlo| gh qhg dv rshudwru h{srqhqwldov ri wklv w|sh1 Li wkh yduldeoh ehlqj vkliwhglv dq dqjoh/ wkhq wkh wudqvodwlrq lv dfwxdoo| d urwdwlrq1 Wkh Khuplwldq rshudwruv zklfk dsshdu lq h{srqhqwv ri wkhvh wudqvodwlrq rshudwruv duh uhihuuhgwr dv wkh jhqhudwruv ri wkh dvvrfldwhgwudqv0 odwlrq1 Wkxv/ K lv wkh jhqhudwru ri wlph wudqvodwlrqv/ zkloh S lv wkh jhqhudwru ri vsdwldo wudqvodwlrqv1 Wkurxjk vlplodu dqdo|vhv/ rqh fdq hvwdeolvk wkh idfw wkdw U lv wkh jhqhudwru ri wudqvodwlrqv lq prphqwxp/ dqgwkdw dq| frpsrqhqw ri wkh dqjxodu prphqwxp rshudwru O jhqhudwhv urwdwlrqv derxw wkdw d{lv1 Wlph0Ghshqghqw Kdplowrqldq 0 Zkhq wkh Kdplowrqldq lv qrw lqghshqghqw ri wlph qr vlpsoh forvhgirup h{suhvvlrq iru wkh hyroxwlrq rshudwru h{lvwv1 Lw lv srvvleoh/ krzhyhu/ wr ghyhors dq lqwhjudo htxdwlrq iru wkh hyroxwlrq rshudwru wkdw lv vrphwlphv txlwh xvhixo1 Wr wklv hqgzh frpelqh wkh Vfkuùglqjhu htxdwlrq g l| m#+w,l @ K+w,m#+w,l +5183;, gw zlwk wkh gh qlqj htxdwlrq m#+w,l @ X+w> w3,m#+w3,l +5183<, iru wkh hyroxwlrq rshudwru wr rewdlq g l| X+w> w ,m#+w ,l @ K+w,X+w> w ,m#+w ,l= +51843, gw 3 3 3 3

Vlqfh wklv lv ydolgiru doo lqlwldo vwdwh yhfwruv m#+w3,l> zh ghgxfh wkdw g l| X+w> w ,@K+w,X+w> w ,> +51844, gw 3 3 vkrzlqj wkdw wkh hyroxwlrq rshudwru lwvhoi reh|v dq rshudwru irup ri wkh Vfkuùglqjhu htxdwlrq1 Xqolnh wkh Vfkuùglqjhu htxdwlrq iru wkh vwdwh yhfwru/ krzhyhu/ wkh rshudwru X kdv d zhoo0gh qhg lqlwldo frqglwlrq/ qdpho|/

X+w3>w3,@4> +51845, 435 Wkh Irupdolvp ri Txdqwxp Phfkdqlfv zklfk vwhpv iurp wkh idfw wkdw olp X+w> w ,m#+w ,l @ m#+w ,l= Wklv erxqgdu| frqglwlrq w$w3 3 3 3 rq X lv vrphwlphv ghvfulehge| vd|lqj wkdw lw lv vprrwko| frqqhfwhgwr wkh xqlw rshudwru1 Zh fdq xvh wklv lqlwldo frqglwlrq wr irupdoo| lqwhjudwh wkh Vfkuùglqjhu htxdwlrq iru X dv iroorzv1 Zh zulwh +wuhdwlqj w3 vlpso| dv d sdudphwhu/ qrw d yduldeoh ri lqwhjudwlrq, g l X+w3>w ,gw3 @ K+w3,X+w3>w ,gw3> +51846, gw3 3 | 3 ru ] ] X++w>w3, l w gX @ K+w3,X+w3>w ,gw3= +51847, | 3 X+w3>w3, w3 Lqwhjudwlqj dqgxvlqj wkh lqlwldo frqglwlrq wklv ehfrphv ] l w X+w> w ,@4 gw3K+w3,X+w3>w ,= +51848, 3 | 3 w3

Wklv lv rqo| d irupdo vroxwlrq ehfdxvh wkh uljkw kdqgvlghfrqwdlqv wkh hyroxwlrq rshudwru lwvhoi> krzhyhu lw grhv kdyh wkh lqlwldo frqglwlrq douhdg| exlow lq/ dqg lw fdq eh lwhudwhg wr rewdlq dq h{sdqvlrq iru X lq srzhuv ri K1 Wr gr wklv/ zh uhshdwhgo| lqvhuw wkh zkroh lqwhjudo iru X lqwr wkh lqwhjudo lq zklfk lw dsshduv % & ] w ] w3 l l 33 X+w> w ,@4 gw3K+w3, 4 gw K+w33,X+w33>w , 3 | | 3 w3 w3 ] w 5 ] w ] w3 l l 33 @ 4 . gw3K+w3,. gw3 gw K+w3,K+w33,. +51849, | | w3 w3 w3 [4 +q, @ X +w> w3, +5184:, q@3 zkhuh wkh qwk whup ri wkh h{sdqvlrq kdv wkh jhqhudo irup

q ] ] ] l w wq w5 X +q,+w> w ,@ gw gw gw K+w ,K+w , K+w ,= +5184;, 3 | q qÃ4 4 q qÃ4 4 w3 w3 w3 Wklv dqgvlplodu h{sdqvlrqv iru wkh hyroxwlrq rshudwru surylghd xvhixo vwduwlqj srlqw iru wkh ghyhorsphqw ri wlph0ghshqghqw shuwxuedwlrq wkhrulhv1 Dv d qdo qrwh/ zh revhuyh wkdw wklv h{sdqvlrq iru X lpsolhv d vlplodu lqwhjudo htxdwlrq dqgh{sdqvlrq iru wkh vwdwh yhfwru/ l1h1/

m#+w,l @ X+w> w3,m#+w3,l ] l w @ 4 . gw3K+w3,X+w3>w , m#+w ,l +5184<, | 3 3 w3 ru ] l w m#+w,l @ m#+w ,l gw3K+w3,m#+w3,l> +51853, 3 | w3 zkhuh wkh vhfrqgwhup rq wkh uljkw0kdqg0vlghri wkh odvw h{suhvvlrq uhsuhvhqwv wkh fkdqjh wkdw kdv rffxuuhglq wkh vwdwh yhfwru ehwzhhq wlphv w3 dqg w1 Fkdswhu 6 WKH KDUPRQLF RVFLOODWRU

Zh qrz frqvlghu dq h{whqghg h{dpsoh zklfk doorzv xv wr dsso|wkh wkhruhwlfdo dssdudwxv frqvwuxfwhg lq suhylrxv vhfwlrqv1 Wkh h{dpsoh zh fkrrvh/ wkdw ri d sduwlfoh vxemhfwhg wr d olqhdu uhvwrulqj irufh 0 wkh vr0fdoohg kduprqlf rvfloodwru 0 lv lpsruwdqw iru vhyhudo uhdvrqv1 Iluvw/ lw lv rqh ri wkh uhodwlyho|vpdoo qxpehu ri txdqwxp phfkdqlfdo sureohpv wkdw fdq eh vroyhg h{dfwo|dqg frpsohwho|1Lq dgglwlrq/ wkh sureohp surylghv d edvlv iru rxu xqghuvwdqglqj ri pdq|lpsruwdqw sk|vlfdo sureohpv/ lqfoxglqj prohfxodu yleudwlrqv/ wkh yleudwlrqdo h{flwdwlrqv ri vrolgv +l1h1/ skrqrqv,/ dqg wkh txdqwl}dwlrq ri wkh hohfwur0 pdjqhwlf hog +skrwrqv,1 Lq d uhdo vhqvh/ wkh rqh0glphqvlrqdo kduprqlf rvfloodwru lv wkh pdlq exloglqj eorfn ri d juhdw ghdo ri txdqwxp hog wkhru|1

614 Vwdwhphqw ri wkh Sureohp Zh frqvlghu d sduwlfoh ri pdvv p vxemhfw wr d olqhdu uhvwrulqj irufh I @ n{/fruuhvsrqg0 lqj wr wkh txdgudwlf srwhqwldo 4 4 Y +{,@ n{5 @ p$5{5 +614, 5 5 s zkhuh $ @ n@p1 Lq wkh Kdplowrqldq ghvfulswlrq ri fodvvlfdo phfkdqlfv/ wkh v|vwhp lv ghvfulehg e|wkh g|qdplfdo yduldeohv i{> sj/ dqg wkh hyroxwlrq lv jryhuqhg e|wkh Kdplowrqldq s5 4 K @ W . Y @ . p$5{5= +615, 5p 5 Kdplowrq*v htxdwlrqv ri prwlrq CK s {b @ @ +616, Cs p CK sb @ @ p$5{ +617, C{ duh/ xsrq wdnlqj d vhfrqg ghulydwlyh/ htxlydohqw wr wkh idploldu Qhzwrqldq htxdwlrqv

{ . $5{ @3s . $5s @3 +618, zkrvh vroxwlrqv ohdg wr wkh idploldu rvfloodwru|ehkdylru

{+w,@D vlq+$w . , +619,

s+w,@Dp$ frv+$w . ,= +61:, 437 Wkh Kduprqlf Rvfloodwru

Lq sdvvlqj iurp d fodvvlfdo wuhdwphqw wr d txdqwxp phfkdqlfdo rqh/ wkh g|qdplfdo ydul0 deohv duh uhsodfhg e|rshudwruv { $ [ +61;, s $ S @ |N zklfk reh|wkh fdqrqlfdo frppxwdwlrq uhodwlrqv

^[> S`@l|= +61<,

Hyroxwlrq ri wkh txdqwxp phfkdqlfdo v|vwhp lv jryhuqhg e| wkh dvvrfldwhg Kdplowrqldq rshudwru S 5 4 K @ . p$5[5= +6143, 5p 5

Vlqfh wkh v|vwhp lv frqvhuydwlyh +CK@Cw @3,/wklvhyroxwlrqlvehvwfrqvlghuhglqwkh edvlv ri wkh hljhqvwdwhv m!ql ri wkh Kdplowrqldq/ zklfk duh dvvxphg wr vsdq wkh vsdfh ri d vlqjoh sduwlfoh prylqj lq rqh0glphqvlrq/ dqg zklfk reh|wkh hqhuj|hljhqydoxh htxdwlrq

+K Hq,m!ql @3= +6144,

Dv zlwk dq|hljhqydoxh sureohp/ zh qhhg dq lqlwldo uhsuhvhqwdwlrq lq zklfk wr zrun1 Lq wkh m{l uhsuhvhqwdwlrq/ dvvrfldwhg zlwk wkh hljhqvwdwhv ri wkh srvlwlrq rshudwru [> wklv ehfrphv d glhuhqwldo htxdwlrq |5 g5! 4 q . p$5{5 H ! +{,@3 +6145, 5p g{5 5 q q iru wkh hljhqixqfwlrqv !q+{,@k{m!ql1 Wkh qrwdwlrq wkdw zh kdyh lqwurgxfhg vxjjhvwv d glvfuhwh vshfwuxp/ dqg/ lqghhg lw fdq eh dqwlflsdwhg wkdw doo ri wkh hljhqvwdwhv ri wkh kduprqlf srwhqwldo pxvw eh erxqg vwdwhv1 Wklv iroorzv iurp wkh revhuydwlrq wkdw wkh srwhqwldo hqhuj|ri wkh rvfloodwru ehfrphv lq qlwh dv m{m$41Dvduhvxow/wkhzdyh ixqfwlrq pxvw jr wr }hur dw odujh glvwdqfhv iurp wkh ruljlq lq rughu iru wkh hqhuj|ri wkh v|vwhp wr uhpdlq qlwh1 Wkxv/ wkh deryh htxdwlrq lv wr eh vroyhg zlwk wkh erxqgdu| frqglwlrq !q+{, $ 3 dv m{m$4/ fkdudfwhulvwlf ri d erxqg vwdwh vroxwlrq1

Ri frxuvh/ lw lv dovr srvvleoh wr vroyh wkh hljhqydoxh htxdwlrq lq wkh zdyh yhfwru ru pr0 phqwxp uhsuhvhqwdwlrq1 Lqghhg/ lq wkh mnl edvlv/ wkh hljhqydoxh htxdwlrq iru wkh kduprqlf rvfloodwru lv dovr d vhfrqg rughu glhuhqwldo htxdwlrq |5n5 4 g5! H ! +n, p$5 q @3> +6146, 5p q q 5 gn5 gxh wr wkh idfw wkdw { @ lg@gn lv d glhuhqwldo rshudwru lq wkdw uhsuhvhqwdwlrq1 Djdlq/ wklv hljhqydoxh htxdwlrq lv wr eh vroyhg xqghu wkh uhtxluhphqw wkdw wkh vroxwlrq ydqlvk dv mnm$4> vr wkdw wkh hqhuj|ri wkh v|vwhp +lq wklv fdvh wkh nlqhwlf hqhuj|,eh qlwh1

D wudglwlrqdo dssurdfk frpprqo|wdnhq wr vroyh hlwkhu ri wkhvh htxdwlrqv lv wkh vr0fdoohg srzhu vhulhv phwkrg/ wkh edvlf vwhsv ri zklfk zh hqxphudwh iru wkh vsdwldo hljhqixqfwlrqv ehorz= Vwdwhphqw ri wkh Sureohp 438

41 Ghwhuplqh iru odujh { wkdw wkh vroxwlrq kdv wkh dv|pswrwlf irup !+{, D+{,hÃ{5 > zkhuh @ p$@5|> dqg D+{, lv vorzo|ydu|lqj lq {= Wklv dv|pswrwlf irup iroorzv vlqfh iru odujh { wkh glhuhqwldo htxdwlrq fdq eh zulwwhq ^fi1 Ht1 +6145,` lq wkh vlpsohu irup g5! p5$5 q {5! +{,@3= +6147, g{5 |5 q 51 Dvvxph d srzhu vhulhv vroxwlrq ri wkh irup [ Ã{5 n !+{,@h dn{ +6148, n wr ghvfuleh wkh vorzo|0ydu|lqj ixqfwlrq D+{,> rewdlqlqj d uhfxuvlrq uhodwlrq iru wkh frh!flhqwv dn= 61 Vkrz wkdw li wkh vhulhv grhv qrw whuplqdwh/ wkh vhulhv surgxfhg zloo |lhog d vroxwlrq wkdw glyhujhv h{srqhqwldoo|dv h {5 iru odujh {= Ghgxfh/ wkhuhe|/ wkdw wkh vroxwlrqv fruuhvsrqglqj wr sk|vlfdoo| dffhswdeoh vroxwlrqv pxvw kdyh vhulhv zklfk whuplqdwh/ l1h1/ iru zklfk D+{, lv d sro|qrpldo lq {1

71 Ghgxfh wkh ydoxhv ri hqhuj| Hq iru zklfk wkh vhulhv whuplqdwhv/ wkhuhe|vroylqj wkh hljhqydoxh sureohp1

Lq zkdw iroorzv zh wdnh d glhuhqw dssurdfk/ gxh wr Gludf/ wkdw doorzv xv xowlpdwho|wr rewdlq doo hljhqixqfwlrqv iurp wkh vroxwlrq wr d vlpsoh uvw0rughu glhuhqwldo htxdwlrq1 Wklv dojheudlf phwkrg xvhv wkh ixqgdphqwdo frppxwdwlrq uhodwlrqv wr gluhfwo|ghgxfh wkh vshfwuxp dqg ghjhqhudf|ri wkh kduprqlf rvfloodwru Kdplowrqldq1

61414 Dojheudlf Dssurdfk wr wkh Txdqwxp Kduprqlf Rvfloodwru Wr idflolwdwh rxu vwxg|zh ehjlq e|lqwurgxflqj vrph vlpsoli|lqj qrwdwlrq1 Zh revhuyh uvw wkdw wkh fodvvlfdo kduprqlf rvfloodwru srvvhvvhv d qdwxudo iuhtxhqf| $= Txdqwxp phfkdqlfdoo|wklv lpsolhv wkh h{lvwhqfh ri d qdwxudo hqhuj|vfdoh %3 @ |$1Wkxv/wkh Kdplowrqldq/ zklfk lwvhoi kdv xqlwv ri hqhuj|/ fdq eh zulwwhq lq wkh irup S 5 4 |$ 4 S 5 4 K @ . p$5[5 @ . p$5[5 +6149, 5p 5 5 |$ p |$ |$ S 5 p$[5 @ . +614:, 5 p|$ | ru pruh vlpso| |$ K @ +s5 . t5, +614;, 5 lq zklfk S s @ s +614<, p|$ dqg u p$ t @ [ +6153, | uhsuhvhqw glphqvlrqohvv prphqwxp dqg srvlwlrq rshudwruv/ uhvshfwlyho|1 +Qrwh wkdw zh duh uhod{lqj rxu frqyhqwlrq ri uhsuhvhqwlqj rshudwruv e|fdslwdo ohwwhuv1, Lw lv uhdglo| yhul hg wkdw wkhvh qhz rshudwruv reh|d glphqvlrqohvv irup ^t> s`@l +6154, 439 Wkh Kduprqlf Rvfloodwru ri wkh fdqrqlfdo frppxwdwlrq uhodwlrqv/ dqg dsduw iurp d voljkwo|glhuhqw qrupdol}dwlrq/ wkh hljhqvwdwhv imtlj dqg imslj ri wkhvh rshudwruv duh hvvhqwldoo|wkrvh ri wkhlu glphqvlrq0 doo|fruuhfw frxqwhusduwv im{lj dqg imnlj= Wkhuh lv d uhsuhvhqwdwlrq dvvrfldwhg zlwk hdfk vhw ri vwdwhv/ vr wkdw ] ] gt mtlktm @ 4 @ gs mslksm +6155,

ktmt3l @ +t t3, ksms3l @ +s s3, lq whupv ri zklfk zh fdq h{sdqg dqg duelwudu|vwdwh ri wkh v|vwhp +zklfk lv wkdw ri d sduwlfoh prylqj lq rqh0glphqvlrq,1 Wkxv/ zh fdq zulwh

] 4 ] 4 m#l @ gt #+t,mtl @ gs #+s,msl> +6156, Ã4 Ã4 zklfk gh qh frqyhqlhqw srvlwlrq dqg prphqwxp zdyhixqfwlrqv/ #+t, dqg #+s,> uhvshf0 wlyho|1 Zh fdq/ pruhryhu/ h{sdqg hdfk edvlv nhw lq whupv ri wkh edvlv yhfwruv ri wkh rwkhu uhsuhvhqwdwlrq mxvw dv zh fdq iru wkh qrupdo vwdwhv ri wkh srvlwlrq dqg zdyhyhfwru uhsuhvhqwdwlrqv=

] 4 msl @+5,Ã4@5 gt hlstmtl Ã4 ] 4 mtl @+5,Ã4@5 gs hÃlstmsl= +6157, Ã4 Ilqdoo|/ lw lv vwudljkwiruzdug wr vkrz wkdw lq wkh mtl uhsuhvhqwdwlrq s dfwv olnh d glhuhqwldo rshudwru/ l1h1/ g#+t, s#+t, $l +6158, gt dqglqwkhmsl uhsuhvhqwdwlrq t dfwv olnh d glhuhqwldo rshudwru g#+s, t#+s, $ l = +6159, gs

Wr surfhhg/ lw lv xvhixo wr qrwh wkdw wkh Kdplowrqldq iru wkh fruuhvsrqglqj fodvvlfdo sureohp lv idfwrul}deoh/ l1h1/ li t dqg s zhuh fodvvlfdo yduldeohv zh frxog zulwh |$ 4 K @ +t5 . s5,@ |$+t . ls,+t ls,= +lq wkh fodvvlfdo olplw,1 +615:, 5 5 Wkh idfw wkdw t dqg s gr qrw frppxwh uhqghuv wklv idfwrul}dwlrq lqydolg/ exw lw grhv ohdg xv wr frqvlghu wkh qrq0Khuplwldq rshudwruv 4 4 d @ s +t . ls, d. @ s +t ls, +615;, 5 5 lq whupv ri zklfk rxu ruljlqdo rshudwruv t dqg s fdq eh zulwwhq 4 l t @ s +d. . d, s @ s +d. d,= +615<, 5 5 Wkh surgxfw ri d. dqg d lv hdvlo|hydoxdwhg= 4 4 d.d @ +t ls,+t . ls,@ ^t5 . s5 . l+ts st,`= +6163, 5 5 Vwdwhphqw ri wkh Sureohp 43:

Uhfrjql}lqj wkh frppxwdwru ^t> s`@l lq wklv odvw h{suhvvlrq zh qg wkdw 4 4 d.d @ +t5 . s5, = +6164, 5 5 Wklv lghqwlw|doorzv xv wr h{suhvv wkh kduprqlf rvfloodwru Kdplowrqldq lq wkh irup |$ 4 K @ +s5 . t5,@|$+d.d . ,= +6165, 5 5 Lqwurgxflqj rqh ixuwkhu elw ri vlpsoli|lqj qrwdwlrq/ zh ghqrwh e|

Q @ d.d +6166, wkh rshudwru surgxfw ri d. dqg d1 Wkxv/ wkh Kdplowrqldq K fdq eh zulwwhq lq wkh iroorzlqj vlpsoh dqg vxjjhvwlyh irup 4 K @+Q . ,|$= +6167, 5

Lw lv reylrxv wkdw wkh hljhqvwdwhv ri wkh +pdqlihvwo|Khuplwldq, rshudwru Q @ d.d duh dovr hljhqvwdwhv ri wkh kduprqlf rvfloodwru Kdplowrqldq1 Lqghhg/ li zh fdq qg d frpsohwh vhw ri hljhqvwdwhv mql vxfk wkdw Qmql @ qmql> +6168, wkhq wkhvh vwdwhv zloo dovr eh hljhqvwdwhv ri wkh Kdplowrqldq/ l1h1/ 4 4 Kmql @+Q . ,|$mql @+q . ,|$mql @ H mql +6169, 5 5 q

4 zkhuh Hq @+q . 5 ,|$= Wkxv/ zh fkdqjh rxu ruljlqdo qrwdwlrq iru wkh hqhuj|hljhqvwdwhv vr wkdw mql @ m!ql= Lw lv lpsruwdqw wr vwuhvv wkdw/ dw wklv srlqw/ zh kdyhq*w uhdoo|grqh dq|wklqj/ vlqfh zh grq*w nqrz zkdw ydoxhv duh lq wkh vshfwuxp ri wkh rshudwru Q @ d.d= Zh kdyh vlpso|wudqvihuuhg wkh hljhqydoxh sureohp wkdw zh kdyh wr vroyh wr wkdw ri wkh rshudwru Q/ udwkhu wkdq wkh rshudwru K= Zh zloo uhihu wr wkh rshudwru Q dv wkh qxpehu rshudwru/ ehfdxvh/ dv zh zloo vhh/ lw frxqwv wkh qxpehu ri hqhuj|txdqwd/ lq xqlwv ri |$> dvvrfldwhg zlwk wkh v|vwhp1 Rxu jrdo lq zkdw iroorzv lv wr xvh wkh frppxwdwlrq uhodwlrqv reh|hg e| wkh qhz rshudwruv d> d.> dqg Q @ d.d> wr ghgxfh wkh vwuxfwxuh ri wkh hqhuj| hljhqvwdwhv ri wklv v|vwhp1

Wkh frppxwdwlrq uhodwlrqv wkdw zh zloo qhhg duh hdvlo|rewdlqhg1 Zh qrwh/ uvw/ wkdw 4 4 l ^d> d.`@ ^t . ls> t ls`@ +l^s> t` l^t> s`, @ +^t> s`.^t> s`, +616:, 5 5 5 zklfk wkh fdqrqlfdo frppxwdwlrq uhodwlrqv uhgxfh wr

^d> d.`@4= +616;,

Rqh frqvhtxhqfh ri wklv uhodwlrq lv wkdw dd. @ d.d .4> vr wkdw zh fdq zulwh

dd. @ Q .4= +616<,

Qh{w/ zh hydoxdwh wkh frppxwdwru

^Q>d`@^d.d> d`@d.^d> d`.^d.>d`d +6173, 43; Wkh Kduprqlf Rvfloodwru zklfk rxu suhylrxv uhvxow uhgxfhv wr

^Q>d`@d= +6174,

Ilqdoo|/ zh hydoxdwh

^Q>d.`@^d.d> d.`@d.^d> d.`.^d.>d.`d +6175, zklfk uhgxfhv wr ^Q>d.`@d.= +6176, Frpelqlqj wkhvh uhodwlrqv/ zh kdyh wkh iroorzlqj forvhg dojheud ri frppxwdwlrq uhodwlrqv

^d> d.`@4 ^Q>d`@d ^Q>d.`@d.= +6177,

61415 Vshfwuxp dqg Hljhqvwdwhv ri wkh Qxpehu Rshudwru Q Xvlqj wkh frppxwdwlrq uhodwlrqv rewdlqhg deryh/ zh qrz ghgxfh d qxpehu ri edvlf surs0 huwlhv dvvrfldwhg zlwk wkh hljhqvwdwhv ri wkh qxpehu rshudwru Q dqg/ khqfh/ ri wkh hljhq0 vwdwhv ri wkh kduprqlf rvfloodwru Kdplowrqldq K1 Lq zkdw iroorzv/ zh ehjlq e|vlpso| dvvxplqj wkh h{lvwhqfh ri dw ohdvw rqh qrq}hur hljhqyhfwru mql ri wkh revhuydeoh Q= Wklv lv d wulyldo dvvxpswlrq/ vlqfh Q lv d vlpsoh ixqfwlrq ri wkh revhuydeoh K> dqg lv wkhuhiruh dq revhuydeoh ri wkh v|vwhp1 Wklv dvvxpswlrq wkhq doorzv xv wr suryh wkh iroorzlqj=

41, Srvlwlylw| ri hljhqydoxhv= Li mql lv d qrq}hur hljhqyhfwru ri wkh rshudwru Q/dqgq lv wkh dvvrfldwhg hljhqydoxh/ wkhq q 31 Lq rwkhu zrugv/ wkh hljhqydoxhv ri Q duh srvlwlyh gh qlwh1 Surri= Wklv mxvw iroorzv iurp wkh reylrxv srvlwlylw|ri wkh rshudwru Q @ d.d> zklfk lpsolhv wkdw li Q mql @ qmql wkhq

kqmQmql @ qkqmql @ kqmd.dmql @ mmdmqlmm5 +6178, vr wkdw mmdmqlmm5 q @ 3= +6179, kqmql zklfk suryhv wkh dvvhuwlrq1

51, Dfwlrq ri d. rq hljhqyhfwruv ri Q= Li mql lv dq hljhqyhfwru ri wkh rshudwru Q zlwk hljhqydoxh q> wkhq wkh yhfwru . mq.l @ d mql +617:, lv dq hljhqyhfwru ri Q zlwk hljhqydoxh q .4= Surri= Zh frqvlghu wkh dfwlrq ri Q rq wkh vwdwh mq.l> dqg xvh wkh frppxwdwlrq uhodwlrq ^Q>d.`@d.> wr ghgxfh wkdw Qd. @ d.Q . d. @ d.+Q .4,> dqg khqfh

. . . . Qmq.l @ Qd mql @ d +Q .4,mql @ d +q .4,mql @+q .4,+d mql, +617;, vr wkdw Qmq.l @+q .4,mq.l +617<, vkrzlqj wkdw mq.l reh|v wkh hljhqydoxh htxdwlrq1 Qrwh wkdw wkh yhfwru mq.l lv qrq}hur/ vlqfh . kq.mq.l @ kqmdd mql @ kqmQ .4mql @+q .4,kqmql A 3 +6183, Vwdwhphqw ri wkh Sureohp 43<

Dv d fruroodu|/ lw iroorzv wkdw li wkhuh h{lvwv rqh qrq}hur hljhqyhfwru mql> wkhq wkhuh qhfhvvdulo|h{lvwv dq lq qlwh vhtxhqfh ri hljhqyhfwruv imql>d.mql> +d.,5mql> jdvvrfldwhg zlwk d fruuhvsrqglqj lqfuhdvlqj vhtxhqfh iq>q.4>q.5> j ri hljhqydoxhv wkdw fdq eh rewdlqhg e|uhshdwhg dssolfdwlrq ri wkh rshudwru d. wr wkh vwdwh mql= Iru wklv uhdvrq/ wkh rshudwru d. lvriwhquhihuuhgwrdvwkhudlvlqj rshudwru ru wkh fuhdwlrq rshudwru/ ehfdxvh/ dv zh zloo vhh/ lw dfwv wr fuhdwh txdqwd ri hqhuj|lq wkh v|vwhp1

61, Dfwlrq ri d rq hljhqyhfwruv ri Q= Li mql lv dq hljhqyhfwru ri Q zlwk hljhqydoxh q> dqg li mqÃl lv gh qhg wkurxjk wkh uhodwlrq

mqÃl @ dmql +6184, wkhq wkhuh duh wzr srvvlelolwlhv/ rqh ri zklfk pxvw eh wuxh/ hlwkhu= +l, wkh nhw mqÃl lv wkh qxoo yhfwru/ lq zklfk fdvh q @3> ru +ll, wkh nhw mqÃl lv dq hljhqyhfwru ri Q zlwk hljhqydoxh q 4= Surriri+l,= Zh qrwh uvw wkdw li mqÃl @3> wkhq

. kqÃmqÃl @3@kqmd dmql @ kqmQmql @ qkqmql +6185, vkrzlqj wkdw q @3> vlqfh mql lv e|dvvxpswlrq qrq}hur1 Frqyhuvho|/li q @3> wkh vdph h{suhvvlrq uhdg iurp uljkw wr ohiw vkrzv wkdw kqÃmqÃl @3> zklfk lpsolhv wkdw mqÃl lv wkh qxoo yhfwru1 Dv d frqvhtxhqfh/ zh vhh wkdw dq|hljhqyhfwru m3l ri Q zlwk hljhqydoxh q @3 reh|v wkh htxdwlrq dm3l @3=

Surriri+ll,= Li q 9@3> wkhq wkh dujxphqw deryh vkrzv wkdw mqÃl lv qrw wkh qxoo yhfwru/ vlqfh wkhq kqÃmqÃl @ qkqmql9@31 Zh wkhq frqvlghu wkh dfwlrq ri Q rq mqÃl dqg xvh wkh frppxwdwlrq uhodwlrq ^Q>d`@d> wr ghgxfh wkdw Qd @ dQ d @ d+Q 4,> vr wkdw

QmqÃl @ Qdmql @ d+Q 4,mql @ d+q 4,mql @+q 4,+dmql, +6186, dqg wkxv QmqÃl @+q 4,mqÃl= +6187,

Wklv odvw htxdwlrq vkrzv wkdw mqÃl lv d qrq}hur hljhqyhfwru ri Q zlwk hljhqydoxh vpdoohu e|rqhwkdqwkdwriwkhhljhqyhfwrumql= Wklv lpsolhv/ dv d fruroodu|/ wkdw li wkhuh h{lvwv rqh qrq}hur hljhqyhfwru mql wkhq wkhuh h{lvwv d vhtxhqfh ri hljhqyhfwruv imql>dmql>d5mql> j dvvrfldwhg zlwk d fruuhvsrqglqj vhtxhqfh ri ghfuhdvlqj hljhqydoxhv iq>q 4>q 5> j> zklfk fdq eh rewdlqhg e|uhshdwhg dssolfdwlrq ri wkh rshudwru d= Iru wklv uhdvrq/ wkh rshudwru d lv riwhq uhihuuhg wr dv wkh orzhulqj rshudwru ru dqqlklodwlrq rshudwru/ ehfdxvh lw dfwv wr dqqlklodwh ru uhgxfh wkh qxpehu ri hqhuj|txdqwd lq wkh v|vwhp1

71, Wkh hljhqydoxhv ri Q duh wkh qrq0qhjdwlyh lqwhjhuv 0 Zh fdq qrz dvvhuw wkdw wkh vshfwuxp ri wkh qxpehu rshudwru Q frqvlvwv ri wkh qxpehuv q @3> 4> 5> 1Wrsuryhwklv/ dvvxph wkdw wkhuh h{lvwv d +qhfhvvdulo|srvlwlyh, hljhqydoxh q 3 ri Q zklfk lv qrw lq wklv vhw1 Jlyhq dq|qrq}hur hljhqyhfwru mql zlwk wklv hljhqydoxh/ zh frxog wkhq surgxfh qrq0 qxoo hljhqvwdwhv ri Q zlwk qhjdwlyh hljhqydoxhv e|uhshdwhg dssolfdwlrq ri wkh orzhulqj rshudwru d= Wklv zrxog ylrodwh rxu surri wkdw Q kdv qr qhjdwlyh hljhqydoxhv1 Rq wkh rwkhu kdqg/ li q lv d srvlwlyh lqwhjhu ru }hur/ wkhq wkh vhtxhqfh ri hljhqydoxhv whuplqdwhv dw }hur ehiruh lw fdq surgxfh qhjdwlyh hljhqydoxhv1 Wkdw lv/ li q @3> wkh vhtxhqfh whuplqdwhv lpphgldwho|/ zlwk dm3l @3= 443 Wkh Kduprqlf Rvfloodwru

+Qrwh wkdw m3l rq wkh ohiw lv wkh hljhqyhfwru zlwk hljhqydoxh 3> zkloh wkh 3 rq wkh uljkw lv wkh qxoo yhfwru1, Li q lv d srvlwlyh lqwhjhu/ wkhq wkh vhtxhqfh whuplqdwhv diwhu h{dfwo| q vwhsv/ l1h1/ dqÃ4mql lv dq hljhqydoxh ri Q zlwk hljhqydoxh 3> vr e|rxu suhylrxv uhvxow dqmql @3= Wkxv wkh rqo|srvvleoh ydoxhv iru wkh hljhqydoxhv ri q duh wkh qrq0qhjdwlyh lqwhjhuv1 Vlqfh rqh ri wkhvh lv/ e|dvvxpswlrq/ qrw qxoo/ zh fdq surgxfh doo wkrvh zlwk orzhu +srvlwlyh, lqwhjhu ydoxhv e|dssolfdwlrq ri d dqg doo wkrvh zlwk kljkhu lqwhjhu ydoxhv e|dssolfdwlrq ri d.1 Wkxv/ doo ydoxhv lq wkh vhw q @3> 4> 5> pxvw eh hljhqydoxhv ri Q1 Wklv vwloo ohdyhv wkh txhvwlrq ri ghjhqhudf|/l1h1/ wkh srvvlelolw|wkdw wkhuh pd|eh pruh wkdq rqh olqhduo|lqghshqghqw hljhqyhfwru iru d jlyhq hljhqydoxh q=

81, Wkh hljhqydoxhv ri Q duh qrqghjhqhudwh 0 Zh zloo uvw vkrz wkdw li wkh hljhqydoxh q lv qrqghjhqhudwh/ wkhq vr lv q .41 Wr vhh wklv/ dvvxph wkdw q lv qrqghjhqhudwh/ dqg ohw m!q.4l dqg m#q.4l eh wzr duelwudu|hljhqvwdwhv ri Q kdylqj hljhqydoxh q .4= Iurp wkhvh vwdwhv zh fdq wkhq surgxfh wkh vwdwhv m!ql @ dm!q.4l dqg m#ql @ dm#q.4l> zklfk zrxogkdyhwrehhljhqvwdwhvriQ dvvrfldwhg zlwk wkh qrqghjhqhudwh hljhqydoxh q> dqg vr duh olqhduo|ghshqghqw1 Iru wzr yhfwruv/ olqhdu ghshqghqfh lpsolhv sursruwlrqdolw|/vr . wkhuh h{lvwv d frqvwdqw vxfk wkdw m!ql @ m#ql= Dfwlqj zlwk wkh udlvlqj rshudwru d wkhq uhyhdov wkdw

. . d m!ql @ d dm!q.4l @ Qm!q.4l @+q .4,m!q.4l . . @ d +m#ql,@d dm#q.4l @ Qm#q.4l @ +q .4,m#q.4l +6188, iurp zklfk zh ghgxfh wkdw m!q.4l @ m#q.4l= Wklv vkrzv wkdw m!q.4l dqg m#q.4l duh qhfhvvdulo|olqhduo|ghshqghqw1 Wkhuh lv dw prvw rqh olqhduo|lqghshqghqw hljhqyhfwru ri Q zlwk hljhqydoxh q .41 Khqfh/ li hljhqydoxh q lv qrqghjhqhudwh/ vr lv q .4=

Wr frpsohwh wkh dujxphqw/ zh qrz vkrz wkdw wkhuh h{lvwv/ lq idfw/ h{dfwo| rqh olqhduo|lqghshqghqw hljhqyhfwru m3l ri Q zlwk hljhqydoxh q @3> iurp zklfk lw iroorzv wkdw doo wkh hljhqydoxhv ri Q duh qrqghjhqhudwh1 Wr gr wklv zh h{solflwo|frqvwuxfw wkh fruuh0 vsrqglqj hljhqixqfwlrq !3+t,@ktm3l lq wkh srvlwlrq uhsuhvhqwdwlrq1 Wklv lv idflolwdwhg e| wkh idfw/ vkrzq deryh/ wkdw dq| hljhqvwdwh m3l ri Q zlwk hljhqydoxh 3 lv dqqlklodwhg e| wkh orzhulqj rshudwru/ l1h1/ lw reh|v wkh htxdwlrq

dm3l @3= +6189,

Xvlqj wkh uhodwlrq d @ s4 +t . ls,> wklv lpsolhv wkdw 5 4 4 g ktmdm3l @ s ktmt . lsm3l @ s +t . ,! +t,@3 +618:, 5 5 gt 3 lq zklfk zh kdyh xvhg wkh glhuhqwldo irup wdnhq e|wkh rshudwru s lq wkh srvlwlrq uhsuh0 vhqwdwlrq1 Wklv uvw rughu glhuhqwldo htxdwlrq ohdgv wr wkh uhodwlrq g! 3 @ tgt> +618;, !3 zklfk fdq eh lqwhjudwhg iurp t @3wr rewdlq 4 oq^! +t,@! +3,` @ t5 +618<, 3 3 5 ru Ã 4 t5 !3+t,@!3+3, h 5 = +6193, Vwdwhphqw ri wkh Sureohp 444

Wkxv/ wkh hljhqixqfwlrqv ri Q zlwk q @3glhu iurp rqh dqrwkhu rqo|wkurxjk dq ryhudoo pxowlsolfdwlyh frqvwdqw1 Wkxv wkhuh lv rqo|rqh olqhduo|lqghshqghqw vroxwlrq zlwk wklv hljhqydoxh1 Wkh hljhqydoxh q @3lv/ wkhuhiruh/ qrqghjhqhudwh dv duh doo wkh hljhqydoxhv ri Q1

Zh qrz vxppdul}h wkh uhvxowv ri wkh suhfhglqj vhulhv ri dujxphqwv1 Wkh vshfwuxp ri wkh qxpehu rshudwru Q lv wkh vhw ri qrq0qhjdwlyh lqwhjhuv vshfwuxp+Q,@i3> 4> 5> j= +6194, Iru hdfk hohphqw lq wklv vhw/ wkhuh h{lvwv h{dfwo|rqh olqhduo|lqghshqghqw hljhqvwdwh mql= Lw iroorzv wkdw wkh vshfwuxp ri wkh kduprqlf rvfloodwru Kdplowrqldq lv wkh vhw ri qrqghjhqhudwh hqhujlhv 4 vshfwuxp+K,@iH @+q . ,|$ m q @3> 4> 5> j= +6195, q 5 Qrwh wkdw wkh hljhqvwdwhv ri K irup d vhw ri htxdoo|vsdfhg ohyhov vwduwlqj dw wkh plqlpxp 4 hqhuj| H3 @ 5 |$> zklfk lv riwhq uhihuuhg wr dv wkh }hur0srlqw hqhuj|ri wkh jurxqg vwdwh1 Wkh uvw h{flwhg vwdwh H4 lv kljkhu lq hqhuj|wkdq wkh jurxqg vwdwh e|rqh txdqwxp H @ |$ ri hqhuj|/ dqg wkh hqhuj| vsdflqj ehwzhhq dgmdfhqw ohyhov lv d frqvwdqw1 Wkh qxpehu rshudwru Q> wkhuhiruh/ frxqwv wkh qxpehu ri hqhuj|txdqwd wkdw kdyh ehhq dgghg wr wkh v|vwhp/ dqg wkh rshudwruv d. dqg d fdq eh ylhzhg dv fuhdwlqj ru dqqlklodwlqj wkhvh hqhuj|txdqwd e|udlvlqj ru orzhulqj wkh ydoxh ri q1

-4 -2 0 0 2 4 q

Xs wr wklv srlqw zh kdyh ghgxfhg hvvhqwldo ihdwxuhv dvvrfldwhg zlwk wkh hljhqvwdwhv dqg hljhqydoxhv ri wkh kduprqlf rvfloodwru Kdplowrqldq1 Zh qrz hvk rxw wkh dqdo|vlv e| h{solflwo|frqvwuxfwlqj dq RQE ri vtxduh0qrupdol}hg hqhuj|hljhqvwdwhv1

61416 Wkh Hqhuj| Edvlv Zh kdyh douhdg|frqvwuxfwhg wkh hljhqvwdwh ri Q zlwk q @3e|ghulylqj wkh irup ri wkh zdyh ixqfwlrq Ã 4 t5 ktm3l @ !3+t,@Dh 5 +6196, wkdw uhsuhvhqwv wklv vwdwh lq wkh srvlwlrq uhsuhvhqwdwlrq1 Wr frpsohwh wkh slfwxuh zh qhhg wr vshfli|wkh qrupdol}dwlrq frqvwdqw D= Fruuhfw qrupdol}dwlrq uhtxluhv wkdw ] 4 ] 4 5 5 Ãt5 k3m3l @ gt m!3+t,m @ mDm gt h @4= +6197, Ã4 Ã4 445 Wkh Kduprqlf Rvfloodwru

s Wkh lqwhjudo dsshdulqj lq wklv frqglwlrq lv zhoo nqrzq dqg kdv wkh ydoxh > iurp zklfk zh ghgxfh wkdw wkh fruuhfwo|qrupdol}hg jurxqg vwdwh zdyh ixqfwlrq kdv wkh irup

Ã4@7 Ã 4 t5 !3+t,@ h 5 = +6198, Lw lv dovr srvvleoh wr h{suhvv wklv lq whupv ri wkh uhdo srvlwlrq yduldeoh {/ udwkhu wkdq wkh s p$ glphqvlrqohvv yduldeoh t @ | {1 Wklv lv prvw hdvlo|grqh e|qrwlqj wkdw/ lq jhqhudo/ qrupdol}dwlrq uhtxluhv wkdw ] ] ] gt 4@ m! +t,m5gt @ m! +t+{,,m5 g{ @ m! +{,m5g{ +6199, q q g{ q vr wkdw u gt p$ 4@7 ! +{,@! ^t+{,` @ ! ^t+{,`= +619:, q q g{ | q Pdnlqj wkh dssursuldwh vxevwlwxwlrq jlyhv wkh jurxqg vwdwh zdyh ixqfwlrq

p$ 4@7 p$ ! +{,@ h{s {5 = +619;, 3 | 5| Wkh uhpdlqlqj hljhqvwdwhv fdq eh jhqhudwhg iurp wkh jurxqg vwdwh e|uhshdwhg dssolfdwlrq ri wkh udlvlqj rshudwru d.= Xqiruwxqdwho|/ d vlpsoh0plqghg dssolfdwlrq ri d. wr wkh jurxqg vwdwh grhv qrw jhqhudwh qrupdol}hg hljhqvwdwhv1 Wr vhh wklv/ ohw xv ghqrwh e| mql dqg mq.4l wkh vtxduh0qrupdol}hg hljhqvwdwhv ri Q zlwk hljhqydoxhv q dqg q .4> uhvshfwlyho|1 Qrz rxu hduolhu dujxphqw vkrzv wkdw vwdwh d.mql lv dovr dq hljhqvwdwh ri Q zlwk hljhqydoxh q .4> dqg vr pxvw eh dw ohdvw sursruwlrqdo wr wkh vwdwh mq .4l> vlqfh wkh hljhqvwdwhv ri Q duh qrqghjhqhudwh1 Wkxv/ wkhuh h{lvwv d frqvwdqw q vxfk wkdw

. d mql @ qmq .4l +619<,

Wdnlqj wkh qrup ri wklv yhfwru uhyhdov wkdw

. 5 kqmdd mql @ mqm +61:3, ru/ xvlqj wkh idfw wkdw dd. @ Q .4> zh vhh wkdw

5 mqm @ q .4= +61:4,

Il{lqjs wkh uhodwlyh skdvh ri rxu edvlv yhfwruv vxfk wkdw q lv uhdo dqg srvlwlyh/ zh rewdlq q @ q .4> iurp zklfk zh ghgxfh wkh edvlf uhodwlrq s d.mql @ q .4mq .4l +61:5, ehwzhhq edvlv yhfwruv zlwk qhljkerulqj hqhuj|hljhqydoxhv1 Iru wkh sxusrvh ri frqvwuxfw0 lqj wkhvh vwdwhv lw lv xvhixo wr zulwh wklv uhodwlrq lq wkh htxlydohqw irup

d.mq 4l mql @ s = +61:6, q

E|uhfxuvlrq/ wklv doorzv xv wr h{suhvv wkh vwdwh mql lq whupv ri wkh jurxqg vwdwh/ l1h1/

d.mq 4l +d.,5mq 5l +d.,qm3l mql @ s @ s @ s +61:7, q q+q 4, q$ Vwdwhphqw ri wkh Sureohp 446 ru +d.,qm3l mql @ s = +61:8, q$

Wr qg wkh zdyh ixqfwlrqv zklfk uhsuhvhqw wkh hljhqvwdwhv lq wkh srvlwlrq uhsuhvhqwdwlrq zh surmhfw wklv rqwr wkh edvlv yhfwruv mtl ri wkdw uhsuhvhqwdwlrq

ktm+d.,qm3l ! +t,@ktmql @ s q q$ 4 g q @ s t ! +t, +61:9, 5qq$ gt 3 ru xvlqj wkh h{solflw irup iru wkh qrupdol}hg jurxqg vwdwh zdyh ixqfwlrq zh qg wkdw Ã4@7 q g Ã 4 t5 ! +t,@ s t h 5 = +61::, q 5qq$ gt

Wkxv/ zh kdyh dq h{solflw suhvfulswlrq iru fdofxodwlqj wkh zdyh ixqfwlrq iru wkh vwdwh q lq wkh srvlwlrq uhsuhvhqwdwlrq1 Dqrwkhu xvhixo irup ri wklv iroorzv iurp wkh uhodwlrq s d.mql @ q .4mq .4l +61:;, ghulyhg hduolhu/ iurp zklfk lw iroorzv wkdw

d.mql mq .4l @ s > +61:<, q .4 zklfk ehfrphv/ lq wkh srvlwlrq uhsuhvhqwdwlrq/ ktmd.mql 4 4 g ktmq .4l @ s @ s s t ! +t, +61;3, q .4 q .4 5 gt q Zh wkxv kdyh d uhfxuvlrq uhodwlrq 4 4 g ! +t,@ s s t ! +t, +61;4, q.4 q .4 5 gt q zklfk doorzv hdfk zdyh ixqfwlrq wr eh frqvwuxfwhg iurp wkh rqh suhfhglqj lw1

H{dpsohv= Zh ghulyh ehorz wkh uvw wkuhh kduprqlf rvfloodwru zdyh ixqfwlrqv

41 Iru q @3zh kdyh Ã4@7 Ãt5@5 !3+t,@ktm3l @ h

51 Dsso|lqj wkh suhvfulswlrq deryh zh qg wkdw iru q @4 4 4 g 5 ! +t,@ktm4l @ s s t Ã4@7hÃt @5 4 5 5 gt Ã4@7 5 @ s +5t, hÃt @5 5 447 Wkh Kduprqlf Rvfloodwru

61 Iru q @5> zh kdyh Ã4@7 4 g 5 ! +t,@ktm5l @ s t s +5t, hÃt @5 5 5 5 gt 5 Ã4@7 5 @ s 5t5 4 hÃt @5 5 Wkhvh zdyh ixqfwlrqv duh judskhg ehorz dv vrolg olqhv/ zlwk wkh dvvrfldwhg sured0 5 elolw|ghqvlwlhv m!qm lqglfdwhg dv gdvkhg olqhv1

0.7

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!q+t, iru q @51 Vwdwhphqw ri wkh Sureohp 448

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@ gt!q+t,#+t> 3,= Chapter 4 MANY PARTICLE SYSTEMS

The postulates of quantum mechanics outlined in previous chapters include no restrictions as to the kind of systems to which they are intended to apply. Thus, although we have considered numerous examples drawn from the quantum mechanics of a single particle, the postulates themselves are intended to apply to all quantum systems, including those containing more than one and possibly very many particles.

Thus, the only real obstacle to our immediate application of the postulates to a system of many (possibly interacting) particles is that we have till now avoided the question of what the linear vector space, the state vector, and the operators of a many-particle quantum mechanical system look like. The construction of such a space turns out to be fairly straightforward, but it involves the forming a certain kind of methematical product of di¤erent linear vector spaces, referred to as a direct or tensor product. Indeed, the basic principle underlying the construction of the state spaces of many-particle quantum mechanical systems can be succinctly stated as follows:

The state vector Ã of a system of N particles is an element of the direct product space j i S(N) = S(1) S(2) S(N) ¢¢¢ formed from the N single-particle spaces associated with each particle.

To understand this principle we need to explore the structure of such direct product spaces. This exploration forms the focus of the next section, after which we will return to the subject of many particle quantum mechanical systems.

4.1 The Direct Product of Linear Vector Spaces

Let S1 and S2 be two independent quantum mechanical state spaces, of dimension N1 and N2, respectively (either or both of which may be in…nite). Each space might represent that of a single particle, or they may be more complicated spaces, each associated with a few or many particles, but it is assumed that the degrees of mechanical freedom represented by one space are independent of those represented by the other. We distinguish states in (1) (2) each space by superscripts. Thus, e.g., Ã represents a state in S1 and Á astateof j i j i S2. To describe the combined system we now de…ne a new vector space

S12 = S1 S2 (4.1)

of dimension N12 = N1 N2 which we refer to as the direct or tensor product of S1 and £ S2. Some of the elements of S12 are referred to as direct or tensor product states,andare 124 Many Particle Systems formed as a direct product of states from each space. In other words, from each pair of (1) (2) states Ã S1 and Á S2 we can construct an element j i 2 j i 2 (1) (2) (1) (2) Ã; Á Ã Á = Ã Á S12 (4.2) j i´j i j i j i j i 2 of S12; in which, as we have indicated, a simple juxtaposition of elements de…nes the tensor product state when there is no possibility of ambiguous interpretation. By de…nition, then, the state Ã; Á represents that state of the combined system in which subsystem 1 is de…nitely inj statei Ã (1) and subsystem 2 is in state Á (2): The linear vector space j i j i S12 , which is intended to describe the combined system, consists precisely of all such directproductstatesaswellasallpossiblelinearcombinationsofthosestates.This direct product of states is assumed to be commutative in the trivial sense that there is nothing special about taking the elements in the reverse order, i.e., Ã; Á Ã (1) Á (2) = Á (2) Ã (1) (4.3) j i´j i j i j i j i except that in the abbreviated notation on the left hand side we agree to chose a distinct ordering of the spaces once and for all and thus associate the …rst symbol in the list with that part of the state arising from S1 and the second symbol for that part of the state arising from S2. In the decoupled form on the right, however, we are free to move the two kets from each space past each other whenever it is convenient. The tensor product is also assumed to be linearly distributive in the sense that if Ã (1) = ® » (1) + ¯ ´ (1), then j i j i j i Ã; Á Ã (1) Á (2) =[® » (1) + ¯ ´ (1)] Á (2) = ® »;Á + ¯ ´; Á ; (4.4) j i´j i j i j i j i j i j i j i (2) and similarly for kets Á which are linear combinations in S2. It is important to j i emphasize that there are many states in the space S12 that are not direct product states, although (by construction) any state in the product space can be written as a linear combination of such states. On the other hand, for any given linear combination Ã = ® »;Á + ¯ ´; Â (4.5) j i j i j i of product states, there may or may not be other states in S1 and S2 which allow Ã to be “factored” into a single direct product of states from each space. If no such factorizationj i exists,thenthestateissaidtobeanentangled state of the combined system. Under such circumstances neither subsystem can be described independently by its own state vector, without consideration of the state of the other. Generally, such entanglements arise as a result of interactions between the component degrees of freedom of each space. The space combined space S12 consists of all possible direct product states as well as all possible entangled states.

We denote by Ã; Á the bra of the dual space S12¤ = S1¤ S2¤ adjoint to the ket Ã; Á . Thus, the combinedh j symbol Ã; Á labeling the state is untouched by the adjoint process:j i [ Ã; Á ]+ =[Ã (1) Á (2)]+ =[ Ã (1) Á (2)]= Ã; Á : (4.6) j i j i j i h j h j h j

Inner products taken between elements of the direct product space are obtained by straightforward linear extension of inner products in each factor space, with the stip- ulation that it is only possible to take inner products between those factors in the same space, i.e.,

Ã; Á ´; » = Ã (1) Á (2) ´ (1) » (2) h j i h j h j j i j i = ³Ã ´ (1) Á » ´³(2): ´ (4.7) h j i h j i The Direct Product of Linear Vector Spaces 125

Thus, kets and bras in one space commute past those of the other to form a bracket with members of their own space. Since any state is expressible as a linear combination of product states, this completely speci…es the inner product in the combined space.

Basis vectors for the product space S12 can be constructed from basis vectors in the (1) factor spaces S1 and S2. Speci…cally, if the states Ái form a discrete ONB for S1 (2) fj i g and the states Â form a discrete ONB for S2, then the set of N1 N2 product fj ji g £ states Á ;Â form an ONB for the tensor product space S12.Wewrite: fj i jig

Á ;Â Á ;Â = ±i;i ±j;j (4.8) h i jj i0 j0 i 0 0

Ái;Âj Ái;Âj =1 (4.9) i;j j ih j X to denote the orthonormality and completeness of the product basis in S12.Anystatein the system can be expanded in such a basis in the usual way, i.e.,

Ã = Ái;Âj Ái;Âj Ã = Ãij Ái;Âj : (4.10) j i i;j j ih j i i;j j i X X

Similar relations hold for direct product bases formed form continuous ONB’s in S1 and

(1) (1) S2: Thus, if » and Â form continuous ONB’s for the two (in…nite dimensional) factor spacesfj theni g the productfj i g space is spanned by the basis vectors »;Â ; for which we can write j i »;Â »0;Â = ±(» »0)±(Â Â0) (4.11) h j 0i ¡ ¡ d» dÂ »;Â »;Â =1 (4.12) j ih j Z Z Ã = d» dÂ »;Â »;Â Ã = Ã(»;Â) »;Â : (4.13) j i j ih j i i;j j i Z Z X Finally, we can also form direct product bases using a discrete basis for one space and a continuous basis for the other. Note, that by unitary transformation in the product space it is generally possible to produce bases which are not the direct products of bases in the factor spaces (i.e., ONB’s formed at least partially from entangled states). Note also, that we have implicitly de…ned operators in the product space through the last relation. More generally, operators in S12 are formed from linear combinations of (what else) the direct product of operators from each space. That is, for every pair of linear operators A of S1 and B of S2 we associate an operator AB = A B which acts in S12 in such a way that each operator acts only on that part of the product state with which it is associated. Thus,

AB Ã; Â = A Ã (1) B Â (2) = Ã ;Â : (4.14) j i j i j i j A Bi ³ ´³ ´

Every operator in the individual factor spaces has a natural extension into the product space, since it can be multiplied by the identity operator of the other space; i.e., the (1) extension of the operator A of S1 into S12 is the operator

A(12) = A(1) 1(2), (4.15) 126 Many Particle Systems

(2) where 1 is the identity operator in S2. Often we will drop the superscripts, since the symbol A represents the same physical observable in S1 and in S12 (but is generally unde…ned in S2). Identical constructions hold for the extension of operators of S2. Clearly, 1(12) = 1(1) 1(2) = 1(1)1(2). Again, as with direct product states, the order of the factors is not important, so that operators of one factor space always commute with operators of the other, while operators from the same space retain the commutation relations that they had in the original space This implies, for example, that if A a (1) = a a (1) and B b (2) = b b (2),then j i j i j i j i AB a; b = BA a; b = A a (1) B b (2) = ab a; b (4.16) j i j i j i j i j i ³ ´³ ´ so that the eigenstates of a product of operators from di¤erent spaces are simply products of the eigenstates of the factors.

As with the states, a general linearoperatorinS12 can be expressed as a linear combination of direct product operators, but need not be factorizeable into such a product itself. A simple example is the sum or di¤erence of two operators, one from each space; again if A a (1) = a a (1) and B b (2) = b b (2),then j i j i j i j i (A B) a; b =(a b) a; b : (4.17) § j i § j i

In general any operator can be expanded in terms of an ONB for the product space in the usual way, e.g.,

H = Á ;Â Hij;i j Á ;Â ; (4.18) j i ji 0 0 h i0 j0 j i;j i ;j X X0 0 where H = Á ;Â H Á ;Â . Note that if H = AB is a product of operators from ij;i0j0 i j i0 j0 each space, thenh the matrixj j elementsi representing H in any direct product basis is just the product of the matrix elements of each operator as de…ned in the factor spaces, i.e.,

Á ;Â H Á ;Â = Á ;Â AB Á ;Â = Á A Á Â B Â h i jj j i0 j0 i h i jj j i0 j0 i h ij j i0 ih jj j j0 i

The resulting N1 N2 dimensional matrix in S12 is then said to be the direct or tensor £ product of the matrices representing A in S1 and B in S2: Finally we note that this de…nition of direct product spaces is easily extensible to treat multiple factor spaces. Thus, e.g., if S1;S2; and S3 are three independent quantum (1) mechanical state spaces, then we can take the 3-fold direct product of states Ã S1 (2) (3) (1) (2) j (3)i 2 and Á S2; and Â S3 to construct elements Ã;Á;Â = Ã Á Â of the directj i product2 space j i 2 j i j i j i j i S123 = S1 S2 S3 with inner products

(1) (2) (2) Ã0;Á0;Â0 Ã; Á; Â = Ã0 Ã Á0 Á Â0 Â h j i h j i h j i h j i and operators ABC Ã; Á; Â = Ã ;Á ;Â : j i j A B C i The Direct Product of Linear Vector Spaces 127

4.1.1 Motion in 3 Dimensions Treated as a Direct Product of Vector Spaces Tomaketheseformalde…nitionsmoreconcreteweconsiderafewexamples.Consider, e.g., our familiar example of a single spinless quantum particle moving in 3 dimensions. It turns out that this space can be written as the direct product

S3D = Sx Sy Sx: (4.19)

of 3 spaces Si, each of which is isomorphic to the space of a particle moving along one cartesian dimension. In each of the factor spaces we have a basis of position states and relevant operators, e.g., in Sx we have the basis states, x and operators X; Kx;Px; fj ig ¢¢¢ ,inSy the basis states y and operators Y;Ky;Py; , and similarly for Sz.Inthe fj ig ¢¢¢ direct product space S3D we can then form, according to the rules outlined in the last section, the basis states ~r = x; y; z = x y z (4.20) j i j i j i j i j i each of which is labeled by the 3 cartesian coordinates of the position vector ~r of R3.Any stateinthisspacecanbeexpandedintermsofthisbasis

Ã = dx dy dz x; y; z x; y; z Ã = dx dy dz Ã(x; y; z) x; y; z ; (4.21) j i j ih j i j i Z Z or in more compact notation

Ã = d3r ~r ~r Ã = d3rÃ(~r) ~r : (4.22) j i j ih j i j i Z Z Thus, the state Ã is represented in this basis by a wave function Ã(~r)=Ã(x; y; z) of 3 variables. Notej i that by forming the space as the direct product, the individual components of the position operator Xi automatically are presumed to commute with one another, since they derive from di¤erent factor spaces. Indeed, it follows that the canonical commutation relations

[Xi;Xj]=0=[Pi;Pj] (4.23)

[Xi;Pj]=i~±i;j (4.24) are automatically obeyed due to the rule for extending operators into the product space. The action of the individual components of the position operator and momentum operators also follow from the properties of the direct product space, i.e.,

X x; y; z = X x 1y y 1z z = x x; y; z (4.25) j i j i j i j i j i

3 @Ã(~r) Px Ã = i~ d r ~r (4.26) j i ¡ @x j i Z and so on. In a similar fashion it is easily veri…ed that all other properties of the space S3D of a single particle moving in 3 dimensions follow entirely from the properties of the tensor product of 3 one-dimensional factor spaces. Of course, in this example, it is merely a question of mathematical convenience whether we view S3 in this way or not. 128 Many Particle Systems

4.1.2 The State Space of Spin-1/2 Particles Another situation in which the concept of a direct product space becomes valuable is in treating the internal, or spin degrees of freedom of quantum mechanical particles. It is a well-established experimental fact that the quantum state of most fundamental particles is not completely speci…ed by properties related either to their spatial coordinates or to their linear momentum. In general, each quantum particle possesses an internal structure characterized by a vector observable S;~ the components of which transform under rotations like the components of angular momentum. The particle is said to possess “spin degrees of freedom”. For the constituents of atoms, i.e., electrons, neutrons, protons, and other spin-1=2 particles, the internal state of each particle can be represented as a superposition of two linearly independent eigenvectors of an operator Sz whose eigenvalues s = 1=2 § (in units of ~) characterize the projection of their spin angular momentum vectors S~ onto some …xed quantization axis (usually taken by convention to be the z axis). A particle whose internal state is the eigenstate with s =1=2 is said to be spin up, with s = 1=2; spin down. ¡ The main point of this digression, of course, is that the state space of a spin-1=2 particle can be represented as direct product

S = Sspatial Sspin spin-1=2 of a quantum space Sspatial describing the particle’s spatial state (which is spanned, e.g., by an in…nite set of position eigenstates ~r ), and a two-dimensional quantum space Sspin describing the particle’s internal structure.j i This internal space is spanned by the eigenstates s of the Cartesian component Sz of its spin observable S;~ with eigenvalues s = 1=2: Thej i direct product of these two sets of basis states from each factor space then generates§ the direct product states ~r; s = ~r s ; j i j i j i which satisfy the obvious orthonormality and completeness relations

3 d r ~r; s ~r; s =1 ~r0;s0 ~r; s = ±(~r ~r0)±s;s : j ih j h j i ¡ 0 s= 1=2 X§ Z An arbitrary state of a spin-1=2 particle can then be expanded in this basis in the form

3 3 Ã = d rÃs(~r) ~r; s = d r [Ã+(~r) ~r; 1=2 + Ã (~r) ~r; 1=2 ] j i j i j i ¡ j ¡ i s= 1=2 X§ Z Z and thus requires a two component wave function (or spinor). In other words, to specify the state of the system we must provide two seprate complex-valued functions Ã+(~r) and 2 Ã (~r); with Ã+(~r) describing the probability density to …nd the particle spin-up at ~r and¡ Ã (~r) 2j describingj the density to …nd the particle spin-down at ~r.Notethat,by j ¡ j 2 construction, all spin related operators (S;S~ z;S ; etc.) automatically commute with all spatial related operators (R;~ K;~ P;~ etc.). Thus, the concept of a direct product space arises in many di¤erent situtations in quantum mechanics and when properly identi…ed as such can help to elucidate the structure of the underlying state space.

4.2 The State Space of Many Particle Systems We are now in a position to return to the topic that motivated our interest in direct product spaces in the …rst place, namely, the construction of quantum states of many The State Space of Many Particle Systems 129 particle systems. The guiding principle has already been state, i.e., that the state vector of a system of N particles is an element of the direct product space formed from the N single-particle spaces associated with each particle.

Thus, as the simplest example, consider a collection of N spinless particles each moving in one-dimension, along the x-axis, say (e.g., a set of particles con…ned to a quantum wire). The ®th particle of this system is itself associated with a single particle state space S(®) that is spanned by a set of basis vectors x® , and is associated with the standard set fj ig (N) of operators X®;K®;P®; etc. The combined space S of all N particles in this system is then the N-fold direct product

S(N) = S(1) S(2) S(N) (4.27) ¢¢¢ of the individual single-particle spaces, and so is spanned by the basis vectors formed from the position eigenstates of each particle, i.e., we can construct the direct product basis

(1) (2) (N) x1;:::;xN = x1 x2 ::: xN : (4.28) j i j i j i j i In terms of this basis an arbitrary N-particle quantum state of the system can be expanded in the form

Ã = dx1 :::dxN x1;:::;xN x1;:::;xN Ã j i j ih j i Z = dx1 :::dxN Ã(x1;:::;xN ) x1;:::;xN : (4.29) j i Z

Thus, the quantum mechanical description involves a wave function Ã(x1;:::;xN ) which is a function of the position coordinates of all particles in the system. This space is clearly isomorphic to that of a single particle moving in N dimensions, but the interpretation is di¤erent. For a single particle in N dimensions the quantity Ã(x1;:::;xN ) represents the amplitude that a position measurement of the particle will …nd it located at the point ~r having the associated cartesian coordinates x1;:::;xN .ForN particles moving in one dimension, the quantity Ã(x1;:::;xN ) represents the amplitude that a simultaneous position measurement of all the particles will …nd the …rst at x1, the second at x2,and so on.

The extension to particles moving in higher dimensions is straightforward. Thus, for example, the state space of N spinless particles moving in 3 dimensions is the tensor product of the N single particle spaces S(®) each describing a single particle moving in 3 dimensions. The ®th such space is now spanned by a set of basis vectors ~r® ,andis fj ig associated with the standard set of vector operators R~ ®; K~ ®; P~®; etc. We can now expand an arbitrary state of the combined system

3 3 Ã = d r1 :::d rN ~r1;:::;~rN ~r1;:::;~rN Ã j i j ih j i Z 3 3 = d r1 :::d rN Ã(~r1;:::;~rN ) ~r1;:::;~rN : (4.30) j i Z in the direct product basis ~r1;:::;~rN of position localized states, each of which describes a distinct con…gurationfj of the Nigparticles. The wave function is then a function of the N position vectors ~r® of all of the particles (or of the 3N cartesian components thereof). A little re‡ection shows that the mathematical description of N particles moving 130 Many Particle Systems in 3 dimensions is mathematically equivalent, both classically and quantum mechanically, to a single particle moving in space of 3N dimensions.

Before discussing other properties of many-particle systems, it is worth pointing out that our choice of the position representation in the examples presented above is arbitrary. The state of a system of N spinless particles moving in 3 dimensions may also be expanded in the ONB of momentum eigenstates

3 3 Ã = d k1 :::d kN ~k1;:::;~kN ~k1;:::;~kN Ã j i j ih j i Z

3 3 = d k1 :::d kN Ã(~k1;:::;~kN ) ~k1;:::;~kN ; j i Z or in any other complete direct product basis. In addition, it should be noted that the rules associated with forming a direct product space ensure that all operators associated with a given particle automatically commute with all the operators associated with any other particle.

4.3 Evolution of Many Particle Systems The evolution of a many particle quantum system is, as the basic postulates assert, governed through the Schrödinger equation

d i~ Ã = H Ã (4.31) dtj i j i where H represents the Hamiltonian operator describing the total energy of the many particle system. For a system of N particles, the Hamiltonian can often be written in the form N P 2 H = ® + V (R~ ; R~ ;:::;R ): (4.32) 2m 1 2 N ®=1 ® X As for conservative single particle systems, the evolution of the system is most easily described in terms of the eigenstates of H, i.e., the solutions to the energy eigenvalue equation H Á = E Á : (4.33) j Ei j Ei Projecting this expression onto the position representation leads to a partial di¤erential equation N 2 ~ 2 ¡ Á + V (~r ;~r ;:::;~r )Á = EÁ (4.34) 2m ® E 1 2 N E E ®=1 ® r X for the many particle eigenfunctions ÁE(~r1;~r2;:::;~rN ).WhenN is greater than two this equation is (except in special cases) analytically intractable (i.e., nonseparable). This analytical intractability includes the important physical case in which the potential energy of the system arises from pairwise interactions of the form 1 V = V (~r® ~r¯): (4.35) 2 ¡ ®;¯ ®X=¯ 6 EvolutionofManyParticleSystems 131

Solutions of problems of this sort are fundamental to the study of atomic and molecular physics when N is relatively small (N 200, typically) and to the study of more general forms of matter (i.e., condensed phases,· liquids, solids, etc.) when N is very large (N 1024). Under these circumstances one is often led to consider the development of approximate» solutions developed, e.g., using the techniques of perturbation theory.

An important, and in principle soluble special case is that of noninteracting particles, for which the potential can be written in the form

N V = V®(~r®) (4.36) ®=1 X corresponding to a situation in which each particle separately responds to its own external potential. In fact, using potentials of this type it is often possible, in an approximate sense, to treat more complicated real interactions such as those described by (4.35). In fact, if all but one of the particles in the system were …xed in some well-de…ned state, then this remaining particle could be treated as moving in the potential …eld generated by all the others. If suitable potentials V®(~r®) could thus be generated that, in some average sense, took into account the states that the particles actually end up in, then the actual Hamiltonian P 2 H = ® + V 2m ® ® X couldberewrittenintheform

P 2 H = ® + V (~r ) +¢V 2m ® ® ® ® X · ¸ = H0 +¢V where

¢V = V V®(~r®) ¡ ® X would, it is to be hoped, represent a small perturbation. The exact solution could then be expanded about the solutions to the noninteracting problem associated with the Hamil- tonian P 2 H = ® + V (~r ) : 0 2m ® ® ® ® X · ¸ In this limit, it turns out, the eigenvalue equation can, in principle, be solved by the method of separation of variables. To obtain the same result, we observe that in this limit the Hamiltonian can be written as a sum

N P 2 N H = ® + V (R~ ) = H ; (4.37) 0 2m ® ® ® ®=1 ® ®=1 X · ¸ X of single particle operators, where the operator H® acts only on that part of the state associated with the single particle space S(®). In each single particle space the eigenstates En of H® form an ONB for the associated single particle space. Thus the many particle j ® i space has as an ONB the simultaneous eigenstates En ;En ;:::;En of the commut- j 1 2 N i ing set of operators H1;H2;:::;HN . These are automatically eigenstates of the total f g 132 Many Particle Systems

Hamiltonian H0, i.e.,

H0 En1 ;En2 ;:::;En® ;:::;EnN = H® En1 ;En2 ;:::;En® ;:::;EnN j i ® j i X = En® En1 ;En2 ;:::;En® ;:::;EnN ® j i X = E En ;En ;:::;En ;:::;En ; (4.38) j 1 2 ® N i where the total energy E = ® En® is just the sum of the single particle energies (as it is classically). The corresponding wave function associated with such a state is P ~r1;~r2;:::;~rN En ;En ;:::;En = Á (~r1)Á (~r2) :::Á (~rN ) (4.39) h j 1 2 N i n1 n2 nN which is just the product of the associated single particle eigenfunctions of the operators H®,thesameresultthatonewould…ndbyusingtheprocessofseparationofvariables. Indeed, the standard process of solving a partial di¤erential equation by the method of separation of variables can be interpreted as the decomposition of an original functional space of several variables into the direct product of the functional spaces associated with each.

4.4 Systems of Identical Particles The developments in this chapter derive their importance from the fact that there exist experimental systems of considerable interest which contain more than one particle. It is useful at this point to consider the implications of another important empirical fact: in many of these systems the particles of interest belong to distinct classes of (apparently) indistinguishable or identical particles. We use names (protons, electrons, neutrons, silver ions, etc.) to distinguish the di¤erent classes of indistinguishable particles from one another. Operationaly, this means that two members of a given class (e.g., two electrons) are not just similar, but are in fact identical to one another, i.e., that there exists no experiment which could possibly distinguish one from the other. This leads us to ask the following question: What constraint, if any, does indistinguishability impose upon the state vector of a system of identical particles?

To explore this question, consider …rst a system of N distinguishable, but physically similar, particles each of which is associated with a state space S(®) which is isomorphic (®) to all the others. If the set of vectors Áº º =1; 2; forms an ONB for the space of the ®th particle then the many particlefj i spacej S(N) is¢¢¢g spanned by basis vectors of the form Á ;Á ;:::;Á = Á (1) Á (2) ::: Á (N) (4.40) j º1 º2 ºN i j º1 i j º2 i j ºN i which corresponds to a state in which particle 1 is in state Áº1 ; particle 2 in state Áº2 ; and so on.

Consider, now, the operation of “interchanging” two of the particles in the system. Formally, we can de…ne a set of N(N 1)=2 unitary exchange operators U®¯ through their action on any direct product basis¡ as follows:

U®¯ Á ;:::;Á ;:::;Á ;:::;Á = Á ;:::;Á ;:::;Á ;:::;Á (4.41) j º1 º® º¯ ºN i j º1 º¯ º® ºN i which puts particle ® in the state formerly occupied by particle ¯, puts particle ¯ in the state formerly occupied by particle ®, and leaves all other particles alone (we assume Systems of Identical Particles 133

® = ¯). We note in passing that each exchange operator is unitary since it maps any direct6 product basis onto itself, but in a di¤erent order. We note also that the product of any exchange operator with itself gives the identity operator, as is easily veri…ed by multiplying the equation above by U®¯,.and which is consistent with the intuitive idea 2 that two consecutive exchanges is equivalent to none. Thus, we deduce that U®¯ = 1:

These properties of the exchange operators aside, the important physical point is that for distinguishable particles an exchange of this sort leaves the system in a physically di¤erent state (assuming º® = º¯). The two states Á ;:::;Á ;:::;Á ;:::;Á and 6 j º1 º¯ º® ºN i U®¯ Á ;:::;Á ;:::;Á ;:::;Á , are linearly independent. j º1 º¯ º® ºN i When we mentally repeat this exercise of exchanging particles with a system of indistinguishable particles, however, we must confront the fact that there can be no experiment which can tell which particle is in which state, since there is no way of distinguishing the di¤erent particles in the system from one another. That is to say, we cannot know that particle ® is in the state Áº® , but only that the state Áº® is occupied by one of the particles. Thus, an orderedj listi enumerating which particlesj arei in which states, such as that labeling the direct product states above, contains more information than is actually knowable. For the moment, let Á ;:::;Á (I) denote the physical state of a system j º1 ºN i of N indistinguishable particles in which the states Áº1 ;:::;ÁºN are occupied. We then invoke the principle of indistinguishability and assert that this state and the one

(I) U®¯ Á ;:::;Á (4.42) j º1 ºN i obtained from it by switching two of the particles must represent the same physical state of the system. This means that they can di¤er from one another by at most a phase factor, i.e., a unimodular complex number of the form ¸ = eiµ. Thus, we assert that there exists some ¸ for which

(I) (I) U®¯ Á ;:::;Á = ¸ Á ;:::;Á : (4.43) j º1 ºN i j º1 ºN i Note, however, that exchanging two particles twice in succession must return the system to its original state, i.e.,

(I) (I) U®¯[U®¯ Á ;:::;Á ]= Á ;:::;Á : (4.44) j º1 ºN i j º1 ºN i This last statement is true whether applied to distinguishable or indistinguishable particles. The implication for the undetermined phase factor, however, is that

U 2 Á ;:::;Á (I)]=¸2 Á ;:::;Á (I) = Á ;:::;Á (I) (4.45) ®¯j º1 ºN i j º1 ºN i j º1 ºN i from which we deduce that ¸2 =1.Thisimpliesthat¸ = 1,sothat § (I) (I) U®¯ Á ;:::;Á = Á ;:::;Á : (4.46) j º1 ºN i §j º1 ºN i Thus, we have essentially proved the following theorem: The physical state of a system of N identical particles is a simultaneous eigenstate of the set of exchange operators U®¯ with eigenvalue equal either to +1 or 1. A state is said to be symmetric under f g ¡ exchange of the particles ® and ¯ if it is an eigenvector of U®¯ with eigenvalue +1; and is antisymmetric if it is an eigenvector of U®¯ with eigenvalue 1.Itistotally symmetric if it is symmetric under all exchanges and totally antisymmetric¡ if it is antisymmetric under all exchanges. Now it is not hard to see that the set of all totally symmetric states of N particles (N) (N) forms a subspace SS of the original product space S , (since any linear combination 134 Many Particle Systems

(N) of such states is still totally symmetric). We shall call SS the symmetric subspace of S(N). Similarly the set of all totally antisymmetric states forms the antisymmetric (N) (N) subspace SA of S . Our theorem shows that all physical states of a system of N (N) (N) indistinguishable particles must lie either in SS or SA . (We can’t have a physical state symmetric under some exchanges and antisymmetric under others, since this would imply a physical di¤erence between some of the particles.) Moreover, if a given class of identical particles had some states that were symmetric and some that were antisymmetric, it would be possible to form linear combinations of each, forming physical states that were neither, and thus violating our theorem. We deduce, therefore, that each class of identical (N) (N) particles can have physical states that lie only in SS or only in SA ; it cannot have some states that are symmetric and some that are antisymmetric.

Experimentally, it is indeed found that the identi…able classes of indistinguishable particles divide up naturally into those whose physical states are all antisymmetric and those whose physical states are all symmetric under the exchange of any two particles in the system. Particles which are antisymmetric under exchange, such as electrons, protons, and neutrons, are referred to as fermions. Particles which are symmetric under exchange, such as photons, are referred to as bosons.

4.4.1 Construction of the Symmetric and Antisymmetric Subspaces We …nd ourselves in an interesting formal position. We have found that it is a straightforward exercise to construct the Hilbert space S(N) of N distinguishable particles as a direct product of N single particle spaces. We see now that the physical states associated with N indistinguishable particles are necessarily restricted to one or the other of two subspaces of the originally constructed direct product space. We still have not said how to actually construct these subspaces or indeed how to actually produce a physical state of such a system. This would be straightforward, of course, if we had at our disposal the projectors PS and PA onto the corresponding symmetric and antisymmetric subspaces, for then we could start with any state in S(N) and simply project away those parts of it which were not symmetric or antisymmetric, respectively. These projectors, if we can construct them, must satisfy the condition obeyed by any projectors, namely, 2 2 PS = PS PA = PA: (4.47)

In addition, if ÃS and ÃA represent, respectively, any completely symmetric or antisymmetric statesj ini S(N)j, theni we must have

PS Ã = Ã PA Ã =0 (4.48) j Si j Si j Si PA Ã = Ã PS Ã =0 (4.49) j Ai j Ai j Ai the right-hand relations follow because an antisymmetric state must be orthogonal to all symmetric states and vice versa. It turns out that the projectors PS and PA are straightforward to construct once one has assembled a rather formidable arsenal of unitary operators referred to as permutation operators, which are very closely related to, and in a sense constructed from, the exchange operators. To this end it is useful to enumerate some of the basic properties of the exchange operators U®¯ : 1. All exchange operators are Hermitian, since, as we have seen, they have real eigenvalues ¸ = 1. § 2. All exchange operators are nonsingular, since they are equal to their own inverses, 2 a fact that we have already used by observing, e¤ectively, that U®¯ =1, hence 1 U®¯¡ = U®¯. Systems of Identical Particles 135

3. All exchange operators are unitary since they are Hermitian and equal to their own + 1 inverses; it follows that U®¯ = U®¯ = U®¯¡ . Thus each one transforms any complete direct product basis for S(N) onto another, equivalent, direct product basis.

4. Di¤erent exchange operators do not generally commute. This makes sense on a physical basis; if we …rst exchange particles ® and ¯ and then exchange particles ¯ (which is the original particle ®)and° we get di¤erent results then if we make these exchanges in the reverse order.

The product of two or more exchange operators is not, in general, simply another exchange operator, but is a unitary operator that has the e¤ect of inducing a more complicated permutation of the particles among themselves. Thus, a product of two or more exchange operators is one of the N! possible permutation operators the members of which set we will denote by the symbol U»,where

1; 2 ;:::; N » (4.50) ´ » ;» ;:::;» µ 1 2 N ¶ denotes an arbitrary permutation, or reordering of the integers (1; 2 ;:::; N) into a new order, denoted by (»1;»2;:::;»N ). Thus the operator U» has the e¤ect of replacing particle 1 with particle »1, particle 2 with particle »2 and so on, i.e.

U» Áº ;:::;Áº = Áº ;:::;Áº j 1 N i j »1 »N i There are N! such permutations of N particles. The set of N! permutation operators share the following properties, some of which are given without proof:

1. The product of any two permutation operators is another permutation operator. In fact, they form a group, the identity element of which is the identity permutation that

maps each particle label onto itself. Symbolically, we can write U»U»0 = U»00 .The group property also insures that this relation applies to the whole set of permutation operators, i.e., for any …xed permutation operator U» the set of products U»U» is f 0 g equivalent to the set U» of permutation operators itself. This important property will be used below. f g

2. Each permutation », or permutation operator U» can be classi…ed as being either “even” or “odd”. An even permutation operator can be written as a product

U» = U®¯U°± U¹º ¢¢¢ of an even number of exchange operators, and odd permutation as a product of an odd number of exchange operators. This factorization of each U» is not unique, since we can obviously insert an even number of factors of U®¯ in the product without changing the permutation (or the even-odd classi…cation ). This is equivalent to the observation that an arbitrary permutation of N particles can be built up through a series of simple exchanges in many di¤erent ways.

3. The exchange parity "» of a given permutation operator is de…ned to be +1 if U» is even and 1 if it is odd. Equivalently, if U» can be written as a product of n ¡ n exchange operators then "» =( 1) : ¡ 136 Many Particle Systems

It turns out that for a given number N of particles, there are an equal number of even and odd permutation operators (indeed multiplying any even permutation operator by one exchange operator makes it an odd permutation operator and vice versa.). It is also fairly easy to see that if a state ÃS is symmetric (i.e., invariant) under all exchanges, it is also symmetric under any productj i thereof, and thus is symmetric under the entire set of permutation operators, i.e., U» Ã = Ã : (4.51) j Si j Si Physically, this means that it is invariant under an arbitrary permutation of the particles in the system. Alternatively, it means that Ã is an eigenstate of U» with eigenvalue 1: j Si

On the other hand, a state ÃA which is antisymmetric under all exchanges will be left unchanged after an even numberj i of exchanges (i.e., after being operated on by an even number of exchange operators), but will be transformed into its negative under an odd number of exchanges. Thus is succinctly expressed by the relation

U» Ã = "» Ã : (4.52) j Ai j Ai

Thus,anantisymmetricstateisaneigenstateofU» with eigenvalue "».

With these properties in hand, we are now ready to display the form of the projectors onto the symmetric and antisymmetric subspaces of S(N). These are expressible as relatively simple sums of the permutation operators as follows; the symmetric projector is essentially the symmetric sum 1 P = U (4.53) S N! » » X of all the permutation operators, while the antisymmetric projector has a form 1 P = U " : (4.54) A N! » » » X that weights each of the permutation operators by its exchange parity of 1.Thus,half the terms in the antisymmetric projector have a 1 andhalfhavea+1. § The proof that these operators do indeed¡ satisfy the basic properties of the projectors as we described earlier is straightforward. First we note that if ÃS is a symmetric state, then j i 1 1 PS Ã = U» Ã = Ã = Ã (4.55) j Si N! j Si N! j Si j Si » » X X where the sum over the N! permutations » eliminates the normalization factor. Similarly, we have 1 1 PA Ã = "»U» Ã = "» Ã =0; (4.56) j Si N! j Si N! j Si » » X X where we have used the fact that there are equal numbers of even and odd permutations to evaluate the alternating sum in the last expression. Similarly, if ÃA is an antisymmetric state, j i 1 1 2 PA Ã = "»U» Ã = " Ã = Ã (4.57) j Ai N! j Ai N! »j Ai j Ai » » X X while 1 1 PS Ã = U» Ã = "» Ã =0: (4.58) j Ai N! j Ai N! j Ai » » X X Systems of Identical Particles 137

2 2 Finally, we must show that PS = PS,andPA = PA. To show this we note …rst that 1 1 U P = U U = U = P ; (4.59) » S N! » »0 N! »00 S » » X0 X00 where we have used the group property which ensures that the set of permutation operators simply reproduces itself when multiplied by any single permutation operator. Thus for the symmetric projector we have 1 1 P P = U P = P = P ; (4.60) S S N! » S N! S S » » X X00 showing that it is indeed a projection operator. Toproveasimilarresultfortheantisymmetricprojector,wenote…rstthatthe exchange parity of the product of any two permutation operators is the product of their individual parities. Thus if U»U»0 = U»00 ; then

"»00 = "»"»0 : (4.61) This can be seen by factorizing both permutation operators in the product into exchange operators. If U» and U»0 contain n and m factors, respectively, then U»00 contains n + m n+m n m factors, so "»00 =( 1) =( 1) ( 1) = "»"»0 . Using this, along with the obvious 2 ¡ ¡ ¡ relation "» =1; it follows that 1 1 1 U P = U U " = " U U " " = " " U » A N! » »0 »0 N! » » »0 » »0 » N! »00 »00 » » » X0 X0 X00 = "»PA; (4.62) Thus for the antisymmetric projector we have 1 1 1 P P = " U P = "2P = P = P ; (4.63) A A N! » » A N! » A N! A A » » » X X X which is the desired idempotency relation for the antisymmetic projector.

Using these projectors, the physical state of a system of N identical particles is constructed (N) (N) by projection. To each state Ã S there corresponds at most one state ÃS SS (N) j i2 j i2 and one state ÃA SA : For bosons, the normalized symmetrical state is given by the projection j i2 PS Ã ÃS = j i (4.64) j i Ã PS Ã h j j i (N) onto SS ; while for fermions we have p

PA Ã ÃA = j i : (4.65) j i Ã PA Ã h j j i It is important to point out that the projectionp may give the null vector if, for example, the original state is entirely symmetric or antisymmetric to begin with. Example: Two Identical Bosons -Let Á and Â be two orthonormal single particle j i j i states. The state Á(1);Â(2) = Á; Â = Ã is in the two-particle direct product space S(2): The projectionj of Ã ontoi thej symmetrici j i subspace of S(2) is j i 1 1 PS Ã = [U12 + U21] Á; Â = [ Á; Â + Â; Á ] : (4.66) j i 2 j i 2 j i j i 138 Many Particle Systems

To normalize we evaluate 1 1 1 1 [ Á; Â + Â; Á ] [ Á; Â + Â; Á ]= [1+0+0+1]= ; (4.67) 2 h j h j 2 j i j i 4 2 so 1 Ã = [ Á; Â + Â; Á ] : (4.68) j Si p2 j i j i

Notice that if Á = Â; then the original state is already symmetric, i.e., PS Á; Á = Á; Á . Thus, it is possible for two (or more) bosons to be in the same single particlej state.i j Alsoi notice that if Á = Â; then the states Á; Â and Â; Á are both projected onto the same 6 j i j i physical state, i.e., PS Á; Â = PS Â; Á . This fact, namely that there are generally many (N) j i j i (N) states in S that correspond to the same physical state in SS is referred to as a lifting of the exchange degeneracy of S(N). Example: Two Identical Fermions - Again let Á and Â be two orthonormal single j i j i particle states, and Á(1);Â(2) = Á; Â = Ã be the associated two-particle state in S(2): The projection of Ãj onto thei antisymmetricj i j i part of S(2) is j i 1 1 PA Ã = ["12U12 + "21U21] Á; Â = [ Á; Â Â; Á ] : (4.69) j i 2 j i 2 j i¡j i To normalize we evaluate 1 1 1 1 [ Á; Â Â; Á ] [ Á; Â Â; Á ]= [1 0 0 + 1] = ; (4.70) 2 h j¡h j 2 j i¡j i 4 ¡ ¡ 2 so 1 Ã = [ Á; Â Â; Á ] : (4.71) j Ai p2 j i¡j i (2) Again, the projection of the states Á; Â and Â; Á onto SA correspond to the same physical state, (a lifting of the exchangej degeneracy)i j althoughi they di¤er from one another i¼ by a phase factor, i.e., PS Á; Â = PA Â; Á = e PS Á; Â . Notice, however, that if Á = Â; then the projectionj of thei original¡ j statei vanishes.j Thus,i two identical fermions cannot occupy the same physical state. This fact, which follows from the symmetrization requirement is referred to as the Pauli exclusion principle. It is possible to write the fermion state derived above in a convenient mathematical form involving a determinant, i.e., if we write 1 Á(1) Á(2) ÃA = j ij i (4.72) j i p2 Â(1) Â(2) ¯ ¯ ¯ j ij i ¯ and formally evaluate the determinant of¯ this odd matrix¯ we obtain ¯ ¯ 1 Ã = Á(1) Â(2) Â(1) Á(2) j Ai p2 j ij i¡j ij i 1 h i = [ Á; Â Â; Á ] (4.73) p2 j i¡j i This determinantal way of expressing the state vector (or the wave function) is referred to as a Slater determinant, and generalizes to a system of N particles. Thus, if

Áº1 ; Áº2 ; ; ÁºN represent a set of N orthonormal single particle states, then the Slaterj i j determinanti ¢¢¢ j i (1) (2) (N) Áº1 Áº1 Áº1 1 j (1)ij(2)i¢¢¢j(N)i Áº2 Áº2 Áº2 ÃA = ¯ j ij i¢¢¢j i ¯ (4.74) j i pN! ¯ ¯ ¯ ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¯ ¯ Á(1) Á(2) Á(N) ¯ ¯ ºN ºN ºN ¯ ¯ j ij i¢¢¢j i ¯ ¯ ¯ ¯ ¯ Systems of Identical Particles 139

is a properly normalized state of N fermions. Note that if any two states Áº® and Áº¯ are the same, then the corresponding rows will be identical and the resultingj i statej willi vanish, automatically satisfying the Pauli exclusion principle. 4.4.2 Number Operators and Occupation Number States We now turn to the task of identifying and constructing ONB’s for the symmetric (or bosonic) subspace describing a collection of N identical bosons and the antisymmetric (or fermionic) subspace describing a collection of N identical fermions. As we have seen, from a given set of single particle states Áº º =1; 2; we can form for the direct product (N) fj ij ¢¢¢g space S of N distinguishable particles an ONB of direct product states Áº1 ;:::;ÁºN , in which particle 1 is in state Á ; particle 2 in state Á ; and so on. It isj useful at thisi j º1 i j º2 i point to introduce a set of operators Nº º =1; 2; associated with this representation which “count” the number of particlesf j that are¢¢¢g in each single-particle state, i.e., by de…nition,

N1 Á ;:::;Á = n1 Á ;:::;Á j º1 ºN i j º1 ºN i N2 Á ;:::;Á = n2 Á ;:::;Á j º1 ºN i j º1 ºN i . .

Nº Á ;:::;Á = nº Á ;:::;Á j º1 ºN i j º1 ºN i . . (4.75)

where, e.g., n1 represents the number of times the symbol Á1 appears in the list (Áº1 ;:::;ÁºN ); i.e., it describes the number of particles in the collection occupying the single-particle state Á1 : Similarly, the eigenvalue j i N

nº = ±º;º® (4.76) ®=1 X is, for this state, the number of particles in the single-particle state Áº : Thus the direct product states of this representation are simultaneous eigenstates ofj thisi set of number operators Nº ; andeachsuchstateischaracterizedbyaspeci…csetofeigenvalues f g nº : The eigenvalues nº are referred to as the occupation numbers associated with thisf g representation of single particle states. Now, typically, the number of single-particle states Á is in…nite, and so most of the occupation numbers nº characterizing any fj ºig f g given basis state Áº1 ;:::;ÁºN are equal to zero. In fact, it is clear that at most N of them are not zero, andj this maximumi can be obtained only if all the particles are in di¤erent single-particle states. On the other hand, at least one of the occupation numbers is not zero, since the sum of the occupation numbers is equal to the total number of particles in the system, i.e., N = nº: (4.77) º X Now for distinguishable particles, the occupation numbers nº = n1;n2; generally do not determine uniquely the corresponding basis states.f Thisg isf because¢¢¢g the occupation numbers, by construction, contain information about which single-particle states are …lled, but contain no information about which particles are in which states. Thus, e.g., if Á ;:::;Á is a direct product state characterized by a certain set of j º1 ºN i occupation numbers nº ; any permutation of the particles in the system among the f g same speci…ed set of single-particle states will leave the system in a state U» Áº1 ;:::;ÁºN having exactly the same set of single-particle states …lled (albeit by di¤erentj particlesi than in the original). The set of occupation numbers for any such state will, therefore, be identical with that of the unpermuted state. If all of the occupied single-particle states 140 Many Particle Systems

Áº1 ;:::;ÁºN are distinct (i.e., if all the associated occupation numbers are either zero or one) , then there will be N! linearly independent states U» Á ;:::;Á associated j º1 ºN i with the same set of occupation numbers nº . If, however, some of the single-particle states are multiply occupied, then any permutationf g that simply rearranges those particles in the same occupied state will leave the system in exactly the same state as before. If, e.g., there are n1 particles in the state Á then there will be n1! permutations that j 1i simply permute these n1 particles among themselves, and so on. Arguing in this way for each multiply-occupied state, we deduce that the number of linearly-independent states

U» Áº1 ;:::;ÁºN of N distinguishable particles associated with a given set nº of occupationj numbersi is given by the expression f g © ª N! g ( nº )= : (4.78) f g n1!n2! ¢¢¢ Recall that 0!=1!=1; so that this expression reduces to N! when there are N distinct single-particle states …lled. This number g ( nº ) de…nes more closely the term exchange degeneracy introduced earlier, since it is, inf fact,g the degeneracy in S(N) associated with the set of simultaneous eigenvalues nº of the number operators Nº , a degeneracy that arises entirely due to the distinguishabilityf g of the particles described.f g (N) Now when the basis vectors Áº1 ;:::;ÁºN of this representation of S are projected onto the symmetric or anti-symmetricj subspaces,i they generate an ONB for each of these two smaller subspaces.© As we mightª expect, however, there is a reduction in the number of linearly-independent basis vectors that survive the projection. In particular, projection onto either the symmetric or anti-symmetric spaces eliminates any information regarding which particle is in which single-particle state. As a consequence, all the basis vectors U» Áº1 ;:::;ÁºN associated with a given set nº of occupation numbers project onto atj most one linearlyi independent basis vectorf ofg each subspace. This dramatic reduction© (which is the essenceª of the removal of the exchange degeneracy that we observed in the two-particle case) allows us to label each such basis vector by the associated set of occupation numbers, and provides us with what is referred to as the occupation number representation associated with a given set of single-particle states. In what follows we describe the projection process independently for the bosonic and fermionic subspaces.

The projection of the basis vector Áº1 ;:::;ÁºN onto the symmetric subspace (N) j i S of N-identical bosons is, by de…nition, the vector PS Á ;:::;Á : On the other S j º1 ºN i hand the projection of the state U» Á ;:::;Á ; which is associated with the same set j º1 ºN i of occupation numbers, is given by the vector PSU» Á ;:::;Á : But, we note that j º1 ºN i 1 1 P U = U U = U = P ; (4.79) S » N! »0 » N! »00 S » » X0 X00 wherewehaveusedthegrouppropertiesofthepermutationoperators.Thuswe…ndthat

PSU» Á ;:::;Á = PS Á ;:::;Á ; (4.80) j º1 ºN i j º1 ºN i (N) and hence deduce that all g ( nº ) basis vectors of S associated with the same set nº f g (N) f g of occupation numbers project onto precisely the same vector in SS : As a consequence, (N) the basis vectors obtained in this way by projection onto SS can be uniquely labeled by the occupation numbers nº that characterize them. We thus denote by f g

PS Áº1 ;:::;ÁºN n1;n2;::: = j i (4.81) j i PS Á ;:::;Á jj j º1 ºN ijj Systems of Identical Particles 141

the symmetric state of N identical bosons containing n1 particles in state Á1;n2 particles in state Á2; and so on. The set of such states with º nº = N span the symmetric subspace S(N) of N identical bosons, and form what is referred to as the occupation S P number representation associated with this set of single particle states. (Note that any such ONB of single particle states generates a similar representation.) For any such representation for the symmetric subspace, therefore, we can write an orthonormality relation

n1;n2;::: n0 ;n0 ;::: = ±n ;n ±n ;n (4.82) h j 1 2 i 1 10 2 20 ¢¢¢ showing that two occupation number states are orthogonal unless they have exactly the same set of occupation numbers, and a completeness relation

n1;n2;::: n1;n2;::: =1 (4.83) j ih j nº fXg for the symmetric space, where the sum is over all sets of occupation numbers consistent with the constraint º nº = N: We note in passing that the number operators Nº form a complete set of commuting observables (CSCO) for this symmetric subspace,f sinceg each basis vector inP this occupation number representation is uniquely labeled by the associated set of eigenvalues. Construction of the occupation number representation for the antisymmetric space of N identical fermions is similar, but some important di¤erences arise. In par- (N) ticular, we note …rst that the projection onto SA of any basis vector Áº1 ;:::;ÁºN of (N) j i SS having more than one particle in any given single particle state vanishes, since the corresponding Slater determinant (4.74) will have repeated rows. Thus most of the basis states of S(N) have no physical counterpart in the antisymmetric subspace. (In this sense, threfore, fermionic spaces are always smaller than bosonic spaces.) In general, only those direct product states Áº1 ;:::;ÁºN with each particle in a distinct single-particle state j i (N) Á ;:::;Á will have a non-vanishing projection PA Á ;:::;Á onto S : In that º1 ºN j º1 ºN i A case, the projection of the N! linearly independent states U» Á ;:::;Á characterized j º1 ºN i bythesameset nº of occupation numbers (which will now all be 0’s or 1’s) will take f g the form PAP» Á ;:::;Á : But this product of operators can also be simpli…ed, i.e., j º1 ºN i 1 " P U = " U U = » " " U U A » N! »0 »0 » N! »0 » »0 » » » X0 X0 1 = "» "» U» = "»PA (4.84) 2N! 00 00 3 » X00 4 5 where we have used the group properties of the permutation operators and the identity 2 "» =1: Thus, we …nd that

PAU» Á ;:::;Á = "»PA Á ;:::;Á = PA Á ;:::;Á : (4.85) j º1 ºN i j º1 ºN i § j º1 ºN i

Thus, the even permutations project onto the vector PA Á ;:::;Á andtheoddper- j º1 ºN i mutations onto its negative PA Áº1 ;:::;ÁºN : Of course, although these two states di¤er by a phase factor of unit modulus¡ j ( 1=ei¼),i they represent precisely the same physical state in Hilbert space. Thus, with a¡ suitable phase convention, we …nd that all of the basis vectors associated with a given acceptable set nº of occupation numbers project onto (N) f g the same basis vector of SA ; and so are uniquely labeled by the occupation numbers that 142 Many Particle Systems

characterize them. Thus, the number operators Nº form a complete set of commuting observables (CSCO) for the antisymmetric subspace,f g as well. We thus denote by

PA Áº1 ;:::;ÁºN n1;n2;::: = j i (4.86) j i PA Á ;:::;Á jj j º1 ºN ijj the antisymmetric state of N identical fermions containing n1 particles in state Á1;n2 particles in state Á ; and so on, where all nº 0; 1 ; and where nº = N: To 2 2f g º unambiguously …x the phase of the associated occupation number basis state n1;n2;::: ; P j i we note that of all the N! states U» Á ;:::;Á associated with the same set of j º1 ºN i occupation numbers nº ; only one of them has the occupied states ordered so that the …rst particle is in thef lowestg occupied© state, the secondª is in the next-to-lowest occupied state, and so on. It is this state (or any even permutation thereof) that we project (N) onto S to produce the basis state. Thus n1;n2;::: is identi…ed with the normalized A j i antisymmetric projection of that state Áº1 ;:::;ÁºN that has the correct set of occupation numbers and for which j i º1 <º2 < <ºN : (4.87) ¢¢¢ Another way of putting it is that we identify the basis state n1;n2;::: with that Slater j i determinant (4.74) in which the indices º® of the single particle states are strictly increasing in going from the top row to the bottom row. For this set of states, we can write completeness and orthonormality relations essentially identical to those that we wrote for the occupation number representation of the bosonic subspace, except for the restriction on the allowed set of occupation numbers to those for which nº 0; 1 : 2f g 4.4.3 Evolution and Observables of a System of Identical Particles The state vector Ã(t) describing a collection of identical bosons or fermions, if it is to continue to describej i such a system for all times, must remain within the bosonic or fermionic space in which it starts. What does this imply about the structure of the corresponding Hamiltonian and of the observables for such a system? To address this question we note that, as for any quantum system, evolution of the state vector is governed by Schrödinger’s equation d i~ Ã(t) = H Ã(t) (4.88) dtj i j i where H is the Hamiltonian governing the many-particle system. Now the state vector of a collection of bosons must remain symmetric under the exchange of any two particles in the system, and the state vector of a collection of identical fermions must remain antisymmetric under any such exchange. Thus, we can write that, for all times t;

U®¯ Ã(t) = ¸ Ã(t) (4.89) j i j i where ¸ =+1for bosons and 1 for fermions. Applying U®¯ to the evolution equation we determine that ¡ d i~¸ Ã(t) = U®¯H Ã(t) dtj i j i + = ¸U®¯HU Ã(t) (4.90) ®¯j i + wherewehaveinsertedafactorof1=U®¯U®¯ between H and the state vector. Canceling the common factor of ¸; and comparing the result to the original evolution equation we …nd that d + i~ Ã(t) = U®¯HU Ã(t) = H Ã(t) (4.91) dtj i ®¯j i j i Systems of Identical Particles 143 which is satis…ed provided that + H = U®¯HU®¯: (4.92) This shows that H, if it is to preserve the exchange symmetry of the state vector, must be invariant under the unitary transformations that exchange particles in the system. An op- + erator A is said to be symmetric under the exchange of particles ® and ¯ if U®¯AU®¯ = A + and is said to be antisymmetric under exchange if U®¯AU®¯ = A. Using this termi- nology, we see that the Hamiltonian of a collection of identical bosons¡ or fermions must be symmetric under all particle exchanges in the system. Another way of expressing the same thing is obtained by multiplying (4.92) through on the right by U®¯ to obtain the result HU®¯ = U®¯H; or, [U®¯;H]=0 (4.93) Thus, any operator that is symmetric under particle exchange commutes with the exchange operators. If H commutes with all of the exchange operators, then it commutes with any product of exchange operators, i.e., with any permutation operator U»; thus

[U»;H]=0: (4.94)

Since the projectors PS and PA onto the bosonic and fermionic subspaces are linear combinations of permutation operators, they must also commute with H, i.e.,

[PS;H]=0=[PA;H] : (4.95)

From the basic theorems that we derived for commuting observables, it follows that the (N) (N) eigenspaces of PS and PA; in particular, the subspaces SS and SA ; must be globally- invariant under the action of the Hamiltonian H.Inotherwords,H connects no state inside either subspace to any state lying outside the subspace in which it is contained. Indeed, this is the way that H keeps the state vector inside the relevant subspace. It follows, for example, that if the symmetric and antisymmetric projectors commute with the Hamiltonian at each instant, then they also commute with the evolution operator U(t; t0) which can be expressed as a function of the Hamiltonian. This means, e.g., that (N) if Ã(t0) is some arbitrary initial state vector lying in S which evolves into the state j i Ã(t) = U(t; t0) Ã(t0) ; then the projection of Ã(t) onto the symmetric subspace can be jwritteni j i j i

Ã (t) = PS Ã(t) = PSU(t; t0) Ã(t0) = U(t; t0)PS Ã(t0) = U(t; t0) Ã (t0) (4.96) j S i j i j i j i j S i and the projection onto the antisymmetric space can be written

Ã (t) = PA Ã(t) = PAU(t; t0) Ã(t0) = U(t; t0)PA Ã(t0) = U(t; t0) Ã (t0) (4.97) j A i j i j i j i j A i which shows that we can project an arbitrary initial state …rst to get an initial state vector with the right symmetry, and then evolve within the subspace, or simply evolve the arbitrary initial state and project at the end of the evolution process to get the state that evolves out of the appropriate projection of the initial state. The fact that the Hamiltonian commutes with PS and PA also means that there exists an orthonormal basis of energy eigenstates spanning each of the two subspaces of interest. This also makes sense from the point of view of the measurement process. If this were not the case, then an arbitrary state in either subspace would have to be represented as a linear combination of energy eigenstates some of which lie outside (or have components that lie outside) the subspace of interest. An energy measurement would then have a nonzero probability of collapsing the system onto one of these inadmissible 144 Many Particle Systems energy eigenstates, i.e., onto a state that is not physically capable of describing a collection of identical particles, since it lies outside the relevant subspace. Clearly, arguments of this sort based upon what can happen during a measurement process must apply as well to any observable of a system of identical particles. That is, in order for a measurement of an observable A not leave the system in an inadmissible state, it must have a complete set of eigenstates spanning the relevant subspace, and must, therefore, be symmetric under all particle exchanges. So what kind of operators are, in fact, symmetric under all particle exchanges? As we have seen, the number operators Nº associated with a given direct product representation of S(N) have the property thatf g they count the number of particles in any given single particle state, but are completely insensitive to which particles are in which state. Indeed, if the state Á ;:::;Á is characterized by a given set of occupation j º1 ºN i numbers nº (i.e., is an eigenstate of the number operators Nº with a particular set of f g f g eigenvalues) then so is any state U» Áº1 ;:::;ÁºN obtained from this one by a permutation of the particles. In other words, itj follows that ifi

Nº Á ;:::;Á = nº Á ;:::;Á (4.98) j º1 ºN i j º1 ºN i then

Nº U» Á ;:::;Á = nº U» Á ;:::;Á j º1 ºN i j º1 ºN i = U»Nº Á ;:::;Á : (4.99) £ ¤ £ j º1 ºN i ¤

Since this holds for each vector of the basis set, it follows that Nº U» = U»Nº ; hence

[U»;Nº ]=0; (4.100) which implies that [PS;Nº]=0=[PA;Nº ] : (4.101) Indeed, the number operators form a CSCO for the two subspaces of interest, and clearly have a complete ONB of eigenvectors (i.e., the occupation number states) spanning each subspace. It follows that any observable that can be expressed as a function of the number operators will also be symmetric under all particle exchanges. It is also possible to form suitable observables as appropriate linear combinations of single-particle operators. To see how this comes about, consider as a speci…c example, a system of two particles, each of which we can associate with a position operator R~ 1 and 2 R~ 2 whose eigenstates in the two particle space S are the direct product position states ~r1;~r2 and have the property that fj ig

R~ 1 ~r1;~r2 = ~r1 ~r1;~r2 R~ 2 ~r1;~r2 = ~r2 ~r1;~r2 (4.102) j i j i j i j i To see what happens under particle exchange, consider the operator ~ ~ + U21R1U21 = U21R1U21; (4.103)

+ where U21 = U21 exchanges particles 1 and 2. Thus, we …nd that

U21R~ 1U21 ~r1;~r2 = U21R~ 1 ~r2;~r1 = U21~r2 ~r2;~r1 j i j i j i = ~r2 ~r1;~r2 = R~ 2 ~r1;~r2 (4.104) j i j i Thus, as we might have anticipated, ~ + ~ U21R1U21 = R2: (4.105) Systems of Identical Particles 145

More generally, in a system of N particles if B® is a single-particle operator associated with particle ®; then its transform under U®¯

+ U®¯B®U®¯ = B¯ (4.106) is the corresponding operator for particle ¯. In this light, any symmetric combination of operators such as R~ ® +R~ ¯ is readily veri…ed to be symmetric under the associated particle exchange, i.e., ~ ~ + ~ ~ U®¯ R® + R¯ U®¯ = R¯ + R® (4.107) whereas any antisymmetric combination,³ such´ as that related to the relative displacement of two particles is antisymmetric, since

+ U®¯ R~ ® R~ ¯ U = R~ ¯ R~ ® = R~ ® R~ ¯ : (4.108) ¡ ®¯ ¡ ¡ ¡ ³ ´ ³ ´ On the other hand, the relative distance between the particles R~ ¯ R~ ® , being an even ¡ function of the R~ R~ is symmetric. Thus any function of¯ a symmetric¯ operator is ¯ ® ¯ ¯ symmetric, while an¡ even function of antisymmetric operators is¯ also symmetric.¯ Extending this to a system of N particles, any observable of a system of identical particles must be symmetric under all particle exchanges and permutations. This would include, e.g., any complete sum N B = B® (4.109) ®=1 X of the corresponding single particle operators for each particle in the system. Examples of operators of this type include the location of the center of mass

1 N R~ = R~ (4.110) N ® ®=1 X (recall that all the particles are assumed identical, so the masses are all the same), and the total linear and angular momentum

N N P~ = P~® L~ = L~ ®: (4.111) ®=1 ®=1 X X As we have seen, a system of noninteracting particles has a Hamiltonian that is of precisely this type, i.e., N H = H® (4.112) ®=1 X where, e.g., P 2 H = ® + V (R ): (4.113) ® 2m ® Actually, although we have constructed these operators as symmetric linear combinations of single-particle operators, it is easy to show that any operator of this type can also represented in terms of the number operators associated with a particular occupation (®) number representation. Suppose, e.g., that the single-particle states Áº are eigenstates j i (®) (®) of the single-particle operator B® with eigenvalues bº: Thus, e.g., B® Á = bº Á : j ºi j ºi 146 Many Particle Systems

Then the direct product states formed from this set will be simultaneous eigenstates of the all the related operators B®;

B® Á ;:::;Á ;:::;Á = bº Á ;:::;Á ;:::;Á (4.114) j º1 º® ºN i ® j º1 º® ºN i and so will be an eigenstate of the symmetric operator B = ® B®; i.e., P B Áº1 ;:::;ÁºN = B® Áº1 ;:::;ÁºN = bº® Áº1 ;:::;Áº® ;:::;ÁºN j i ® j i ® j i X X = b Á ;:::;Á ;:::;Á (4.115) j º1 º® ºN i where the collective eigenvalue b is the sum of the individual single-particle eigenvalues

N N

b = bº® = bº±º;º® = bºnº (4.116) ®=1 ®=1 º º X X X X where we have reexpressed the sum over the particle index ® by a sum over the state index º; and collected all the eigenvalues associated with the nº particles in the same single-particle state Áº together. Thus we can reexpress the eigenvalue equation above as

B Áº1 ;:::;ÁºN = bºnº Áº1 ;:::;ÁºN j i º j i X

= bºNº Áº1 ;:::;ÁºN (4.117) º j i X and, through a simple process of identi…cation, reexpress the operator B in the form

B = bºNº: (4.118) º X Thus, e.g., the noninteracting particle Hamiltonian above can be expressed in terms of thesingleparticleenergyeigenstates Áº obeying the equation H® Áº = "º Áº in the form j i j i j i H = "ºNº: (4.119) º X There is a real sense in which this way of expressing the Hamiltonian is to be preferred over the earlier form, particularly when we are dealing with identical particles. The operators Nº are not labeled by particle numbers, but by the single-particle states that can be occupied or not. Thus in expressing the Hamiltonian in this fashion, we are not using a notation that suggests (erroneously!) that we can actually label the individual particles in the system, unlike, e.g., (4.112), which explicitly includes particle labels as well as the number of particles in the system. The operator (4.119), on the other hand , makes no explicit reference either to particle labels or to the number of particles in the system, and therefore has exactly the same form in any of the spaces S(N) associated with any number N =1; 2; of identical particles. Operators¢¢¢ of this type, which can be expressed as a sum of single-particle operators, or equivalently, as a simple linear function of the number operators associated with a particular occupation number representation, we will refer to as one-body operators, since they really depend only on single-particle properties. In addition to these, there are also operators that depend upon multiple-particle properties. For example, interactions between particles are often represented by “two-body” operators of the form 1 Vint = V (R~ ® R~ ¯) (4.120) 2 ¡ ®;¯ ®X=¯ 6 Systems of Identical Particles 147 which is a symmetrized sum of operators that each depend upon the properties of just two particles. It turns out that operators of this type can usually be expressed as a simple bilinear function of the number operators associated with some set of single-particle states. Typically, however, the Hamiltonian contains both one-body and two-body parts, and the representation in which the one-body part is expressible in the form (4.119) is not one in which the two-body part is expressible as a simple function of the number operators. Conversely, a representation in which the interactions are expressible in terms of number operators is not one in which the one-body part is, also. The reason for this is that, typically, the interactions induce transitions between single-particle eigenstates, i.e., they take particles out of the single-particle states that they occupy and place them back into other single particle states. In the process, they change the occupation numbers characterizing the state of the system. It is useful, therefore, to de…ne operators that are capable of inducing transitions of this type. We have, in a sense, already encountered operators that do this sort of thing in our study of the harmonic oscillator. The energy eigenstates n of the 1D harmonic oscillator are each characterized by an integer n 0; 1; 2; fjthatig can be viewed as an “occupation number” characterizing the number of2f vibrational¢¢¢g quanta (now considered as a kind of “particle”) in the system. The annihilation, creation, and number operators a; a+; and N = a+a decrease, increase, and count the number of these quanta. In the 3D harmonic isotropic oscillator, which is separable in Cartesian coordinates, the energy eigenstates nx;ny;nz are characterized by a set of three occupation fj ig numbers nº º =1; 2; 3 characterizing the number of vibrational quanta associated with each Cartesianf j degree ofg freedom, and there are a set of annihilation, creation, and number + operators aº, aº ; and Nº for each axis. The di¤erent annihilation and creation operators obey characteristic commutation relations

+ + [aº ;aº ]=0= a ;a (4.121) 0 º º0 + £ ¤ aº;a = ±º;º (4.122) º0 0 that, as we have seen, completely determine£ the¤ associated integer spectrum of the number + operators Nº = aº aº: + Within this context, we now note that product operators of the form ax ay have the e¤ect + a ay nx;ny;nz = pnx +1pny nx +1;ny 1;nz (4.123) x fj ig fj ¡ ig of transferring a quantum of vibrational excitation from one axis to another, i.e., of inducing transitions in the states of the quanta. It is precisely operators of this type, that are capable of creating, destroying, and counting material particles in di¤erent single particle states that we wish to de…ne for a collection of identical particles. In order to carry this plan out, however, we need to realize that these creation and annhilation operators have the e¤ect of taking the system out of the space of N-particles, and into a space containing N +1; or N 1 particles. Thus, we need to expand our space in a way that allows us to put together, in¡ the same space, states of the system containing di¤erent particle numbers. The mathematical procedure for doing this is to combine the di¤erent N-particle spaces together in what is referred to as a direct sum of vector spaces which is, in a way, similar to that associated with the direct product of vector spaces that we have already encountered.. The de…nition and details associated with this procedure are explored in the next section. 4.4.4 Fock Space as a Direct Sum of Vector Spaces The idea of expressing a vector space as a sum of smaller vector spaces is actually implicitly contained in some of the concepts that we have already encountered. Consider, e.g., an 148 Many Particle Systems arbitrary observable A of a linear vector space S: By de…nition, the observable A possesses a complete orthonormal basis a; ¿ of eigenstates spanning the space. The eigenstates fj ig associated with a particular eigenvalue a form a subspace Sa of S; the vectors of which are orthogonal to the vectors in any of the other eigenspaces of A: Also, any vector Ã in S can be written as a linear combination of vectors taken from each of the orthogonalj i subspaces Sa; i.e.,

Ã = Ãa (4.124) j i a j i X where

Ãa = Ãa;¿ a; ¿ = Pa Ã : (4.125) j i ¿ j i j i X Under these circumstances, we say that the space S can be decomposed into a direct sum of the eigenspaces associated with the observable A; and symbolically represent this decomposition in the form S = Sa Sa Sa (4.126) © 0 © 00 ©¢¢¢ The vector space S has a dimension equal to the sum of the dimensions of all the eigenspaces Sa; as can be seen by counting up the basis vectors needed to span each orthogonal subspace. This idea of decomposing larger spaces into direct sums of smaller spaces can be reversed. Thus, given two separate vector spaces S1 and S2 of dimension N1 and N2; respectively, de…ned on the same …eld of scalars, we produce a larger vector space S of dimension N = N1 + N2 as the direct sum

S = S1 S2: (4.127) ©

The space S then contains, by de…nition, all vectors in S1; all vectors in S2; and all possible linear combinations of the vectors in S1 and S2 (the latter two spaces now being relegated to the role of orthogonal subspaces of S), with each vector in S1 orthogonal, by construction, to each vector in S2: It is this procedure that we carry out with the di¤erent N-particle spaces associated with a collection of identical particles. In particular, we de…ne the Fock Space of a collection of identical bosons or fermions, respectively, as the space obtained by forming the direct sum of the corresponding spaces describing N =0; 1; 2; particles. In particular, the Fock space of a set of identical bosons is written as the direct¢¢¢ sum

0 1 2 3 SS = S S S S (4.128) S © S © S © S ©¢¢¢ (N) 0 of the symmetric spaces SS for each possible value of N: The space SS of zero particles is assumed to contain exactly one linearly-independent basis vector, referred to as the vacuum and denoted by 0 ; and vectors from di¤erent N-particle spaces are assumed to be automatically orthogonalj i to each other. Similarly, the Fock space associated with a collection of identical fermions is written as the direct sum

0 1 2 3 SA = S S S S (4.129) A © A © A © A ©¢¢¢ (N) of the antisymmetric spaces SS for each value of N; where, again, the space of zero particles contains one basis vector, the vacuum, denoted by 0 ; and the di¤erent N- particle subspaces are assumed orthogonal. In what follows wej i explore separately the di¤erent structure of the bosonic and fermionic Fock spaces. Systems of Identical Particles 149

4.4.5 The Fock Space of Identical Bosons As we have seen, the symmetric occupation number states

n1;n2;::: nº = N (4.130) j i º X (N) associated with a given set of single-particle states Áº forms an ONB for the space SS of N identical bosons. The collection of such states,j therefore,i with no restriction on the sum of the occupation numbers, forms a basis for the Fock space SS of a set of identical bosons. In particular, the vacuum state is associated, in this (and any) occupation number representation with the vector 0 = 0; 0;::: (4.131) j i j i in which nº =0for all single particle states Áº . The single-particle states Áº themselves can be expressed in this representation in the form j i

Á = 1; 0; 0;::: j 1i j i Á = 0; 1; 0;::: j 2i j i . . (4.132) and so on. These basis states of Fock space are all simultaneous eigenvectors of the number operators Nº associated with this set of single particle states, and are uniquely labeled by the associated set of occupation number nº : Thus, the operators Nº form a CSCO for Fock space, just as they do for each off theg symmetric N-particlef subspacesg (N) SS . Thus, we can write a completeness relation for this representation of the form

1 1 n1;n2;::: n1;n2;::: =1: (4.133) n =0 n =0 ¢¢¢ j ih j X1 X2 We now wish to introduce operators that change the occupation numbers in a way similar to that associated with the harmonic oscillator. Because we are interested at present in describing bosons we want the spectrum of each number operator Nº to be exactly the same as for the harmonic oscillator, i.e., nº 0; 1; 2; : Thus, we can actually model the bosonic creation and annihilation operators2fdirectly¢¢¢gon those of the oscillator system, i.e., we introduce for each single particle state Áº an annihilation and creation operator, whose action is de…ned on the occupation numberj i states of this representation such that

+ a n1;:::;nº;::: = pnº +1n1;:::;nº +1;::: (4.134) º j i j i and aº n1;:::;nº;::: = pnº n1;:::;nº 1;::: : (4.135) j i j ¡ i It follows from this de…nition, as special cases, that the single-particle state Áº (which lies in the subspace containing just one boson) can be written in a form j i

Á = a+ 0 = a+ 0; 0;::: (4.136) j ºi º j i º j i + in which it is created “from nothing” by the operator aº : It also follows that any annihilation operator acting on the vacuum

aº 0 =0 (4.137) j i destroys it, i.e., maps it onto the null vector (not the vacuum!). Thus, the operator + aº creates a boson in the state Áº and the operator aº removes a boson from that state. 150 Many Particle Systems

+ Expressing the same idea in a somewhat more pedestrian fashion, the operators aº and aº (N) simply connect states in adjacent subspaces SS of the bosonic Fock space SS. Consistent with the de…nitions above, annihilation and creation operators associated with di¤erent single particle states commute with one another, allowing us to write the commutation relations for the complete set of such operators in the form

+ + [aº ;aº ]=0= a ;a (4.138) 0 º º0 + aº;a = ±£º;º ; ¤ (4.139) º0 0 which are usually referred to as “boson£ commutation¤ relations”. It also follows from the de…nition given above that

+ a aº n1;:::;nº;::: = nº n1;:::;nº;::: (4.140) º j i j i which allows us to identify this product of operators with the associated number operator, + i.e., Nº = aº aº; which obey the commutation relations

[Nº;Nº0 ]=0 (4.141) + + [Nº;aº ]= aº ±º;º Nº ;a = aº ±º;º (4.142) 0 ¡ 0 º0 0 Finally, just as it is possible to create the£ single¤ particle state Áº from the vacuum, so it is possible to express any of the occupation number states ofj thisi representation in a form in which they are created out of the vacuum by an appropriate product of creation operators, i.e., it is readily veri…ed that

+ n1 + n2 + n3 (a1 ) (a2 ) (a3 ) n1;:::;nº ;::: = ¢¢¢ 0 ; (4.143) j i pn1!pn2!pn3! j i ¢¢¢ the normalization factors taking the same form, for each single particle state, as in the simple harmonic oscillator. 4.4.6 The Fock Space of Identical Fermions The construction of the Fock space of a set of identical fermions proceeds in a similar fashion, but some interesting di¤erences arise as a result of the antisymmetric structure of the states associated with this space. For the fermionic space, we know that the antisymmetric occupation number states

n1;n2;::: nº = N (4.144) j i º X (N) generated from a given set of single-particle states Áº forms an ONB for the space SS of N identical fermions, provided that, as required byj thei exclusion principle, all occupation numbers only take the values nº =0or nº =1. Lifting the restriction on the sum of the occupation numbers, we obtain a basis for the Fock space SA of this set of fermions. As in the bosonic case, the vacuum state is associated with the vector

0 = 0; 0;::: (4.145) j i j i in which all nº =0, and the single-particle states Á can be written as j º i Á = 1; 0; 0;::: j 1i j i Á = 0; 1; 0;::: j 2i j i . . (4.146) Systems of Identical Particles 151

and so on. These basis states n1;n2;::: of the fermionic Fock space are also simultaneous j i eigenvectors of the operators Nº; and are uniquely labeled by their occupation numbers nº : Thus, the operators Nº form a CSCO for the fermionic Fock space as well. The fcompletenessg relation di¤ersf fromg that of the bosonic space only by the limits on the summations involved, i.e.,

1 1 n1;n2;::: n1;n2;::: =1: (4.147) n =0 n =0 ¢¢¢ j ih j X1 X2 As in the bosonic case, we now wish to introduce operators that change the occupation numbers. Clearly, however, we cannot model the fermion annihilation and creation operators directly on those of the harmonic oscillator, since to do so would result in a fermion number spectrum inappropriately identical to that of the bosons. In the fermion case we require that the number operator have eigenvalues restricted to the set 0; 1 : It is a remarkable fact that a very slight modi…cation to the commutationf g relations associated with the annihilation and creation operators of the harmonic oscillator yield a set of operators that have precisely the properties that we need. To partially motivate this modi…cation we note that the commutator [A; B] of two operators simply gives, if it is known, a rule or prescription for reversing the order of a product of these two operators whenever it is convenient to do so. Thus if we know the operator [A; B] then we can replace the operator AB wherever it appears with the operator BA +[A; B] : Any rule that allows us to perform a similar reversal would serve the same purpose. For example, it sometimes occurs that the anticommutator A; B of two operators, de…ned as the sum f g A; B = AB + BA (4.148) f g rather than the di¤erence of the operator product taken in each order, is actually a simpler operator than the commutator. Under these circumstances, we can use the anticommutator to replace AB whenever it occurs with the operator BA + A; B : With this in mind, we now consider an operator a¡and its adjointf g a+ which obey anticommutation relations that have the same structure as the commutation relations of those associated with the harmonic oscillator, i.e., suppose that

a; a =0= a+;a+ (4.149) f g © ª a; a+ =1: (4.150) Let us also de…ne, as in the oscillator© case,ª the positive operator N = a+a and let us assume that N has at least one nonzero eigenvector º ; square normalized to unity, with eigenvalue º; i.e., N º = º º : The following statementsj i are then easily shown: j i j i 1. a2 =0=(a+)2

2. aa+ =1 N ¡ 3. Spectrum(N)= 0; 1 f g 4. If º =0; then a 0 =0and a+ 0 = 1 ; i.e., a+ 0 is a square normalized eigenvector of N with eigenvaluej i 1: j i j i j i

5. If º =1; then a+ 1 =0and a 1 = 0 ; i.e., a 1 is a square normalized eigenvector of N with eigenvaluej i 0: j i j i j i 152 Many Particle Systems

The …rst item follows directly upon expanding the anticommutation relations a; a =0= a+;a+ . The second item follows from the last anticommutation relation fa; a+g = aa+f + a+ag =1; which implies that aa+ =1 a+a =1 N: To show that thef spectrumg can only contain the values 0 an 1; we multiply¡ the relation¡ aa+ + a+a =1 by N = a+a and use the fact that aa =0to obtain N 2 = N; which implies that N is a projection operator and so can only have the eigenvalues speci…ed . To show that both of these actually occur, we prove the last two items, which together show that if one of the eigenvalues occurs in the spectrum of N then so does the other. These statements follow from the following observations

a 0 2 = 0 a+a 0 = 0 N 0 =0 (4.151) jj j ijj h j j i h j j i a+ 0 2 = 0 aa+ 0 = 0 (1 N) 0 = 0 0 =1 (4.152) jj j ijj h j j i h j ¡ j i h j i

Na+ 0 = a+aa+ 0 = a+ (1 N) 0 j i j i ¡ j i = a+ 0 (4.153) j i which prove the fourth item, and

a+ 1 2 = 1 aa+ 1 = 1 (1 N) 1 =0 (4.154) jj j ijj h j j i h j ¡ j i a 1 2 = 1 a+a 1 = 1 N 1 = 1 1 =1 (4.155) jj j ijj h j j i h j j i h j i Na 1 = a+aa 1 =0 (4.156) j i j i which proves the …nal item. It should be clear that operators of this type have precisely the properties we require for creating and destroying fermions, since they only allow for occupation numbers of 0 and 1. We thus de…ne, for the fermion Fock space, a complete set of annihilation, + + creation, and number operators aº;aº ; and Nº = aº aº that remove, create, and count fermions in the single particle states Áº : To de…ne these operators completely, we need to specify the commutation propertiesj i obeyed by annihilation and creation operators associated with di¤erent single particle states. It turns out the antisymmetry of the states in this space under particle exchange require that operators associated with di¤erent states anticommute, so that, collectively, the operators associated with this occupation number representation obey the following fermion commutation relations

+ + aº;aº =0= aº ;a f 0 g º0 + © ª aº;a = ±º;º ; (4.157) º0 0 which are just like those for bosons,© exceptª for replacement of the commutator bracket with the anticommutator bracket. It is importnat to emphasize that, according to these de…ntions, operators associated with di¤erent single-particle states do not commute, they anticommute, which means for example that

+ + + + aº a = a aº : (4.158) º0 ¡ º0 Thus, the order in which particles are created in or removed from various states makes a di¤erence. This reversal of sign is reminiscent of, and stems from the same source, as the sign change that occurs when di¤erent direct product states in S(N) are projected into the anti-symmetric subspace, i..e., it arises from the antisymmetry exhibited by the states under particle exchange. Systems of Identical Particles 153

The action of these fermion annihilation and creation operators on the vacuum is essentially the same as for the boson operators, i.e.,

+ a 0 = Á aº 0 =0; (4.159) º j i j º i j i and in a similar fashion we can represent an arbitrary occupation number state of this representation in terms of the vacuum state through the expression

+ n1 + n2 + n3 n1;:::;nº ;::: =(a ) (a ) (a ) 0 : (4.160) j i 1 2 3 ¢¢¢j i Note that normalization factors are not necessary in this expression, since all the occupation numbers are equal to either 0 or 1: Also, the order of the creation operators in the expression is important, with higher occupied states (states with large values of º) …lled …rst, since the creation operators for such states are closer to the vacuum state being acted upon than those with lower indices. This way of representing the occupation number state n1;:::;nº;::: as an ordered array of creation operators acting on the vacuum is often referredj to as standardi form. When an annihilation or creation operator acts upon an arbitrary occupation number state the result depends upon the whether the corresponding single particle states is already occupied, as well as on the number and kind of single particle states already occupied. Speci…cally, it follows from the anticommutation relations above that

0 if nº =0 aº n1;:::;nº;::: = (4.161) j i 8 m ( 1) n1;:::;nº 1;::: if nº =1 < ¡ j ¡ i and : m ( 1) n1;:::;nº +1;::: if nº =0 + ¡ j i a n1;:::;nº;::: = (4.162) º j i 8 < 0 if nº =1 where : m = nº0 (4.163) º <º X0 m are the number of states with º0 <ºalready occupied. The phase factors ( 1) are ¡+ easily proven using the anticommutation relations. For example, the action of aº on the state n1;:::;nº ;::: can be determined by expressing the latter in standard form, and then anticommutingj i the creation operator through those with indices less than º until it is sitting in its standard position. Each anticommutation past another creation operator + + + + generates a factor of 1; since aº a = a aº : The product of all these factors gives 1 ¡ º0 ¡ º0 ¡ raised to the power m = nº : º0<º 0 It is worth noting that, although the fermion creation and annihilation operators P + anticommute, the corresponding number operators Nº = aº aº; actually commute with one another. This follows from the basic de…nition of these operators in terms of their action on the occupation number states, but is also readily obtained using the anticommutation relations. To see this, consider

+ + Nº Nº = aº aºa aº with º = º0; (4.164) 0 º0 0 6 and note that each time a+ is moved to the left one position it incurs a minus sign. When º0 it moves two positions, all the way to the left, we have a product of two minus signs, so + + Nº Nº = a a aºaº : But now we do the same thing with aº , moving it two positions to 0 º0 º 0 the left and so …nd that + + NºNº = a aº a aº = Nº Nº : (4.165) 0 º0 0 º 0 154 Many Particle Systems

+ + In a similar fashion, e.g., we deduce, using the anticommutator aº0 aº + aº aº0 = ±º;º0 ; that

+ + Nº aº0 = aº aºaº0 = aº aº0 aº + + ¡ = a aº aº aº±º;º = aº (Nº ±º;º ) (4.166) º0 0 ¡ 0 0 ¡ 0 which shows that [Nº;aº ]= aº ±º;º (4.167) 0 ¡ 0 0 which is the same commutation relation as for bosons. In a similar fashion it is readily + + established that aºa = ±ºº a aº º0 0 ¡ º0 + + + + + Nºa = a aºa = a (±ºº a aº) º0 º º0 º 0 º0 + + + ¡ + + = a ±ºº + a a aº = a ±ºº + a Nº (4.168) º0 0 º0 º º0 0 º0 which shows that + + Nº;a =+a ±º;º (4.169) º0 º0 0 also as for bosons. £ ¤ 4.4.7 Observables of a System of Identical Particles Revisited Having compiled an appropriate set of operators capable of describing transitions between di¤erent occupation number states we now reconsider the form that general observables of a system of identical particles take when expressed as operators of Fock space. Recall that the problem that led to our introduction of Fock space was basically that di¤erent parts of the same operator (e.g. the Hamiltonian) are expressible as simple functions of number operators in di¤erent occupation number representations. The problem is similar to that encountered in simpler quantum mechanical problems where the question often arises as to whether to work in the position representation, the momentum representation, or some other representation altogether. We are thus led to consider how the occupation number representations associated with di¤erent sets of single particle states are related to one another. Recall that any orthonormal basis of single-particle states generates its own occupation number representation. Thus, e.g. an ONB of states Á º =1; 2; fj º ij ¢¢¢g generates a representation of states n1;:::;nº;::: that are expressible in terms of a set j i+ + of creation, annihilation, and number operators aº ;aº; and Nº = aº aº ; while a different ONB of states Â ¹ =1; 2; generates a di¤erent representation of states fj ¹ij ¢¢¢g n~1;:::;n~¹;::: ; say, expressible in terms of a di¤erent set of creation, annihilation, and j i + + number operators b¹ ;b¹; and Nº = b¹ b¹: We know, on the other hand, that the two sets of single-particle states are related to one another through a unitary transformation U such that, e.g., Á = U Â º =1; 2; (4.170) j ºi j ºi ¢¢¢ with matrix elements U¹º = Â U Â = Â Á (4.171) h ¹j j º i h ¹j ºi that are the inner products of one basis set in terms of the other. This allows us to write, e.g., that

Â¹ = Áº Áº Â¹ = U¹º¤ Áº (4.172) j i º j ih j i º j i X X where U¹º¤ = Â¹ Áº ¤ = Áº Â¹ : But we also know that the single-particle states Áº can be expressedh inj termsi h of thej i vacuum state through the relation j i

Á = a+ 0 : (4.173) j ºi º j i Systems of Identical Particles 155

Using this in the expression for Â ; we …nd that j ¹i + Â¹ = U¹º¤ Áº = U¹º¤ aº 0 j i º j i º j i X X = b+ 0 (4.174) ¹ j i wherewehaveidenti…edtheoperator

+ + b¹ = U¹º¤ aº (4.175) º X that creates Â¹ out of the vacuum. The adjoint of this relation gives the corresponding annihilationj operatori

b¹ = U¹º aº: (4.176) º X Thus the annihilation/creation operators of one occupation number representation are linear combinations of the annihilation/creation operators associated with any other occupation number representation, with the coe¢cients being the matrix elements of the unitary transformation connecting the two sets of single particle states involved. It is straightforward to show that this unitary transformation of annihilation and creation operators preserves the boson or fermion commuation relation that must be obeyed for each type of particle. To treat both types of particles simultaneously, we introduce the notation [A; B] = AB BA; where the minus sign stands for the commutator (and applies to boson operators)§ § and the plus sign stands for the anticommutator (which applies to + fermion operators). Thus, if the operators aº and aº obey the relations

+ + [aº ;aº0 ] =0= aº ;aº § 0 + £ ¤ aº;a = ±º;º (4.177) º0 0 § Then, using the transformation law£ we can¤ calculate the corresponding relation for the + operators b¹ and b¹ : Thus, e.g.,

[b¹;b¹0 ] = b¹b¹0 b¹0 b¹ § § = U¹ºU¹ º (aº aº aº aº ) 0 0 0 § 0 º;º X0

= U¹ºU¹0º0 [aº;aº0 ] =0 (4.178) § º;º X0 whereinthelastlinewehaveusedtheappropriaterelationsforeachtypeofparticle. The adjoint of this relation shows that b+;b+ =0; as well. In a similar fashion we see ¹ ¹0 that h i§

+ + b¹;b = U¹º U ¤ aº;a = U¹º U ¤ ±º;º = U¹ºU ¤ ¹0 ¹0º0 º0 ¹0º0 0 ¹0º § º;º § º;º º h i X0 £ ¤ X0 X = Â Á Á Â = Â Â = ± (4.179) ¹ º º ¹0 ¹ ¹0 ¹;¹0 º h j ih j i h j i X wherewehaveusedthecompletenessofthestates Áº and the orthonormality of the + j i states Â¹ : Thus, the b’s and b ’s obey the same kind of commutation/anticommutation relationsj asi the a’s and a+’s. 156 Many Particle Systems

We are now in a position to see what di¤erent one-body and two-body operators look like in various representations. Suppose, e.g., that H1 is a one-body operator that is represented in the occupation number states generated by the single-particle states Â¹ in the form j i + H1 = "¹N¹ = "¹b¹ b¹: (4.180) ¹ ¹ X X This implies, e.g., that in the space of single-particle, the states Â are the associated j ¹i eigenstates of H1; i.e., H1 Â¹ = "¹ Â¹ : To …nd the form that this takes in any other occupation number representationj i wej simplyi have to express the annihilation and creation operators in (4.180) as the appropriate linear combinations of the new annihilation and creation operators, i.e.,

+ H1 = "¹U¹º¤ U¹0º0 aº aº0 º;º ¹ X0 + = aº Hºº0 aº0 (4.181) º;º X0 where

H = U ¤ " U = Á Â " Â Á ºº0 ¹º ¹ ¹0º0 º ¹ ¹ ¹ º0 ¹ ¹ h j i h j i X X = Á H(1) Á (4.182) h ºj 1 j º0 i in which we have identi…ed the expansion

(1) H1 = Â¹ "¹ Â¹ (4.183) ¹ j i h j X of the operator H1; as it is de…ned in the one-particle subspace. Although we have expressed this in a notation suggestive of Hamiltonians and energy eigenstates, the same considerations apply to any one-body operator. Thus, a general one body operator can be represented in Fock space in an arbitrary occupation number representation in the form

+ (1) B1 = a Bºº aº Bºº = Á B Á : (4.184) º 0 0 0 h ºj 1 j º0 i º;º X0 In the special case when B is diagonal in the speci…ed single particle representation, the double sum collapses into a single sum, and the resulting operator is reduced to a simple function of the number operators of that representation. Thus, a one-body operator B induces single particle transitions, taking a particle out of state Á and putting it into º0 state Áº with amplitude Bºº0 : As we noted earlier, two body operators are often associated with interactions betweenparticles.Often,arepresentationcanbefoundinwhichsuchanoperatorcanbe expressed in the following form 1 1 H2 = N¹N¹ V¹¹ + N¹(N¹ 1)V¹¹ (4.185) 2 0 0 2 ¡ ¹;¹0 ¹ ¹X=¹0 X 6 where in the …rst term V¹¹0 is the interaction energy between a particle in the state Â¹ and another particle in a di¤erent state Â : The second term includes the interactionsj i ¹0 between particles in the same states, andj takesi this form because a particle in a given Systems of Identical Particles 157

1 state does not interact with itself (Note that N¹(N¹ 1) is the number of distinct pairs 2 ¡ of N¹ particles). Both terms can be combined by inserting an appropriate Kronecker delta, i.e., 1 H2 = N¹(N¹ ±¹¹ )V¹¹ : (4.186) 2 0 ¡ 0 0 ¹;¹ X0

This can be simpli…ed further. Using the commutation laws [N¹0 ;b¹]= b¹±¹¹0 obeyed by both fermion and boson operators (see the discussion in the last section)¡ it follows that

N¹ b¹ = b¹(N¹ ±¹¹ ): (4.187) 0 0 ¡ 0 + Multiplying this on the left by b¹ ; we deduce that

+ + + b¹ N¹ b¹ = b¹ b b¹ b¹ = N¹(N¹ ±¹¹ ); (4.188) 0 ¹0 0 0 ¡ 0 which allows us to write 1 + + H2 = b¹ b V¹¹ b¹ b¹: (4.189) 2 ¹0 0 0 ¹;¹ X0 Thus, in this form the two-body interaction is a sum of products involving two annihilation and two creation operators, but it only involves two summation indices, since each annihilation operator is paired o¤ with a creation operator of each type. To see what this looks like in any other occupation number representation we just have to transform each of the annihilation and creation operators in the sum. Thus we …nd that in a representation associated with a set of states Á = U Â ; j º i j º i

1 + + H2 = b¹ b V¹¹ b¹ b¹ 2 ¹0 0 0 ¹;¹ X0 1 + + = U¹q¤ U¹¤ rU¹ sU¹tV¹¹ aq a asat: (4.190) 2 0 0 0 r0 q;r;s;t ¹;¹ X X0 wherewehaveusedtheromanindicesq; r; s; and t to avoid the proliferation of multiply- primed º’s. To simplify this we now re-express the matrix elements of the unitary transformation in terms of the inner products between basis vectors

U ¤ U¤ U¹ sU¹tV¹¹ = Á Â Á Â V¹¹ Â Á Â Á (4.191) ¹q ¹0r 0 0 h qj ¹ih rj ¹0 i 0 h ¹0 j sih ¹j ti ¹;¹ ¹;¹ X0 X0 and notice that each pair of inner products on the right and left of V¹¹0 can be expressed as a single inner product between direct product states in the space S(2) of just two particles, i.e.,

Á Â Á Â V¹¹ Â Á Â Á = Á ;Á Â ;Â V¹¹ Â ;Â Á ;Á h qj ¹ih rj ¹0 i 0 h ¹0 j sih ¹j ti h q rj ¹ ¹0 i 0 h ¹ ¹0 j t si ¹;¹ ¹;¹ X0 X0 = Á ;Á H(2) Á ;Á (4.192) h q rj 2 j t si where the order of the terms has been chosen to reproduce the original set of four inner products, and where we have identi…ed

(2) H = Â ;Â V¹¹ Â ;Â (4.193) 2 j ¹ ¹0 i 0 h ¹ ¹0 j ¹;¹ X0 158 Many Particle Systems as the form that this operator takes in the space S(2) of just two particles. Working our way back up, we …nd that in an arbitrary occupation number representation, a general two-body interaction can be written in the form 1 H = a+a+V a a (4.194) 2 2 q r qrts s t q;r;s;t X where (2) Vqrts = Á ;Á H Á ;Á (4.195) h q rj 2 j t si is the matrix element of the operator taken between states of just two (distinguishable) particles. Thus, to construct such an operator for an arbitrary occupation number representation we simply need to be able to takes its matrix elements with respect to the corresponding set of two-particle direct-product states. As an example, we note that the Coulomb interaction e V (~r1;~r2)= (4.196) ~r1 ~r2 j ¡ j between particles can be written in the form

1 + + V = aq a Vqrtsasat (4.197) 2 r0 q;r;s;t X where the matrix elements are evaluated, e.g., in the two-particle position representation as

Vqrts = Á ;Á V Á ;Á h q rj j t si 3 3 e = d r d r Á¤(~r)Á¤(~r ) Á (~r)Á (~r ) (4.198) 0 q r 0 ~r ~r t s 0 Z Z j ¡ 0j in terms of the wave functions associated with this set of single particle states. 4.4.8 Field Operators and Second Quantization The use of creation and annihilation operators of the type we have just considered is often referred to as the method of second quantization . The “…rst quantization” implied by this phrase is that developed, e.g., by Schrödinger, in which the dynamical variables xi and pi of classical mechanics are now viewed as operators, and the state of the system is characterized by wave functions Ã(xi;t) or Ã(pi;t) of one or another of this set of variables. On the other hand, there are other systems studied by classical mechanics that cannot be described as particles, e.g., waves traveling on a string or through an elastic medium, or Maxwell’s electric and magnetic …eld equations, where the classical dynamical variables are just the …eld amplitudes Ã(x; t) at each point, which like the position and momentum of classical particles always have a well-de…ned value, and where x is now simply viewed as a continuous index labeling the di¤erent dynamical variables Ã(x) that are collectively needed to fully describe the con…guration of the system. In a certain sense, Schrödinger’s wave function Ã(x; t) for a single particle can also be viewed “as a sort of classical …eld” and Schrödinger’s equation of motion can be viewed as simply the “classical” wave equation obeyed by this …eld. One can then ask what happens when this classical system is quantized, with the corresponding …eld amplitudes being associated with operators. It is actually possible to follow this path from classical …elds to quantum ones through a detailed study of the objects of classical …eld theory, including Lagrangian densities, conjugate …elds, and so on. Systems of Identical Particles 159

As it turns out, however, the end result of such a process is actually implicitly contained in the mathematical developments that we have already encountered. The key to seeing this comes from the realization that the procedure for transforming between di¤erent occupation number representations applies, in principle, to any two sets of single-particle states. Until now we have focused on transformations between discrete ONB’s, e.g., Áº and Â¹ ; but it is possible to consider transformations that include continuouslyfj indexedig basisj setsi as well. A case of obvious© interestª would be the basis states ~r of the position representation. The transformation law between these states andfj thoseig of some other single- particle representation, e.g., Á takes the form fj ºig

3 3 3 Á = d r ~r ~r Á = d rÁ (~r) ~r = d rUº (~r) ~r (4.199) j ºi j ih j ºi º j i j i Z Z Z and

~r = Áº Áº ~r = Áº¤(~r) Áº = Uº¤(~r) Áº (4.200) j i º j ih j i º j i º j i X X X where Uº (~r)= ~r Áº and Uº¤(~r)= Áº ~r are simply the wave functions (and conjugates) for the single particleh j i states Á inh thej i position representation. Expressing, the single- j º i particle state Áº in terms of the vacuum state, and substituting into the expansion for the state ~r ; wej …ndi that j i + ~r = Áº¤(~r)aº 0 : (4.201) j i º j i X This allows us to identify the operator that creates out of the vaccum a particle at the point ~r; (i.e., in the single-particle state ~r ) as a linear combination of creation operators j i associated with the states Áº . We will denote this new creation operator with the symbol + j i Ã^ (~r); i.e., ^+ + Ã (~r)= Áº¤(~r)aº : (4.202) º X + The adjoint of the operator Ã^ (~r) gives the corresponding annihilation operator

^ Ã(~r)= Áº(~r)aº (4.203) º X These two families of operators are referred to as …eld operators, since they de…ne an + operator-valued …eld of the real space position parameter ~r.ThusÃ^ (~r) creates a particle at ~r; and Ã^(~r) destroys or removes a particle from that point. The fact that the basis states ~r of this single-particle representation are not square-normalizable leads to some slightj buti fairly predictable di¤erences between the …eld operators and the annihilation and creation operators associated with discrete representations. For example, the commutation/anticommutation relations obeyed by the …eld operators now take a form more appropriate to the continuous index associated with this set of operators. The transformation law is derived as in the discrete case, and we …nd that

^ ^ Ã(~r); Ã(~r 0) = Áº(~r)Áº (~r 0)[aº;aº0 ] =0 (4.204) 0 § § º;º h i X0

+ + + + Ã^ (~r); Ã^ (~r 0) = Á¤(~r)Á¤ (~r 0) a ;a =0 (4.205) º º0 º º0 § º;º § h i X0 £ ¤ 160 Many Particle Systems and

+ + + Ã^(~r); Ã^ (~r 0) = Á (~r)Á¤ (~r 0) a ;a º º0 º º0 § º;º § h i X0 £ ¤ = Á (~r)Á¤ (~r 0)± = Á (~r)Á¤(~r 0) º º0 ºº0 º º º;º º;º X0 X = ~r Áº Áº ~r 0 = ~r ~r 0 º;º h j ih j i h j i X = ±(~r ~r 0) (4.206) ¡ + Also, as a consequence of the normalization, the product of Ã^ (~r) and Ã^(~r) does not give a number operator, but a number density operator, i.e.,

+ n^(~r)=Ã^ (~r)Ã^(~r) (4.207) counts the number of particles per unit volume at the point ~r. That the product has the correct units to describe a number density follows directly from the commutation relations just derived. From this we can de…ne number operators N that count the number of particles in any region of space ; as the integral

3 + N = d r Ã^ (~r)Ã^(~r) (4.208) ~r Z 2 with the total number operator N obtained by extending the integral to all space. Finally, we can express various one-body and two body operators using this representation, by sightly extending our results obtained with discrete representations. Thus, e.g., a general one-body operator can be expressed in terms of the …eld operators through the expression

3 3 + (1) H1 = d r d r0 Ã^ (~r)H1(~r;~r 0)Ã^(~r) H1(~r;~r 0)= ~r H ~r 0 : (4.209) h j 1 j i Z Z For a collection of noninteracting identical particles moving in a common potential, e.g.,

2 2 P ~ 2 H(~r;~r 0)= ~r ~r 0 + ~r V ~r 0 = ±(~r ~r 0)+V (~r)±(~r ~r 0) (4.210) h j2mj i h j j i ¡2mr ¡ ¡ which reduces the previous expression to the familiar looking form

2 H = d3r Ã+(~r) ~ 2Ã(~r)+Ã+(~r)V (~r)Ã(~r) (4.211) ¡2m r Z · µ ¶ ¸ which, it is to be emphasized is an operator in Fock space, although it looks just like a simple expectation value. Similarly, a general two-body operator can be expressed in the somewhat more cumbersome form

3 3 3 3 + + H2 = d r1 d r2 d r3 d r4 Ã^ (~r1)Ã^ (~r2) ~r1;~r2 H2 ~r3;~r4 Ã^(~r4)Ã^(~r3): (4.212) h j j i Z Z Z Z The form that this takes for the Coulomb interaction is left as an exercise for the reader. Chapter 5 APPROXIMATION METHODS FOR STATIONARY STATES

As we have seen, the task of prediciting the evolution of an isolated quantum mechanical can be reduced to the solution of an appropriate eigenvalue equation involving the Hamiltonian of the system. Unfortunately, only a small number of quantum mechanical systems are amenable to an exact solution. Moreover, even when an exact solution to the eigenvalue problem is available, it is often useful to understand the behavior of the system in the presence of weak external ﬁelds that my be imposed in order to probe the structure of its stationary states. In these situations an approximate method is required for calculating the eigenstates of the Hamiltonian in the presence of a perturbation that renders an exact solution untenable. There are two general approaches commonly taken to solve problems of this sort. The ﬁrst, referred to as the variational method, is most useful in obtaining information about the ground state of the system, while the second, generally referred to as time-independent perturbation theory, is applicable to any set of discrete levels and is not necessarily restricted to the solution of the energy eigenvalue problem, but can be applied to any observable with a discrete spectrum.

5.1 The Variational Method Let H be a time-independent observable (e.g., the Hamiltonian) for a physical system having (for convenience) a discrete spectrum. The normalized eignestates φn of H each satisfy the eigenvalue equation {| i}

H φ = En φ (5.1) | ni | ni where for convenience in what follows we assume that the eigenvalues and corresponding eigenstates have been ordered, so that

E0 E1 E2 . (5.2) ≤ ≤ ··· Under these circumstances, if ψ is an arbitrary normalized state of the system it is straightforward to prove the following| i simple form of the variational theorem:the mean value of H with respect to an arbitrary normalized state ψ is necessarily greater than the actual ground state energy (i.e., lowest eigenvalue) of H| ,i i.e.,

H ψ = ψ H ψ E0. (5.3) h i h | | i ≥ The proof follows almost trivially upon using the expansion

H = φn En φn (5.4) n | i h | X of H in its own eigenstates to express the mean value of interest in the form

2 H ψ = ψ φn En φn ψ = ψn En, (5.5) h i n h | i h | i n | | X X 134 Approximation Methods for Stationary States

2 2 and then noting that each term in the sum is itself bounded, i.e., ψn En ψn E0, so that | | ≥ | | 2 2 ψn En ψn E0 = E0 (5.6) n | | ≥ n | | X X 2 where we have used the assumed normalization ψ ψ = n ψn =1of the otherwise arbitrary state ψ . Note that the equality holds onlyh | ifi ψ is actually| | proportional to the ground state of|Hi. | iP Thus, the variational theorem proved above states that the ground state minimizes the mean value of H taken with respect to the normalized states of the space. This has interesting implications. It means, for example, that one could simply choose random vectors in the state space of the system and evaluate the mean value of H with respect to each. The smallest value obtained then gives an upper bound for the ground state energy of the system. By continuing this random, or “Monte Carlo”, search it is possible, in principle, to get systematically better (i.e., lower) estimates of the exact ground state energy. It is also possible to prove a stronger statement that includes the simple bound given above as a special case: the mean value of H is actually stationary in the neighborhood of each of its eigenstates φ . This fact, which is a more complete and precise statement of the variational theorem,| i is compactly expressed in the language of the cal- culus of variations through the relation

δ H φ =0. (5.7) h i To see what this means physically, let φ be a normalizable state of the system about which we consider a family of kets | i

φ(λ) = φ + λ η (5.8) | i | i | i that diﬀer by a small amount from the original state φ , where η is a ﬁxed but arbitrary normalizeable state and λ is a real parameter allowing| i us to parameterize| i the small but arbitrary variations δ φ = φ(λ) φ = λ η (5.9) | i | i − | i | i of interest about the ket φ = φ(0) . Let us now denote| i by | i

φ(λ) H φ(λ) ε(λ)= H λ = h | | i (5.10) h i φ(λ) φ(λ) h | i the mean value of H with respect to the varied state φ(λ) , inwhichwehaveincluded the normalization in the denominator so that we do not| havei to worry about constraining the variation to normalized states. With these deﬁnitions, then, we wish to prove the following: the state φ is an eigenstate of H if and only if, for arbitrary η , | i | i ∂ε =0. (5.11) ∂λ ¯λ=0 ¯ ¯ To prove the statement we ﬁrst compute¯ the derivative of ε(λ) using the chain rule, i.e., introducing the notation

∂ φ(λ) ∂ φ(λ) φ0 = | i = η φ0 = h | = η (5.12) | i ∂λ | ih| ∂λ h | The Variational Method 135 we have (since H is independent of λ)

∂ε φ0 H φ φ H φ0 φ H φ = h | | i + h | | i h | | i φ0 φ + φ φ0 . (5.13) ∂λ φ φ φ φ − φ φ 2 h | i h | i ¯λ=0 h | i h | i h | i ¯ £ ¤ ¯ This can be multiplied¯ through by φ φ and the identity φ0 = η used to obtain the relation h | i | i | i ∂ε φ H φ φ φ = η H φ + φ H η h | | i [ η φ + φ η ] . (5.14) h | i ∂λ h | | i h | | i − φ φ h | i h | i ¯λ=0 ¯ h | i Now denote by E the mean¯ value of H with respect to the unvaried state, i.e., set ¯ φ H φ E = ε(0) = h | | i (5.15) φ φ h | i and write E [ η φ + φ η ]= η E φ + φ E η toputtheaboveexpressionintheform h | i h | i h | | i h | | i ∂ε φ φ = η (H E) φ + φ (H E) η . (5.16) h | i ∂λ h | − | i h | − | i ¯λ=0 ¯ ¯ We now note that if φ is an¯ actual eigenstate of H its eigenvalue must be equal to E = ε(0), in which case| i the right hand side of the last expression vanishes (independent of the state η ). Since φ is nonzero, we conclude that the derivative in (5.11) and (5.16) vanishes for| arbitraryi variations| i δ φ = λ η about any eigenstate φ of H. To prove the converse we| notei that,| i if the derivative of ε|(λi) with respect to λ does indeed vanish for arbitrary kets η , then it must do so for any particular ket we choose; for example, if we pick | i η =(H E) φ . (5.17) | i − | i then, (5.16) above reduces to the relation 0= η (H E) φ + φ (H E) η = φ (H E)2 φ . (5.18) h | − | i h | − | i h | − | i Because, by assumption, H is Hermitian and E real we can now interpret this last equation as telling us that φ (H E)2 φ = (H E) φ 2 =0, (5.19) h | − | i || − | i|| which means that the vector (H E) φ must vanish, and that φ is therefore an eigenstate of H with eigenvalue E whenever− the| i derivative (5.11) vanishes| i for arbitrary variations δ φ = λ η , completing the proof. | i In| i practice, use of this principle is referred to as the variational method, the basic steps of which we enumerate below: 1. Choose an appropriate family φ(α) of normalized trial kets which depend pa- {| i} rameterically on a set of variables α = α1, α2, , αn , referred to as variational parameters. { ··· } 2. Calculate the mean value H(α) = φ(α) H φ(α) (5.20) h i h | | i as a function of the parameters α. 3. Minimize E(α)= H(α) with respect to the variational parameters by ﬁnding the h i values α0 for which ∂ H h i =0 i =1, 2, ,n. (5.21) ∂αi ··· ¯α=α0 ¯ ¯ ¯ 136 Approximation Methods for Stationary States

The value E(α0) so obtained is the variational estimate of the ground state energy with respect to this family of trial kets, and the corresponding state φ(α0) provides the corresponding variational approximation to the ground state. | i It should be noted that if the family φ(α) of trial kets actually contains the ground state (or any excited state), the variational{| i} principle shows that the technique described above will find it and the corresponding energy exactly. This can sometimes be exploited. For example, if symmetry properties of the ground state are known (parity, angular momentum, etc.) it is often possible to choose a family of trial kets that are orthogonal to the exact ground state of the system. In this situation, the variational method will then yield an upper bound to the energy of the lowest lying excited state of the system that is not orthogonal to the family of trial kets employed. We also note, in passing, that our derivation of the variational principle shows that an approximation to the ground state φ0 that is correct to order ε (i.e., φ(ε) = φ0 + ε η ) will yield an | i | i | 2 i | i estimate of the ground state energy E0 which is correct to order ε . This follows from the fact that in an expansion ∂E ε2 ∂2E E (ε)=E0 + ε + + (5.22) ∂ε 2! ∂ε2 ··· ¯ε=0 ¯ε=0 ¯ ¯ ¯ ¯ of the mean energy about that of the actual¯ ground state,¯ the linear term vanishes due to the stationarity condition derived above. This explains the often observed phenomenon that a rather poor approximation to the eigenstate can yield a relatively good estimate of the ground state energy. A particulalry useful application of the variational method involves what is referred to as the Rayleigh-Ritz method, which overcomes to some extent the usual difficulty of dealing with an infinite dimensional space. Suppose for example that we were to take as a trial ket a state φ = φ i (5.23) | i i| i i X expanded in terms of some orthonormal set of vectors i , and take the expansion {| i} coefficients φi (or their real and imaginary parts) as our variational parameters. If the set i is complete, then the family φ of trial kets includes all physical vectors in the state {|space,i} and the resulting variational{| procedurei} will just generate the exact eigenvectors of H. Suppose, on the other hand, that the states i are not complete, but span some {| i} N dimensional subspace SN . The variational procedure would then search through this finite− dimensional subspace to find those states that are closest to being actual eigenstates of the full system. The resulting vectors would then extremize the mean value

H = φ H φ = φ∗Hijφ (5.24) h i h | | i i j i,j X taken with respect to the states φ in this subspace. In this last expession, Hij = i H j denotes the matrix elements of H| iwith respect to the orthonormal states i spanningh | | i {| i}(S) the subspace SN . Suppose, however, we introduce a new Hermitian operator H deﬁned only on the subspace SN and having the same matrix elements

(S) H = i H j = Hij (5.25) ij h | | i as H within that subspace (but which vanishes outside of SN );theN eigenvectors of this (S) restricted operator H will be those states φ in SN which extremize the mean value | i (S) (S) H = φ∗H φ = φ∗Hijφ , (5.26) h i i ij j i j i,j i,j X X Perturbation Theory for Nondegenerate Levels 137 i.e., they will be precisely the variational eigenstates of the full Hamiltonian H that we are looking for. Thus, in this case, application of the variational method simply amounts to diagonalizing the matrix representing H restricted to some ﬁnite subspace of interest. Moreover, as implied by our previous comments on the variational method, the Rayleigh- Ritz method described above will exactly ﬁnd any actual eigenvectors of H that lie entirely within the chosen subspace SN .

5.2 Perturbation Theory for Nondegenerate Levels We now turn to a more general and systematic method for determining the eigenvector and eigenvalues for observables with a discrete spectrum. As in the last section we use the language of energy eigenstates and Hamiltonia even though the method itself is perfectly applicable to other observables. For the purposes of stating the initial problem of interest, however, we consider the eigenvalue problem for a time-independent Hamiltonian H = H(0) + H(1) = H(0) + λV (5.27) having a discrete nondegenerate spectum.InwritingtheHamiltonianinthisform,the eigenvalue problem for the operator H(0), which will be referred to as the “unperturbed part” of the Hamiltonian, is assumed to have been solved, and the perturbation H(1) = λV is presumed to be, in some sense, small compared to H(0). Our goal is to obtain expressions for the eigenstates n = n(λ) and eigenvalues εn = εn(λ) of H as an expansion in powers of the small, real| parameteri | iλ. These eigenstates of the full Hamiltonian H are to be expressed as linear combinations and simple functions of the known eigenstates n(0) and (0) (0) | i eigenvalues εn of the unperturbed Hamiltonian H . Thus, the exact and unperturbed states of the system are assumed to satisfy the equations

(H εn) n =0 n m = δn,m n n =1 (5.28) − | i h | i n | ih | X (0) (0) (0) (0) (0) (0) (0) (H εn ) n =0 n m = δn,m n n =1, (5.29) − | i h | i n | ih | X the two relations on the right of the last two lines indicating that both sets of states form an ONB for the space of interest. We wish to identify, in particular, the unperturbed eigenstates n(0) as those to which the exact states n tend as λ 0. This still leaves the relative| phasei of the two sets of basis vector undetermined,| i as we→ could multiply the basis vectors of one set by an arbitrary set of phases eiφn without aﬀecting the validity of the equations above. For nonzero values of λ, therefore, we further ﬁx the relative phase between these two sets of states by requiring that the inner product n n(0) between corresponding elements of these two basis sets be real and positive. h | i Now, by assumption, there exist expansions of the full eigenstate n and the | i corresponding eigenenergy εn of the form n = n(0) + λ n(1) + λ2 n(2) + (5.30) | i | i | i | i ··· (0) (1) 2 (2) εn = ε + λε + λ ε + (5.31) n n n ··· k (k) k (k) We will refer to the terms λ εn and λ n as the kth order correction to the nth eigenenergy and eigenstate, respectively. The| correspondingi correction to the energy is also generally referred to as the kth order energy shift, for obvious reasons. To determine these corrections, we will simply require that the exact eigenstate n satisfy the appropriate eigenvalue equation | i

(H εn) n =0 (5.32) − | i 138 Approximation Methods for Stationary States

to all orders in λ. Upon substitution of the expansions for n and εn into the eigenvalue equation we obtain | i

∞ ∞ ∞ ∞ (H(0) + λV ) λk n(k) = λkε(k) λj n(j) = λj+kε(k) n(j) (5.33) | i n | i n | i k=0 k=0 j=0 k,j=0 X X X X or ∞ ∞ λkH(0) n(k) + V λk+1 n(k) λj+kε(k) n(k) =0. (5.34)  | i | i − n | i k=0 j=0 X X For this equation to hold for small but arbitrary values of λ, the coefficients of each power of that parameter must vanish separately. The reason for this is essentially that the k polynomials fk(λ)=λ form a linearly independent set of functions on R,soanyrelation k of the form k∞=0 ckλ =0can only be satisfied for all λ in R if ck =0for all k.Applying this requirement to the last equation generates an infinite heirarchy of coupled equations, one for eachP power of λ. The equation generated by setting the coefficient of λk equal to zero is referred to as the kth order equation. Collecting coefficients of the first few powers of λ we obtain after a little rearrangement the zeroth order equation

(H(0) ε(0)) n(0) =0, (5.35) − n | i the ﬁrst order equation

(H(0) ε(0)) n(1) +(V ε(1)) n(0) =0, (5.36) − n | i − n | i the second order equation

(H(0) ε(0)) n(2) +(V ε(1)) n(1) ε(2) n(0) =0, (5.37) − n | i − n | i − n | i the third order equation

(H(0) ε(0)) n(3) +(V ε(1)) n(2) ε(2) n(1) ε(3) n(0) =0, (5.38) − n | i − n | i − n | i − n | i and ﬁnally, after inspecting those which precede it, we deduce for k 2 theformofthe general kth order equation ≥

k (0) (0) (k) (1) (k 1) (j) (k j) (H ε ) n +(V ε ) n − ε n − =0. (5.39) − n | i − n | i − n | i j=2 X As we will demonstrate, the structure of these equations allows for the general kth order solutions to be obtained from those solutions of lower order, allowing for the development of a systematic expansion of the eigenstates and eigenenergies in powers of λ.Tobegin the demonstration we note that the zeroth order equation (5.35) is already satisfied, by assumption. From knowledge of the unperturbed states and eigenenergies, Eq. (5.36) can (1) be solved to give the first order correction εn to the energy. This is most easily done by simply multiplying (5.36) on the left by the unperturbed eigenbra n(0) , i.e., h | n(0) (H(0) ε(0)) n(1) + n(0) (V ε(1)) n(0) =0. (5.40) h | − n | i h | − n | i (0) (0) (0) (0) (0) Since H is Hermitian (and εn therefore real) it follows that n (H εn )=0, and so the first order equation reduces to the relation h | −

ε(1) = n(0) V n(0) λε(1) = n(0) H(1) n(0) . (5.41) n h | | i n h | | i Perturbation Theory for Nondegenerate Levels 139

Thus, the first order correction to the energy eigenvalue for the nth level is simply the mean value of the perturbing Hamiltonian taken with respect to the unperturbed eigenfunctions. Note that the first order correction to the energy comes from a mean value taken with respect to the zeroth order approximation to the state, consistent with the remarks made earlier in the context of the variational method. As we will see, a similar structure persists to all orders of perturbation theory, namely, an approximation of the state to kth order generates an approximation to the energy that is correct to order k +1. (1) Now that we have εn , we can put it back in to the first order equation (5.36) to find an expansion for the first order correction n(1) totheeigenstate.Sincewewantto express this correction as an expansion | i

n(1) = m(0) m(0) n(1) (5.42) | i m | ih | i X in unperturbed eigenstates m(0) of H(0), we obviously need to evaluate the expansion coeffiicients m(0) n(1) . Thus,| wei now take inner products of the first order equation (5.36) with theh other| i members of this complete set of states, i..e., for the states with m = n. Multiplying (5.36) on the left by m(0) we obtain 6 h | m(0) (H(0) ε(0)) n(1) + m(0) (V ε(1)) n(0) =0 (5.43) h | − n | i h | − n | i and observe that for the first term on the left of this expression m(0) (H(0) ε(0)) n(1) =(ε(0) ε(0)) m(0) n(1) , (5.44) h | − n | i m − n h | i while orthogonality of the unperturbed states implies that, for the second term, m(0) ε(1) n(0) = ε(1) m(0) n(0) =0 for m = n. (5.45) h | n | i n h | i 6 Thus, we obtain after a little rearrangement the following result

(0) (0) (0) (1) m V n m n = h (0) | | (0) i m = n. (5.46) h | i − εm εn 6 − for the expansion coefficients of interest. This procedure for obtaining the expansion coefficients for the state n(1) does not work for the term with m = n, since it just leads, | (1) i (0) (0) again, to the expression εn = n V n for the first order energy correction. As it turns out, however, the one remainingh | | expansioni coefficient n(0) n(1) can be evaluated from the normalization condition n n =1and our alreadyh chosen| i phase convention. The normalization condition impliesh the| i expansion

1= n(0) + λ n(1) + λ2 n(2) + [ n(0) + λ n(1) + λ2 n(2) + ] h | h | h | ··· | i | i | i ··· h i = n(0) n(0) + λ n(1) n(0) + n(0) n(1) + O(λ2). (5.47) h | i h | i h | i h i Since, by assumption, n(0) n(0) =1, all of the remaining terms on the right-hand side of this expansion musth vanish,| term-by-term.i Thus, to ﬁrst order normalization of the eigenstates requires that

0= n(1) n(0) + n(0) n(1) =2Re n(0) n(1) . (5.48) h | i h | i h | i On the other hand, we have chosen our phase convention³ so that the´ inner product

n n(0) = n(0) + λ n(1) + λ2 n(2) + n(0) h | i h | h | h | ··· | i = hn(0) n(0) + λ n(1) n(0) + λ2 n(2)i n(0) + (5.49) h | i h | i h | i ··· 140 Approximation Methods for Stationary States is real and positive. Setting the imaginary part equal to zero gives the condition

∞ λk Im n(k) n(0) =0, (5.50) h | i k=1 X h i which again requires, for arbitrary λ, that each inner product in the sum be separately real. Combining this with (5.48) we deduce, therefore, that n(0) n(1) =0. (5.51) h | i By our simple choice of phase, then, the first order correction n(1) is forced to be orthogonal to the unperturbed eigenstate n(0) . Using this fact we| theni end up with the following expansion | i m(0) m(0) V n(0) n(1) = | ih | | i (5.52) | i − (0) (0) m=n εm εn X6 − for the first order correction, and similar expansions m(0) λV n(0) n = n(0) h | | i m(0) + O(λ2) (5.53) | i | i − (0) (0) | i m=n εm εn X6 − m(0) H(1) n(0) n = n(0) h | | i m(0) + O(λ2) (5.54) | i | i − (0) (0) | i m=n εm εn X6 − for the full eigenstate, correct to first order in the perturbation. This expression shows that the pertubation H(1) “mixes” the eigenstates of H(0), bywhichisreferredtothe fact that the eigenstates of H are linear combinations of the unperturbed eigenstates. (0) (0) Note also that the presence of the “energy denominators” εm εn appearing in the expansion coefficients in this expression tend to mix together states− close together in energy more strongly than states that are energetically disparate. This makes it clear why we assumed from the outset that the unperturbed spectrum was non-degenerate, since the method we have developed clearly must fail when applied to perturbations that connect degenerate states. It is also clear that an implicit condition for the perturbation expansion to converge, i.e., that the correction terms be sufficiently small is that

m(0) H(1) n(0) ε(0) ε(0) (5.55) |h | | i| ¿ m − n ¯ ¯ (0) (0) ¯ ¯ for all states m connected to the state n ¯by the perturbing¯ Hamiltonian. When| thei ﬁrst order correction to| thei energy vanishes, or higher accuracy is required, it is necessary to go to higher order in the perturbation expansion. To obtain the second order energy correction we proceed as follows: multiply the second order equation (5.37) on the left by the unperturbed eigenbra n(0) to obtain h | n(0) (H(0) ε(0)) n(2) + n(0) (V ε(1)) n(1) ε(2) n(0) n(0) =0. (5.56) h | − n | i h | − n | i − n h | i (0) Again using the fact that n is an eigenbra of H0, along with the orthogonality relation n(0) n(1) =0deduced above,h | we ﬁnd that h | i ε(2) = n(0) V n(1) . (5.57) n h | | i Inserting the expansion deduced above for n(1) , we then obtain the second order energy shift | i (0) (0) (0) (0) 2 n λV m m λV n λVmn λ2ε(2) = h | | ih | | i = | | (5.58) n − (0) (0) − (0) (0) m=n εm εn m=n εm εn X6 − X6 − Perturbation Theory for Nondegenerate Levels 141

(0) (0) (1) where the quantities λVmn = m λV n = Hmn are just the matrix elements of the perturbation between the unperturbedh | | eigenstates.i Thus, the full eigenenergies, correct to second order, are given by the expression

(1) 2 (0) (1) Hmn 3 εn = ε + H | | + O(λ ). (5.59) n nn − (0) (0) m=n εm εn X6 − In problems involving weak perturbations it usually suffiices to determine corrections and energy shifts to lowest non-vanishing order in the perturbation, and so it is unusal that one needs to go beyond second order for “simple” problems in perturbation theory. Exceptions to this general observation arise quite often when dealing with many-body problems, where diagramatic methods have been developed that take the ideas of perturbation theory to an extrememly high level, and where it is not uncommon to find examples where effects of the perturbation are calculated to all orders. It is worth pointing out, however, that the first order correction to the energy contains no information in it about any changes that occur in the eigenstates of the system as a result of the perturbation. This information appears for the first time in the second order energy shift, as the derivation above makes clear. In many cases it is not possible to perform the sum in (5.59) exactly, and so it is useful to develop simple means for estimating the magnitude of the second order energy shift. As it turns out, it is often straightforward to develop upper and lower bounds for the magnitude of the change in energy that occurs in any given eigenstate. For example, a general upper bound for the second order shift can be obtained for any nondegenerate level by observing that

(1) 2 (1) 2 Hmn Hmn ε(2) = | | | | (5.60) n (0) (0) ≤ (0) (0) m=n εn εm m=n εn εm X6 − X6 − ¯ ¯ ¯ ¯ where in the right hand side we have a sum of positive¯ definite¯ terms that will always be larger in magnitude than a similar sum in which some of the corresponding terms are positive and some negative, depending upon where the energy of each level lies relative to the one of interest. Moreover, each term in the sum on the right can itself be bounded, since the energy denominators are bounded from below by that associated with the level (0) closest in energy to the state n . If we denote by ∆εn the energy spacing between level | i (0) (0) n and the state closest in energy to it, then εn εm ∆εn and so − ≥ ¯ ¯ ¯ 2 ¯ (2) 1 (1)¯ 2 λ ¯ 2 εn Hmn = Vmn ≤ ∆εn | | ∆εn | | m=n m=n X6 X6 We can perform the infinite sum by “removing” the restriction on the summation index. 2 2 We do this latter trick by adding and subtracting the quantity Vnn = V n , where (0) (0) | | |h i | V n = Vnn = n V n is the mean value of the perturbation taken with respect to h i h | | i (1) the unperturbed eigenstate. (It is, essentially, just the first order energy correction εn ). Performing this operation allows us to write the upper bound above in the form

λ2 ε(2) m(0) V n(0) 2 V 2 . (5.61) n ∆ε n ≤ n " m |h | | i| − |h i | # X where the sum is now unrestricted. But since the unperturbed states form a complete set 142 Approximation Methods for Stationary States of states we can now rewrite the sum as

m(0) V n(0) 2 = n(0) V m(0) m(0) V n(0) m |h | | i| m h | | ih | | i X X(0) (0) 2 = n V n = V n (5.62) h | | i h i which is just the mean value of the square of the perturbation taken with respect to the unperturbed eigenstate. Making this substition above and recognizing the root-mean- square statistical uncertainty ∆2V = V 2 V 2 (5.63) h i − h i associated with the perturbing operator taken with respect to the unperturbed state of interest, we obtain our ﬁnal result for the upper bound

2 2 2 (1) (2) λ ∆ V ∆ H εn = (5.64) ≤ ∆εn ∆εn in the second order energy shift. We note in passing that this gives another intuitively reasonable measure for determining the validity of the perturbation expansion, which (2) (1) requires for the smallness of εn that the uncertainty in H be small relative to the (1) energy level spacing associated with the unperturbed states, i.e., that ∆H /∆εn << 1. For the ground state energy (or more generally the extremal eigenvalue) it is also possible to determine a lower bound on the magnitude of the second order energy shift. For the ground state, such a bound follows from the fact that in this case the second order energy shift (1) 2 (2) Hmn ε = | | (5.65) 0 − (0) (0) m=n εm ε0 X6 − is always negative (the change in state always leads to a lower energy, consistent with the variational principle), because the energy denominators are always positive. We can thus write (1) 2 (1) 2 (2) Hmn Hmn ε = | | | | (5.66) 0 (0) (0) ≥ (0) (0) m=n εm ε0 εm ε0 ¯ ¯ 6 − − ¯ ¯ X where in the right-hand side¯ we¯ have used the fact that any single term in the sum is less than or equal to the total sum of all the positive defnitite terms therein. The maximal term in the sum (which is usually one of the low-lying excited states closest in energy to the ground state) can thus be used to provide a reasonable lower bound for the the second order shift in the ground state energy. As an application of the techniques of nondegenerate perturbation theory we consider the example of a harmonically bound electron to which a uniform electric ﬁeld is applied. Thus, we take for our Hamiltonian

H = H0 + Vˆ (5.67) where P 2 1 H = + mω2X2 (5.68) 0 2m 2 is a simple one-dimensional harmonic oscillator describing the bound electron, and the perturbing ﬁeld is described by the potential

Vˆ = eEX = fX. (5.69) − − Perturbation Theory for Nondegenerate Levels 143

Our goal is to treat Vˆ as a small perturbation and calculate relevant corrections to the energy levels and eigenstates in the presence of the applied electric ﬁeld. To this end we recall the standard transformations mω P q = Xp= (5.70) √ r ~ m~ω 1 1 a = (q + ip) a+ = (q ip) N = a+a (5.71) √2 √2 − that allow us to put the harmonic oscillator part of the problem in a simpler, dimensionless form

1 ˆ ~ λ + H0 = N + ~ω V = f q = λq = (a + a) (5.72) 2 − mω − −√2 · ¸ r where λ = f ~ (5.73) rmω is a (presumed small) measure of the strength of the applied ﬁeld. With these deﬁnitions, the unperturbed states of H0 are the usual oscillator states n which obey | i 1 (0) N n = n n H0 n =(n + )~ω n = εn n n n0 = δn,n0 . (5.74) | i | i | i 2 | i | ih| i We also have the relations

a+ n = √n +1n +1 a n = √n n 1 (5.75) | i | i | i | − i in terms of which we readily determine that the ﬁrst order energy shift due to the applied ﬁeld

λ ε(1) = n Vˆ n = − n (a+ + a) n n h | | i √2h | | i λ λ = n a+ n + n a n = √n +1 n n +1 + √n n n 1 −√2 h | | i h | | i −√2 h | i h | − i =0 © ª © ª(5.76) vanishes due to the orthogonality of the unperturbed states. So the first order energy shift vanishes and we must go to second order to calculate the energy shift. Physically this vanishing of the first order energy shift occurs for the unperturbed states because they have equal weight on each side of the origin, and so the net change in energy due to the linear applied potential vanishes. We can anticipate that the second order correction will cause a lowering of the energy as the electron displaces in the presence of the field, lowering its potential energy in the process. To see this we first calculate the first order correction to the eigenstates. In the present problem we will denote by nˆ the exact eigenstates of the system that are presumed to have an expansion | i

nˆ = n + λ n (1) + λ2 n (2) + (5.77) | i | i | i | i ··· in powers of the small parameter λ. The ﬁrst order correction is given, according to the results of the last section, by the expression

m Vˆ n λ m (a+ + a) n λ n (1) = h | | i m = h | | i m . (5.78) | i (0) (0) | i −√ (n m) ω | i m=n εn εm 2 n =n ~ X6 − X06 − 144 Approximation Methods for Stationary States

Letting a and a+ act to the right we ﬁnd after a little calculation that, except for the ground state, this reduces to a sum of just two terms

λ √n +1 √n λ n (1) = n +1 + n 1 | i −√2 ~ω | i ~ω | − i · − ¸ λ = √n +1n +1 √n n 1 . (5.79) √2~ω | i − | − i £ ¤ Thus, to ﬁrst order the exact eigenstates of H can be written λ nˆ = n + √n +1n +1 √n n 1 + O(λ2). (5.80) | i | i √2~ω | i − | − i £ ¤ Thus, the perturbation mixes only the states immediately above and below the unperturbed level. Note that the corresponding expression for the ground state does not contain the second term in the above expression, i.e., λ 0ˆ = 0 + 1 + O(λ2). (5.81) | i | i √2~ω | i We now consider the second order energy shift

2 m Vˆ n ε(2) = h | | i (5.82) n ¯ (0) (0)¯ m=n ¯εn εm¯ X6 ¯ − ¯ which will also (except for the ground state) contain just two terms:

2 + 2 2 (2) λ n +1a n n 1 a n εn = |h | | i| + |h − | | i| 2 ~ω ~ω ( − ) λ2 n +1 n λ2 f 2 = + = = . (5.83) 2 − ω ω −2 ω −2mω2 ½ ~ ~ ¾ ~ Thus, to second order, we ﬁnd

2 2 (0) λ f εn = εn = . (5.84) − 2~ω −2mω2 We note that for this particular problem the energy shift is the same for all states, that is, all of the energies of the system are lowered by the same amount in the presence of the field. It turns out that the second order energy correction for this problem gives the exact eigenenergies (even though the first order correction to the state does not give the exact eigenstates). This result is physically inuitive, since it corresponds to the fact that a classical mass-spring system when hung in a gravitationl field simply stretches, or displaces, to a new equilibrium position, thereby lowering its potential energy, but continues to oscillate with the same frequency as it would if it were left unperturbed. To establish this result in the present context we perform a canonical transform to a new set of variables λ qˆ = q pˆ = p (5.85) − ~ω which has the position coordinate now centered at the new force center at q = λ/~ω. This transformation preserves the commutation relations

[ˆq, pˆ]=[q, p]=i (5.86) Perturbation Theory for Nondegenerate Levels 145 and allows us to write the Hamiltonian in terms of the new coordinate and momenta in the form ω H = ~ (q2 + p2) λq 2 − ω 2λ λ2 λ = ~ qˆ2 + qˆ+ +ˆp2 λ qˆ+ 2 ω 2ω2 − ω · ~ ~ ¸ µ ~ ¶ ω λ2 = ~ (ˆq2 +ˆp2) . (5.87) 2 − 2~ω We can now introduce operators

qˆ+ ipˆ qˆ ipˆ aˆ = aˆ+ = − √2 √2

Nˆ =ˆa+aˆ in the usual way, so that the Hamiltonian takes the form

1 λ2 H =(Nˆ + )~ω 2 − 2~ω 2 of an oscillator of frequency ω lowered uniformly in energy by an amount λ /2~ω. Of course this oscillator has its equilibrium position at qˆ = q λ/~ω =0, i.e., shifted with respect to the unperturbed oscillator, and its energy levels− are in exact agreement with those found to second order in the applied ﬁeld using perturbation theory. In fact, this displacement of the oscillator under the action of the ﬁeld makes it clear that the exact eigenstates of the system satisfy the equation

qˆ nˆ = φn(ˆq)=φn(q λ/~ω)=φn(q ε), h | i − − where ε = λ/~ω, i.e., they are just the unperturbed oscillator states shifted along the x-axis, and centered at the new equilibrium position q = ε. The unitary operator which eﬀects this transformation is the corresponding translation operator

ipε T (ε)=e− which, in the position representation has the eﬀect of displacing the wave function, i.e., T (ε)ψ(q)=ψ(q ε). Thus, we expect that the unperturbed and perturbed eigenstates are related through− the relation

ipε nˆ = T (ε) n = e− n . | i | i | i For small ε (or small λ) we can expand the exponential as T (ε) 1 ipε so that ' − nˆ (1 ipε) n = n ipε n . | i' − | i | i − | i Using the fact that ipε = ε(a+ a)/√2 and substituting back in the deﬁnition of − − ε = λ/~ω we recover the result λ λ nˆ n + n +1 n 1 | i'| i √2~ω | i − √2~ω | − i that we obtained using the techniques of ﬁrst order perturbation theory. 146 Approximation Methods for Stationary States

5.3 Perturbation Theory for Degenerate States The expressions that we derived above for the ﬁrst order correction to the ground state and the second order correction to the energy are clearly inappropriate to situations in which degenerate or nearly-degenerate eigenstates are connected by the perturbation, since the corresponding corrections all diverge as the spacing between the energy levels goes to zero. This divergence is an indication of the strength with which the perturbation tries to mix together states that are very-nearly degenerate (or exactly so), and suggest that we might wish to treat diﬀernyl those states that are known in advance to be very closely related in energy. In this section, therefore, we discuss the general approach taken to deal with problems of this sort. We assume, as before, that the Hamiltonian of interest can be separated into two parts, which we now write in the simplied form

H = H0 + V (5.88) where, since we will not be developing a systematic expansion in powers of the perturbation we have no need for the more complex notation used previously. We again seek the exact eigenstates and eigenenergies of H, expressed as an expansion in eigenstates of H0, the latter of which are assumed to be at least partially degenerate. We will denote by φ , τ an arbitrary othonormal basis of eigenstates of H0, where the index τ is in- {| n i} cluded to distinguish between the diﬀerent linearly independent basis states of H0 having (0) the same unperturbed energy εn .Thebasisstates

φ , τ τ =1, ,Nn (5.89) {| n i| ··· } (0) (0) with ﬁxed energy εn form a basis for an eigensubspace S(εn ) of H0 corresponding to that particular degenerate energy. The dimension Nn of this subspace is just the (0) degeneracy of the corresponding eigenvalue εn of H0. It is important to point out that, due to the degeneracy, our choice of the basis set φn, τ is not unique; any unitary {| i} (0) transformation carried out within any one of the eigenspaces S(εn ) generates a new basis χn, τ that can be used as readily as any other for expanding the exact eigenstates of H{| . Anyi} such basis set will satisfy the obvious eigenvalue, orthogonality, and completeness relations

(0) H φ , τ = ε φ , τ φ , τ 0 φ , τ = δ δ φ , τ φ , τ =1 0 n n n n0 n n0,n τ 0,τ n n | i | ih | i n,τ | ih | X (0) H χ , τ = ε χ , τ χ , τ 0 χ , τ = δ δ χ , τ χ , τ =1(5.90). 0 n n n n0 n n0,n τ 0,τ n n | i | ih | i n,τ | ih | X We now observe that the divergences that render the formulae of nondegenerate perturbation theory inapplicable really only arise if the perturbation actually connects states within each eigensubspace, i.e., if there exists non-zero matrix elements Vnτ,n0τ 0 = φn,τ V φn,τ 0 of the perturbation connecting basis states of the same energy. Thus, our hprevious| | formulaei can, in fact be applied (at least to the level that we have developed them), under two conceivable circumstances, one involving the diagonal matrix elements of V and one involving the oﬀ-diagonal elements:

(1) 1. If the ﬁrst order correction εn,τ = φn, τ V φn, τ is distinct for all the basis states (0) h | | i φn, τ in each eigenspace S(εn ), then the degeneracy is “lifted” in the ﬁrst order of the| perturbation.i Provided the magnitude of this splitting of the energy levels by the perturbation is large compared to the matrix elements of V that connect these states Perturbation Theory for Degenerate States 147

wecanthensimply“redeﬁne” what we call the unperturbed and the perturbing part of the Hamiltonian. In other words, although we orginally decomposed the Hamiltonian in the form H = H0 + V , where, in a representation of eigenstates of H0,

(0) H0 = φn, τ εn φn, τ n,τ | i h | X V = φ , τ Vnτ,n τ φ , τ 0 , | n i 0 0 h n0 | n,τ;n ,τ X0 0

we can now include the diagonal part of V in a redeﬁned Hˆ0 such that H = Hˆ0 +V,ˆ but now,

ˆ (0) (1) H0 = φn, τ [εn + εn,τ ] φn, τ n,τ | i h | X Vˆ = φ , τ Vnτ,n τ φ , τ 0 | n i 0 0 h n0 | n,τ=n ,τ X6 0 0 where the perturbation Vˆ now has no diagonal matrix elements in this representa- (0) tion. In this situation, states within S(εn ) are now no longer degenerate, so we can proceed as before to apply the formulae of non-degenerate perturbation theory, with the energy denominators now including the ﬁrst order shifts, so no divergences occur. 2. If, on the other hand, the oﬀ-diagonal part of the perturbation Vˆ just happens to (0) vanish between all the basis states within a given eigensubspace S(εn ), then (at least to second order) the problematic terms in the perturbation expansion never

actually arise; thus if the submatrix [V ]n representing the perturbation within the degenerate subspace is diagonal, we can actually proceed as though there were no degeneracy.

In passing we might comment regarding the first of these circumstances that the act of including the diagonal part of the perturbation V in a redefined Hˆ0 can always be performed, even when it does not entirely lift the degeneracy. We may therefore assume without loss of generality in what follows that such an operation has already been carried out, and hence that the perturbation has no diagonal components in the basis of interest. Regarding the second circumstance mentioned above, it might be thought that, (0) in actual practice, the vanishing of the off-diagoanl matrix elements of V within S(εn ) would occur in so few circumstances that it hardly merits attention. To the contrary, thereisasenseinwhichitalways can be made to occur. To understand this comment, and in a the process reveal the basic technique that is generally employed for dealing with degenerate states, we note that the off-diagonal matrix elements of the perturbation V taken between basis states in a given degenerate subspace of H0 depend upon which set of basis states of H0 we choose to begin with. If, e.g., we choose a set φn, τ we get one set of matrix elements {| i} Vnτ;nτ = φ , τ V φ , τ , 0 h n | | n i defining a certain submatrix [V ]n,whileifwechoose,instead,anyotherset χn, τ we obtain a completely different set of matrix elements {| i}

Vñτ;nτ = χ , τ V χ , τ 0 0 h n | | n i 148 Approximation Methods for Stationary States defining a different submatrix [V˜ ] representing the perturbation V within this degenerate eigensubspace. ˜ Thus, as we would expect, the submatrix [V ]n or [V ]n representing the pertur- (0) bation within the eigenspace S(εn ) depends upon the particular basis set ( φ , τ or {| n i} χn, τ ) that we chose to work in. In light of circumstance 2, above, the question that arises{| isi} the following: under what circumstances can we find a representation of basis (0) states χ , τ within S(εn ) for which the submatrix V˜ representing the perturbation {| n i} in that subspace is strictly diagonal? h i Insofar as the perturbation V itself is presumed to be an observable (and thus Hermitian), any submatrix [V ]n or [V˜ ]n representing V within such a subspace must itself be a Hermitian, and related to any of the other matrices representing V within this subspace by a unitary (sub)tranformation. For a finite Nn dimensional subspace, however, we know that it is always possible to find a representation that diagonalizes any Hermitian matrix. For each subspace we just have to go through the usual procedure of finding the roots ˜εn,τ to the characteristic equation

det ([V ]n ε)=0 − (0) for the Nn dimensional submatrix [V ]n that represents V within a given eigenspace S(εn ), and then solve the resulting linear equations to find those combinations χ , τ of the {| n i} original basis vectors φ , τ that are also eigenvectors of the submatrix [V ]n. In this {| n i} new representation, by construction, no elements of the new basis set χn, τ having the same energy unperturbed energy are connected to one another by{| nonzeroi} matrix elements of the perturbation. Moreover, the diagonalization of V within each eigenspace (0) S(εn ) provides a new set of eigenvalues ˜εn,τ (the roots of the characteristic equation det [V ε]n computed within the subspace) which will form the diagonal elements of the matrix− representing V in this representation. These diagonal elements can then be combined with those of H0 to obtain new unperturbed eigenenergies (correct to first order) (0) εn,τ = εn +˜εn,τ that will themselves often at least partially lift the degeneracy. The particular states found during the diagonalization can then be chosen as a new set of zeroth order states with which to pursue higher order corrections, according to the techniques of nondegenerate perturbation theory. Thus, the basic result of degenerate perturbation theory is not an explicit formula, as it is in the nondegenerate case. Rather it is a simple prescription: diagonalize the perturbation V within the degenerate subspaces of H0 to determine a new basis for proceeding, if necessary, with the determination of higher order corrections using standard techniques. 5.3.1 Application: Stark Effect of the n =2Level of Hydrogen We consider as an application of the ideas developed above the splitting of the spectral lines observed in the absorbtion and emission spectra of the hydrogen atom when it is placed in a uniform DC electric field, the so-called Stark effect. The relevant Hamiltonian can be written in the expected form

H = H0 + V, where p2 e2 H = 2m − r is the usual one describing a single electron bound to the proton of a hydrogen atom, and the perturbation V = Fz = Frcos θ Perturbation Theory for Degenerate States 149 describes the constant force F = eE exerted on the electron by a uniform ﬁeld oriented along the negative z-axis. We initially take as our unperturbed states the standard bound eigenstates n, l, m of the hydrogen atom | i (0) (0) ε0 H0 n, l, m = ε n, l, m ε = | i n | i n −n2 which have both a rotational and an accidental degeneracy of the energy levels. The degeneracy gn of the nth eigenenergy (or the dimension Nn of the associated eigenspace Sn) is given by the expression

n 1 2 − gn = n = (2` +1). `=0 X The first order correction to the energy of the state n, l, m due to the applied electric field vanishes, i.e., | i ε(1) = F nlm Z nlm =0 nlm h | | i reflecting the fact that the mean position of the electron in any of the standard hydrogen atom eigenstates is the origin. To use perturbation theory to find non-vanishing corrections to the energy due to the applied field we must handle the degeneracies. Consider, e.g., the four-fold degenerate n =2level, which is spanned by the four nlm states | i 2, 0, 0 2, 1, 0 2, 1, 1 2, 1, 1 . | i| i| i|− i

Within the subspace S2 the submatrix representing H0 is, of course, diagonal

(0) ε2 000 (0) 0 ε2 00 [H0]= (0)  . 00ε2 0  (0)   000ε   2    To proceed we need to construct the matrix [V ] representing the perturbation within this subspace. Thus we need to evaluate the matrix elements

3 2,l,mz 2,l0,m0 = d r ψ∗ zψ . h | | i 2,`,m 2,`0,m0 Z But the perturbing operator is clearly just the z component of the vector operator R.~ According to the Wigner-Eckart theorem such an operator can only connect states having the same z-component of angular momentum, i.e., those for which m = m0. Thus, of the states in the n =2manifold, the only nonzero matrix elements for this perturbation occur between the state 2, 0, 0 and the state 2, 1, 0 . Hence, within this subspace the matrix of interest has the| form i | i 0 η 00 η 000 [V ]= ∗ ,  0000  0000   where η = 2, 0, 0 Fz 2, 1, 0 . This latter integral is readily evaluated in the position representation,h using| the| knowni form

1 r r/2a 0 ~r 2, 0, 0 = ψ2,0,0(~r)= (2 )e− Y0 (θ, φ) h | i √8a3 − a 150 Approximation Methods for Stationary States

1 1 r r/2a 0 ~r 2, 1, 0 = ψ2,1,0(~r)= e− Y1 (θ, φ) h | i √8a3 √3 a of the hydrogen n =2wave functions. Substituting into the integral of interest we ﬁnd after a short calculation that π F ∞ 4 r r/a 2 η = r (2 )e− dr sin θ cos θdθ = 3Fa. 16a4 − a − ·Z0 ¸·Z0 ¸ Thus, 0 3Fa 00 3Fa − 000 [V ]= .  − 0000  0000   Diagonalizing [V ] we set det(V ε)= ε2[ε2 (3Fa)2]=0 and ﬁnd the eigenvalues − − −

ˆε2,1,1 =ˆε2,1, 1 =0 − ˆε2,+ =+3Fa

ˆε2, = 3Fa − − which we can add to the n =2hydrogenic energies to provide the first order corrections. Thus, to first order, the n =2eigenenergies in the presence of the field take the form

(0) ε2,+ = ε2 +3Fa (0) ε2, = ε2 3Fa − − (0) ε2,1,1 = ε2,1, 1 = ε2 − which correspond, respectively, to new zeroth order states

2, 0, 0 + 2, 0, 1 2, + = | i | i | i √2 2, 0, 0 + 2, 0, 1 2, = | i | i | −i √2 2, 1, 1 | i 2, 1, 1 . | − i Qualitatively we see that the four-fold degenerate n =2hydrogenic level is split by the field into three separate levels, with the nondegenerate lower and higher energy states splitting off linearlyt in the applied field from the remaining two-fold degenerate subspace corresponding to the unperturbed energies. With these new basis states (in which ` is no longer a necessarily good quantum) one can, in principle, investigate higher order corrections to the energy. Chapter 5 ANGULAR MOMENTUM AND ROTATIONS

In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian components Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one another. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. Indeed, it is this lack of commutivity that imparts to angular momentum observables their rich characteristic structure and makes them quite useful, e.g., in classi- fying the bound states of atomic, molecular, and nuclear systems containing one or more particles, and in decomposing the scattering states of such systems into components associated with di¤erent angular momenta. Just as importantly, the existence of internal “spin” degrees of freedom, i.e., intrinsic angular momenta associated with the internal structure of fundamental particles, provides additional motivation for the study of angular momentum and to the general properties exhibited by dynamical quantum systems under rotations.

5.1 Orbital Angular Momentum of One or More Particles The classical orbital angular momentum of a single particle about a given origin is given by the cross product ~` = ~r ~p (5.1) £ of its position and momentum vectors. The total angular momentum of a system of such structureless point particles is then the vector sum ~ L~ = `® = ~r® p~® (5.2) ® ® £ X X 162 Angular Momentum and Rotations of the individual angular momenta of the particles making up the collection. In quantum mechanics, of course, dynamical variables are replaced by Hermitian operators, and so we are led to consider the vector operator

~` = R~ P~ (5.3) £ or its dimensionless counterpart

~` ~l = R~ K;~ = ; (5.4) £ ~ either of which we will refer to as an angular momentum (i.e., we will, for the rest of this chapter, e¤ectively be working in a set of units for which ~ =1). Now, a general vector operator B~ can always be de…ned in terms of its operator components Bx;By;Bz along f g any three orthogonal axes. The component of B~ along any other direction, de…ned, e.g., by the unit vector u;^ is then the operator B~ u^ = Bxux + Byuy + Bzuz.Soitiswiththe ¢ operator ~l; whose components are, by de…nition, the operators

lx = YKz ZKy ly = ZKx XKz lz = XKy YKx: (5.5) ¡ ¡ ¡ The components of the cross product can also be written in a more compact form

li = "ijkXjKk (5.6) j;k X in terms of the Levi-Civita symbol 1 if ijk is an even permutation of 123 "ijk = 1 if ijk is an odd permutation of 123 .(5.7) 8 ¡ < 0 otherwise Although the normal: product of two Hermitian operators is itself Hermitian if and only if they commute, this familiar rule does not extend to the cross product of two vector operators. Indeed, even though R~ and K~ do not commute, their cross product ~l is readily shown to be Hermitian. From (5.6),

+ + + li = "ijkKk Xj = "ijkKkXj = "ijkXjKk = li; (5.8) j;k j;k j;k X X X wherewehaveusedthefactthecomponentsofR~ and K~ are Hermitian and that, since "ijk =0if k = j; only commuting components of R~ and K~ appear in each term of the cross product: It is also useful to de…ne the scalar operator

l2 = ~l ~l = l2 + l2 + l2 (5.9) ¢ x y z which, being the sum of the squares of Hermitian operators, is itself both Hermitian and positive. So the components of ~l; like those of the vector operators R~ and P;~ are Hermitian. We will assume that they are also observables. Unlike the components of R~ and P;~ however, the components of ~l along di¤erent directions do not commute with each other. This is readily established; e.g.,

[lx;ly]=[YKz ZKy;ZKx KzX] ¡ ¡ = YKx [Kz;Z]+KyX [Z; Kz]

= i (XKy YKx)=ilz: ¡ Orbital Angular Momentum of One or More Particles 163

The other two commutators are obtained in a similar fashion, or by a cyclic permutation of x; y; and z; giving

[lx;ly]=ilz [ly;lz]=ilx [lz;lx]=ily; (5.10) which can be written more compactly using the Levi-Civita symbol in either of two ways,

[li;lj]=i "ijklk; (5.11) k X or "ijklilj = ilk; i;j X the latter of which is, component-by-component, equivalent to the vector relation

~l ~l = i~l: (5.12) £ These can also be used to derive the following generalization

~l a;^ ~l ^b = i~l a^ ^b (5.13) ¢ ¢ ¢ £ h i ³ ´ involving the components of ~l along arbitrary directions a^ and ^b. It is also straightforward to compute the commutation relations between the components of ~l and l2,i.e.,

2 2 lj;l = lj;li = li [lj;li]+ [lj;li] li i i i £ ¤ X £ ¤ X X = i ("ijklilk + "ijklkli)=i ("ijklilk + "kjililk) i;k i;k X X = i "ijk(lilk lilk)=0 (5.14) ¡ i;k X where in the second line we have switched summation indices in the second sum and then ~ 2 used the fact that "kji = "ijk: Thus each component of l commutes with l : We write ¡ 2 ~l; l =0 [li;lj]=i "ijklk: (5.15) k h i X The same commutation relations are also easily shown to apply to the operator representing the total orbital angular momentum L~ of a system of particles. For such a system, the state space of which is the direct product of the state spaces for each particle, the operators for one particle automatically commute with those of any other, so that

[Li;Lj]= [li;®;lj;¯ ]=i "ijk ±®;¯ lk;® = i "ijk lk;® ®;¯ k ®;¯ k ® X X X X X = i "ijkLk (5.16) k X Similarly, from these commutation relations for the components of L~ ,itcanbeshown 2 ~ that Li;L =0using the same proof as above for l. Thus, for each particle, and for the total orbital angular momentum itself, we have the same characteristic commutation relations£ ¤ 2 [Li;Lj]=i "ijkLk L;~ L =0: (5.17) k X h i 164 Angular Momentum and Rotations

As we will see, these commutation relations determine to a very large extent the allowed spectrum and structure of the eigenstates of angular momentum. It is convenient to adopt the viewpoint, therefore, that any vector operator obeying these characteristic commutation relations represents an angular momentum of some sort. We thus generally say that an arbitrary vector operator J~ is an angular momentum if its Cartesian components are observables obeying the following characteristic commutation relations

2 [Ji;Jj]=i "ijkJk J;J~ =0: (5.18) k X h i It is actually possible to go considerably further than this. It can be shown, under very general circumstances, that for every quantum system there must exist a vector operator J~ obeying the commutation relations (5.18), the components of which characterize the way that the quantum system transforms under rotations. This vector operator J~ can usually, in such circumstances, be taken as a de…nition of the total angular momentum of the associated system. Our immediate goals, therefore, are twofold. First we will explore this underlying relationship that exists between rotations and the angular momentum of a physical system. Then, afterwards, we will return to the commutation relations (5.18), and use them to determine the allowed spectrum and the structure of the eigenstates of arbitrary angular momentum observables.

5.2 Rotation of Physical Systems A rotation R of a physical system is a distance preserving mapping of R3 onto itself that leaves a single point O; and the handedness of coordinate systems invariant. This de…nition excludes, e.g., re‡ections and other “improper” transformations, which always invert coordinate systems. There are two di¤erent, but essentially equivalent ways of mathematically describing rotations. An active rotation of a physical system is one in which all position and velocity vectors of particles in the system are rotated about the …xed point O; while the coordinate system used to describe the system is left unchanged. A passive rotation, by contrast, is one in which the coordinate axes are rotated, but the physical vectors of the system are left alone. In either case the result, generally, is a change in the Cartesian components of any vector in the system with respect to the coordinate axes used to represent them. It is important to note, however, that a clockwise active rotation of a physical system about a given axis is equivalent in terms of the change it produces on the coordinates of a vector to a counterclockwise passive rotation about thesameaxis. There are also two di¤erent methods commonly adopted for indicating speci…c rotations, each requiring three independent parameters. One method speci…es particular rotations through the use of the so-called Euler angles introduced in the study of rigid bodies. Thus, e.g., R(®; ¯; °) would indicate the rotation equivalent to the three separate rotations de…ned by the Euler angles (®; ¯; °): Alternatively, we can indicate a rotation by choosing a speci…c rotation axis, described by a unit vector u^ (de…ned, e.g., through its polar angles µ and Á), and a rotation angle ®: Thus, a rotation about u^ throughanangle® (positive or negative, according to the right-hand-rule applied to u^) would be written Ru^(®): We will, in what follows, make more use of this latter approach than we will of the Euler angles. Independent of their means of speci…cation, the rotations about a speci…ed point O in three dimensions form a group, referred to as the three-dimensional rotation group. Recall that a set G of elements R1;R2; ; that is closed under an associative binary operation, ¢¢¢ RiRj = Rk G for all Ri;Rj G; (5.19) 2 2 Rotation of Physical Systems 165 is said to form a group if (i) there exists in G an identity element 1 such that R1 = 1R = R 1 for all R in G and (ii) there is in G; for each R; an inverse element R¡ ; such that 1 1 RR¡ =R¡ R = 1. For the rotation group Ru^(®) the product of any two rotations is just the f g rotation obtained by performing each rotation in sequence, i.e., Ru^(®)Ru^0 (®0) corresponds toarotationofthephysicalsystemthroughanangle®0 about u^0; followed by a rotation through ® about u:^ The identity rotation corresponds to the limiting case of a rotation of ® =0about any axis (i.e., the identity mapping). The inverse of Ru^(®) is the rotation

1 R¡u^ (®)=Ru^( ®)=R u^(®); (5.20) ¡ ¡ that rotates the system in the opposite direction about the same axis. It is readily veri…ed that, in three dimensions, the product of two rotations generally depends upon the order in which they are taken. That is, in most cases,

Ru^(®)Ru^ (®0) = Ru^ (®0)Ru^(®): (5.21) 0 6 0 The rotation group, therefore, is said to be a noncommutative or non-Abelian group. There are, however, certain subsets of the rotation group that form commutative subgroups (subsets of the original group that are themselves closed under the same binary operation). For example, the set of rotations Ru^(®) …xed u^ about any single …xed axisformsanAbeliansubgroupofthe3D rotationf group,j since theg product of two rotations in the plane perpendicular to u^ corresponds to a single rotation in that plane through an angle equal to the (commutative) sum of the individual rotation angles,

Ru^(®)Ru^(¯)=Ru^(® + ¯)=Ru^(¯)Ru^(®): (5.22)

The subgroups of this type are all isomorphic to one another. Each one forms a realization of what is referred to for obvious reasons as the two dimensional rotation group. Another commutative subgroup comprises the set of in…nitesimal rotations. A rotation Ru^(±®) is said to be in…nitesimal if the associated rotation angle ±® is an in…nitesimal (it being understood that quantities of order ±2® are always to be neglected with respect to quantities of order ±®). The e¤ect of an in…nitesimal rotation on a physical quantity of the system is to change it, at most, by an in…nitesimal amount. The general properties of such rotations are perhaps most easily demonstrated by considering their e¤ect on normal vectors of R3. The e¤ect of an arbitrary rotation R on a vector ~v of R3 is to transform it into a new vector ~v0 = R [~v] : (5.23) Because rotations preserves the relative orientations and lengths of all vectors in the system, it also preserves the basic linear relationships of the vector space itself, i.e.,

R [~v1 + ~v2]=R [~v1]+R [~v2] : (5.24)

Thus, the e¤ect of any rotation R on vectors in the R3 can be described through the action of an associated linear operator AR; such that

R [~v]=~v0 = AR~v: (5.25)

This linear relationship can be expressed in any Cartesian coordinate system in component form vi0 = Aij vj (5.26) j X 166 Angular Momentum and Rotations

A systematic study of rotations reveals that the 3 3 matrix A representing the linear £ operator AR must be real, orthogonal, and unimodular, i.e.

T T Aij = Aij¤ A A = AA = 1 det(A)=1: (5.27)

We will denote by Au^(®) the linear operator (or any matrix representation thereof, depending upon the context) representing the rotation Ru^(®).TherotationsRu^(®) and the orthogonal, unimodular matrices Au^(®) representing their e¤ect on vectors with respect to a given coordinate system are in a one-to-one correspondence. We say, therefore, that the set of matrices Au^(®) forms a representation of the 3D rotation group. The group formed by the matricesf themselvesg is referred to as SO3, which indicates the group of “special” orthogonal 3 3 matrices (special in that it excludes those orthogonal matrices that have determinant of£ 1; i.e., it excludes re‡ections and other improper transformations). In this group, the matrix¡ representing the identity rotation is, of course, the identity matrix, while rotations about the three Cartesian axes are e¤ected by the matrices

10 0 cos µ 0sinµ Ax(µ)= 0cosµ sin µ Ay(µ)= 010 0 0sinµ ¡cos µ 1 0 sin µ 0cosµ 1 ¡ @ A @ A cos µ sin µ 0 ¡ Az(µ)= sin µ cos µ 0 (5.28) 0 0011 Now it is intuitively clear that@ the matrix associatedA with an in…nitesimal rotation barely changes any vector that it acts upon and, as a result, di¤ers from the identity matrix by an in…nitesimal amount, i.e.,

Au^(±®)=1 + ±® Mu^ (5.29) where Mu^ is describes a linear transformation that depends upon the rotation axis u^ but is independent of the in…nitesimal rotation angle ±®: The easily computed inverse

1 A¡ (±®)=Au^( ±®)=1 ±®Mu^ (5.30) u^ ¡ ¡ and the orthogonality of rotation matrices

1 T T A¡u^ (±®)=Au^ (±®)=1 + ±®Mu^ (5.31) leads to the requirement that the matrix

T Mu^ = M (5.32) ¡ u^ be real and antisymmetric. Thus, under such an in…nitesimal rotation, a vector ~v is taken onto the vector ~v0 = ~v + ±® Mu^~v: (5.33) Rotation of Physical Systems 167

δα

dv = v δα sin θ v v’

Figure 1 Under an in…nitesimal rotation Ru^(±®); the change d~v = ~v0 ~v in a vector ~v is perpendicular to both u^ and ~v; and has magnitude d~v = ~v ±® sin µ. ¡ j j j j But an equivalent description of such an in…nitesimal transformation on a vector can be determined through simple geometrical arguments. The vector ~v0 obtained by rotating the vector ~v about u^ through an in…nitesimal angle ±® is easily veri…ed from Fig. (1) to be given by the expression

~v0 = ~v + ±®(^u ~v) (5.34) £ or, in component form

vi0 = vi + ±® "ijkujvk (5.35) j;k X A straightforward comparison of (5.33) and (5.34) reveals that, for these to be consistent, the matrix Mu^ must have matrix elements of the form Mik = j "ijkuj; i.e., P 0 uz uy ¡ Mu^ = uz 0 ux ; (5.36) 0 ¡ 1 uy ux 0 ¡ @ A where ux;uy; and uz are the components (i.e., direction cosines) of the unit vector u^. Note that we can write (5.36) in the form

Mu^ = uiMi = uxMx + uyMy + uzMz (5.37) i X wherethethreematricesMi that characterize rotations about the three di¤erent Cartesian 168 Angular Momentum and Rotations axes are given by

00 0 001 0 10 ¡ Mx = 00 1 My = 000 Mz = 100 : (5.38) 0 01¡ 01 0 1001 0 0001 ¡ @ A @ A @ A Returning to the point that motivated our discussion of in…nitesimal rotations, we note that

Au^(±®)Au^0 (±®0)=(1 + ±®Mu^)(1 + ±®0Mu^0 )

= 1 + ±®Mu^ + ±®0Mu^0 = Au^0 (±®)Au^(±®0); (5.39) which shows that, to lowest order, a product of in…nitesimal rotations always commutes. This last expression also reveals that in…nitesimal rotations have a particularly simple combination law, i.e., to multiply two or more in…nitesimal rotations simply add up the parts corresponding to the deviation of each one from the identity matrix. This rule, and the structure (5.37) of the matrices Mu^ implies the following important theorem: an in…nitesimal rotation Au^(±®) about an arbitrary axis u^ can always be built up as a product of three in…nitesimal rotations about any three orthogonal axes, i.e.,

Au^(±®)=1 + ±®Mu^

= 1 + ±®uxMx + ±®uyMy + ±®uzMz which implies that Au^(±®)=Ax(ux±®)Ay(uy±®)Az(uz±®): (5.40) Since this last property only involves products, it must be a group property associated with thegroupSO3ofrotationmatrices Au^(®) ; i.e., a property shared by the in…nitesimal rotations that they represent, i.e., f g

Ru^(±®)=Rx(ux±®)Ry(uy±®)Rz(uz±®): (5.41)

We will use this group relation associated with in…nitesimal rotations in determining their e¤ect on quantum mechanical systems.

5.3 Rotations in Quantum Mechanics Any quantum system, no matter how complicated, can be characterized by a set of observables and by a state vector Ã ; which is an element of an associated Hilbert space. A rotation performed on a quantumj i mechanical system will generally result in a transformation of the state vector and to a similar transformation of the observables of the system. To make this a bit more concrete, it is useful to imagine an experiment set up on a rotatable table. The quantum system to be experimentally interrogated is described by some initial suitably-normalized state vector Ã : The experimental apparatus might be arranged to measure, e.g., the component ofj thei momentum of the system along a particular direction. Imagine, now, that the table containing both the system and the experimental apparatus is rotated about a vertical axis in such a way that the quantum system “moves” rigidly with the table (i.e., so that an observer sitting on the table could distinguish no change in the system). After such a rotation, the system will generally be in a new state Ã0 ; normalized in the same way as it was before the rotation. More- over, the apparatusj thati has rotated with the table will now be set up to measure the momentum along a di¤erent direction, as measured by a set of coordinate axes …xed in the laboratory.. Rotations in Quantum Mechanics 169

Such a transformation clearly describes a mapping of the quantum mechanical state space onto itself in a way that preserves the relationships of vectors in that space, i.e., it describes a unitary transformation. Not surprisingly, therefore, the e¤ect of any rotation R on a quantum system can quite generally be characterized by a unitary operator UR; i.e., Ã0 = R [ Ã ]=UR Ã : (5.42) j i j i j i Moreover, the transformation experienced under such a rotation by observables of the system must have the property that the mean value and statistical distribution of an observable Q taken with respect to the original state Ã willbethesameasthemean j i value and distribution of the rotated observable Q0 = R [Q] taken with respect to the rotated state Ã0 ; i.e., j i + Ã Q Ã = Ã0 Q0 Ã0 = Ã U Q0UR Ã (5.43) h j j i h j j i h j R j i From (5.43) we deduce the relationship

+ Q0 = R [Q]=URQUR: (5.44)

Thus, the observable Q0 is obtained through a unitary transformation of the unrotated observable Q using the same unitary operator that is needed to describe the change in the state vector. Consistent with our previous notation, we will denote by Uu^(®) the unitary transformation describing the e¤ect on a quantum system of a rotation Ru^(®) about u^ through angle ®. Just as the 3 3 matrices Au^(®) form a representation of the rotation group £ f g Ru^(®) , so do the set of unitary operators Uu^(®) andsoalsodothesetofmatrices frepresentingg these operators with respect tof any giveng ONB for the state space. Also, as with the case of normal vectors in R3; an in…nitesimal rotation on a quantum system will produce an in…nitesimal change in the state vector Ã : Thus, the unitary operator j i Uu^(±®) describing such an in…nitesimal rotation will di¤er from the identity operator by an in…nitesimal, i.e., Uu^(±®)=1 + ±®M^u^ (5.45) 3 where M^ u is now a linear operator, de…ned not on R but on the Hilbert space of the quantum system under consideration, that depends on u^ but is independent of ±®. Similar to our previous calculation, the easily computed inverse

1 U¡ (±®)=Uu^( ±®)=1 ±®M^ u^ (5.46) u^ ¡ ¡ 1 + and the unitarity of these operators (U ¡ = U ) leads to the result that, now, M^ u^ = M^ + is anti-Hermitian. There exists, therefore, for each quantum system, an Hermitian ¡ u^ operator Ju^ = iM^ u^; such that

Uu^(±®)=1 i±® Ju^: (5.47) ¡

The Hermitian operator Ju^ is referred to as the generator of in…nitesimal rotations about the axis u:^ Evidently, there is a di¤erent operator Ju^ characterizing rotations about each direction in space. Fortunately, as it turns out, all of these di¤erent operators Ju^ can be expressed as a simple combination of any three operators Jx;Jy; and Jz describing rotations about a given set of coordinate axes. This economy of expression arises from the combination rule (5.41) obeyed by in…nitesimal rotations, which implies a corresponding rule Uu^(±®)=Ux(ux±®)Uy(uy±®)Uz(uz±®) 170 Angular Momentum and Rotations for the unitary operators that represent them in Hilbert space. Using (5.47), this fundamental relation implies that

Uu^(±®)=(1 i±® uxJx)(1 i±® uyJy)(1 i±® uzJz) ¡ ¡ ¡ = 1 i±® (uxJx + uyJy + iuzJz): (5.48) ¡ Implicit in the form of Eq. (5.48) is the existence of a vector operator J~,withHermitian components Jx;Jy; and Jz that generate in…nitesimal rotations about the corresponding coordinate axes, and in terms of which an arbitrary in…nitesimal rotation can be expressed in the form Uu^(±®)=1 i±® J~ u^ = 1 i±® Ju (5.49) ¡ ¢ ¡ where Ju = J~ u^ now represents the component of the vector operator J~ along u^. From¢ this form that we have deduced for the unitary operators representing in…n- itesimal rotations we can now construct the operators representing …nite rotations. Since rotations about a …xed axis form a commutative subgroup, we can write

Uu^(® + ±®)=Uu^(±®)Uu^(®)=(1 i±® Ju)Uu^(®) (5.50) ¡ which implies that

dUu^(®) Uu^(® + ±®) Uu^(®) = lim ¡ = iJuUu^(®): (5.51) d® ±® 0 ±® ¡ ! The solution to this equation, subject to the obvious boundary condition Uu^(0) = 1; is the unitary rotation operator

Uu^(®)=exp( i®Ju)=exp i®J~ ~u : (5.52) ¡ ¡ ¢ ³ ´ We have shown, therefore, that a description of the behavior of a quantum system under rotations leads automatically to the identi…cation of a vector operator J;~ whose components act as generators of in…nitesimal rotations and the exponential of which generates the unitary operators necessary to describe more general rotations of arbitrary quantum mechanical systems. It is convenient to adopt the point of view that the vector operator J~ whose existence we have deduced represents, by de…nition, the total angular momentum of the associated system. We will postpone until later a discussion of how angular momentum operators for particular systems are actually identi…ed and constructed. In the meantime, however, to show that this point of view is at least consistent we must demonstrate that the components of J~ satisfy the characteristic commutation relations (5.18) that are, in fact, obeyed by the operators representing the orbital angular momentum of a system of one or more particles.

5.4 Commutation Relations for Scalar and Vector Operators The analysis of the last section shows that for a general quantum system there exists a vector operator J;~ to be identi…ed with the angular momentum of the system, that is essential for describing the e¤ect of rotations on the state vector Ã and its observables j i Q. Indeed, the results of the last section imply that a rotation Ru^(®) of the physical system will take an arbitrary observable Q onto a generally di¤erent observable

+ i®Ju i®Ju Q0 = Uu^(®)QUu^ (®)=e¡ Qe : (5.53)

For in…nitesimal rotations Uu^(±®), this transformation law takes the form

Q0 =(1 i±® Ju)Q(1 + i±® Ju) (5.54) ¡ Commutation Relations for Scalar and Vector Operators 171 which, to lowest nontrivial order, implies that

Q0 = Q i±® [Ju;Q] : (5.55) ¡ Now, as in classical mechanics, it is possible to classify certain types of observables of the system according to the manner in which they transform under rotations. Thus, an observable Q is referred to as a scalar with respect to rotations if

Q0 = Q; (5.56) for all R: For this to be true for arbitrary rotations, we must have, from (5.53), that

+ Q0 = URQUR = Q; (5.57) which implies that URQ = QUR; or

[Q; UR]=0: (5.58) Thus, for Q to be a scalar it must commute with the complete set of rotation operators for the space. A somewhat simpler expression can be obtained by considering the in…nitesimal rotations, where from (5.55) and (5.56) we see that the condition for Q to be a scalar reduces to the requirement that [Ju;Q]=0; (5.59) for all components Ju; which implies that

J;Q~ =0: (5.60) h i Thus, by de…nition, any observable that commutes with the total angular momentum of the system is a scalar with respect to rotations. A collection of three operators Vx;Vy; and Vz canbeviewedasformingthecom- ponents of a vector operator V~ if the component of V~ along an arbitrary direction a^ is ~ ~ Va = V a^ = i Viai: By construction, therefore, the operator J is a vector operator, since its¢ component along any direction is a linear combination of its three Cartesian components withP coe¢cients that are, indeed, just the associated direction cosines. Now, after undergoing a rotation R; a device initially setup to measure the component Va of a vector operator V~ along the direction a^ will now measure the component of V~ along the rotated direction a^0 = ARa;^ (5.61) where AR is the orthogonal matrix associated with the rotation R: Thus, we can write

+ + R [Va]=URVaU = UR(V~ a^)U = V~ a^0 = Va : (5.62) R ¢ R ¢ 0 Again considering in…nitesimal rotations Uu^(±®); and applying (5.55), this reduces to the relation V~ a^0 = V~ a^ i±® J~ u;^ V~ a^ : (5.63) ¢ ¢ ¡ ¢ ¢ h i But we also know that, as in (5.34), an in…nitesimal rotation Au^(±®) about u^ takes the vector a^ onto the vector

a^0 =(1 + ±®Mu^)â =â + ±® (û a^) : (5.64) £ Consistency of (5.63) and (5.64) requires that

V~ a^0 = V~ a^ + ±®V~ (^u a^)=V~ a^ i±® J~ u;^ V~ a^ (5.65) ¢ ¢ ¢ £ ¢ ¡ ¢ ¢ h i 172 Angular Momentum and Rotations i.e., that J~ u;^ V~ a^ = iV~ (^u a^): (5.66) ¢ ¢ ¢ £ Taking u^ and a^ along the ith andh jth Cartesiani axes, respectively, this latter relation can be written in the form [Ji;Vj]=i "ijkVk; (5.67) k X or more speci…cally

[Jx;Vy]=iVz [Jy;Vz]=iVx [Jz;Vx]=iVy (5.68) which shows that the components of any vector operator of a quantum system obey commutation relations with the components of the angular momentum that are very similar to those derived earlier for the operators associated with the orbital angular momentum, itself. Indeed, since the operator J~ is a vector operator with respect to rotations, it must also obey these same commutation relations, i.e.,

[Ji;Jj]= i"ijkJk: (5.69) k X Thus, our identi…cation of the operator J~ identi…ed above as the total angular momentum of the quantum system is entirely consistent with our earlier de…nition, in which we identi…ed as an angular momentum any vector operator whose components obey the characteristic commutation relations (5.18).

5.5 Relation to Orbital Angular Momentum To make some of the ideas introduced above a bit more concrete, we show how the generator of rotations J~ relates to the usual de…nition of angular momentum for, e.g., a single spinless particle. This is most easily done by working in the position representation. For example, let Ã(~r)= ~r Ã be the wave function associated with an arbitrary state Ã of a single spinless particle.h j i Under a rotation R; the ket Ã is taken onto a new j i j i ket Ã0 = UR Ã described by a di¤erent wave function Ã0(~r)= ~r Ã0 : The new wave j i j i h j i function Ã0, obtained from the original by rotation, has the property that the value of the unrotated wavefunction Ã at the point ~r mustbethesameasthevalueoftherotated wave function Ã0 at the rotated point ~r0 = AR~r: This relationship can be written in several ways, e.g., Ã(~r)=Ã0(~r0)=Ã0(AR~r) (5.70) 1 which can be evaluated at the point AR¡ ~r to obtain 1 Ã0(~r)=Ã(AR¡ ~r): (5.71)

Suppose that in (5.71), the rotation AR = Au^(±®) represents an in…nitesimal rotation about the axis u^ through and angle ±®, for which

AR~r = ~r + ±®(^u ~r): (5.72) £ 1 The inverse rotation AR¡ is then given by 1 A¡ ~r = ~r ±®(^u ~r) : (5.73) R ¡ £ Thus, under such a rotation, we can write

1 Ã0(~r)=Ã(A¡ ~r)=Ã [~r ±®(û ~r)] R ¡ £ = Ã (~r) ±®(û ~r) ~ Ã(~r) (5.74) ¡ £ ¢ r Relation to Orbital Angular Momentum 173 wherewehaveexpandedÃ [~r ±® (û ~r)] about the point ~r; retaining …rst order in…ni- tesimals. We can use the easily-proven¡ £ identity

(^u ~r) ~ Ã =^u (~r ~ )Ã £ ¢ r ¢ £ r which allows us to write

Ã0(~r)=Ã (~r) ±® u^ (~r ~ )Ã(~r) ¡ ¢ £ r ~ = Ã (~r) i±® u^ ~r r Ã(~r) (5.75) ¡ ¢ Ã £ i ! In Dirac notation this is equivalent to the relation

~r Ã0 = ~r Uu^(±®) Ã = ~r 1 i±® u^ ~` Ã (5.76) h j i h j j i h j ¡ ¢ j i ³ ´ where `~ = R~ K:~ (5.77) £ Thus, we identify the vector operator J~ for a single spinless particle with the orbital angular momentum operator `:~ This allows us to write a general rotation operator for such a particle in the form

Uu^(®)=exp i®~` u^ : (5.78) ¡ ¢ ³ ´ Thus the components of `~ form the generators of in…nitesimal rotations. Now the state space for a collection of such particles can be considered the direct or tensor product of the state spaces associated with each one. Since operators from di¤erent (1) spaces commute with each other, the unitary operator UR that describes rotations of one particle will commute with those of another. It is not di¢cult to see that under these circumstances the operator that rotates the entire state vector Ã is the product of the rotation operators for each particle. Suppose, e.g., that Ã is aj directi product state, i.e., j i Ã = Ã ;Ã ; ;Ã : j i j 1 2 ¢¢¢ N i Under a rotation R; the state vector Ã is taken onto the state vector j i

Ã0 = Ã10 ;Ã20 ; ;ÃN0 j i j (1) ¢¢¢(2) i (N) = U Ã0 U Ã0 U Ã0 R j 1i R j 2i¢¢¢ R j N i = U (1)U (2) U(N) Ã ;Ã ; ;Ã R R ¢¢¢ R j 1 2 ¢¢¢ N i = UR Ã j i where (1) (2) (N) UR = U U U R R ¢¢¢ R is a product of rotation operators for each part of the space, all corresponding to the same rotation R: Because these individual operators can all be written in the same form, i.e.,

(¯) U =exp i®~`¯ u^ ; R ¡ ¢ ³ ´ where `~¯ it the orbital angular momentum for particle ¯, it follows that the total rotation operator for the space takes the form

UR = exp i®~`¯ u^ =exp i®L~ u^ ¡ ¢ ¡ ¢ ¯ Y ³ ´ ³ ´ 174 Angular Momentum and Rotations where ~ L~ = `¯: ¯ X Thus, we are led naturally to the point of view that the generator of rotations for the whole system is the sum of the generators for each part thereof, hence for a collection of spinless particles the total angular momentum J~ coincides with the total orbital angular momentum L;~ as we would expect. Clearly, the generators for any composite system formed from the direct product of other subsystems is always the sum of the generators for each subsystem being combined. That is, for a general direct product space in which the operators J~1; J~2; J~N are the vector operators whose components are the generators of rotations for each¢¢¢ subspace, the corresponding generators of rotation for the combined space is obtained as a sum

N J~ = J~® ®=1 X of those for each space, and the total rotation operator takes the form

Uu^(®)=exp i®J~ u^ : ¡ ¢ ³ ´ For particles with spin, the individual particle spaces can themselves be considered direct products of a spatial part and a spin part. Thus for a single particle of spin S~ the generator of rotations are the components of the vector operator J~ = L~ + S;~ where L~ takes care of rotations on the spatial part of the state and S~ doesthesameforthespinpart.

5.6 Eigenstates and Eigenvalues of Angular Momentum Operators Having explored the relationship between rotations and angular momenta, we now under- take a systematic study of the eigenstates and eigenvalues of a vector operator J~ obeying angular momentum commutation relations of the type that we have derived. As we will see, the process for obtaining this information is very similar to that used to determine the spectrum of the eigenstates of the harmonic oscillator. We consider, therefore, an arbitrary angular momentum operator J~ whose components satisfy the relations

2 [Ji;Jj]=i "ijkJk J;J~ =0: (5.79) k X h i We note, as we did for the orbital angular momentum L;~ that, since the components Ji do not commute with one another, J~ cannot possess an ONB of eigenstates, i.e., states which are simultaneous eigenstates of all three of its operator components. In fact, one can show that the only possible eigenstates of J~ are those for which the angular momentum is identically zero (an s-state in the language of spectroscopy). Nonetheless, since, according to (5.79), each component of J~ commutes with J2; it is possible to …nd an ONB of eigenstates common to J2 and to the component of J~ along any chosen direction. Usually the component of J~ along the z-axis is chosen, because of the simple form taken by the di¤erential operator representing that component of orbital angular momentum in spherical coordinates. Note, however, that due to the cyclical nature of the commutation relations, anything deduced about the spectrum and eigenstates of J2 and 2 Jz must also apply to the eigenstates common to J and to any other component of J:~ Thus the spectrum of Jz mustbethesameasthatofJx;Jy; or Ju = J~ u:^ 2 2 2 ¢ We note also, that, as with l ; the operator J = i Ji is Hermitian and positive de…nite and thus its eigenvalues must be greater than or equal to zero. For the moment, P Eigenstates and Eigenvalues of Angular Momentum Operators 175 we will ignore other quantum numbers and simply denote a common eigenstate of J2 and Jz as j; m ; where, by de…nition, j i Jz j; m = m j; m (5.80) j i j i J2 j; m = j(j +1)j; m : (5.81) j i j i which implies that m is the associated eigenvalue of the operator Jz; while the label j is intended to simply identify the corresponding eigenvalue j(j +1)of J2: The justi…cation for writing the eigenvalues of J2 in this fashion is only that it simpli…es the algebra and the …nal results obtained. At this point, there is no obvious harm in expressing things in this fashion since for real values of j the corresponding values of j(j +1) include all values between 0 and ; and so any possible eigenvalue of J2 can be represented in the form j(j +1) for some1value of j. Moreover, it is easy to show that any non-negative value of j(j +1)can be obtained using a value of j which is itself non-negative. Without loss of generality, therefore, we assume that j 0: We will also, in the interest of brevity, refertoavector j; m satisfying the eigenvalue¸ equations (5.80) and (5.81) as a “vector of angular momentumj i (j; m)”. To proceed further, it is convenient to introduce the operator

J+ = Jx + iJy (5.82) formed from the components of J~ along the x and y axes. The adjoint of J+ is the operator

J = Jx iJy; (5.83) ¡ ¡ in terms of which we can express the original operators 1 i Jx = (J+ + J ) Jy = (J J+) : (5.84) 2 ¡ 2 ¡ ¡ 2 Thus, in determining the spectrum and common eigenstates of J and Jz ,we will …nd 2 it convenient to work with the set of operators J+;J ;J ;Jz rather than the set ¡ J ;J ;J ;J2 : In the process, we will require commutation relations for the opera- x y z © ª tors in this new set. We note …rst that J ; being a linear combination of Jx and Jy; must © ª 2 § by (5.79) commute with J : The commutator of J with Jz is also readily established; we …nd that § [Jz;J ]=[Jz;Jx] i [Jz;Jy]=iJy Jx (5.85) § § § or [Jz;J ]= J : (5.86) § § § Similarly, the commutator of J+ and J is ¡

[J+;J ]=[Jx;Jx]+[iJy;Jx] [Jx;iJy] [iJy;iJy] ¡ ¡ ¡ =2Jz: (5.87) Thus the commutation relations of interest take the form

2 2 [Jz;J ]= J [J+;J ]=2Jz J ;J =0= J ;Jz : (5.88) § § § ¡ § It will also be necessary in what follows to express£ the operator¤ J2£ in terms¤ of the new “components” J+;J ;Jz rather than the old components Jx;Jy;Jz : To this end we note that J2 fJ2 = J¡2 +gJ2; and so f g ¡ z x y 2 2 J+J =(Jx + iJy)(Jx iJy)=Jx + Jy i [Jx;Jy] ¡ ¡ ¡ 2 2 2 = J + J + Jz = J Jz(Jz 1) (5.89) x y ¡ ¡ 176 Angular Momentum and Rotations and

2 2 J J+ =(Jx iJy)(Jx + iJy)=Jx + Jy + i [Jx;Jy] ¡ ¡ 2 2 2 = J + J Jz = J Jz(Jz +1): (5.90) x y ¡ ¡ These two results imply the relation

2 1 2 J = [J+J + J J+]+Jz : (5.91) 2 ¡ ¡ With these relations we now deduce allowed values in the spectrum of J2 and 2 Jz. We assume, at …rst, the existence of at least one nonzero eigenvector j; m of J and j i Jz with angular momentum (j; m); where, consistent with our previous discussion, the eigenvalues of J2 satisfy the inequalities

j(j +1) 0 j 0: (5.92) ¸ ¸ Using this, and the commutation relations, we now prove that the eigenvalue m must lie, for a given value of j; in the range

j m j: (5.93) ¸ ¸¡

To show this, we consider the vectors J+ j; m and J j; m ; whose squared norms are j i ¡j i 2 J+ j; m = j; m J J+ j; m (5.94) jj j ijj h j ¡ j i 2 J j; m = j; m J+J j; m : (5.95) jj ¡j ijj h j ¡j i Using (5.89) and (5.90) these last two equations can be written in the form

2 2 J+ j; m = j; m J Jz(Jz +1)j; m =[j(j +1) m(m +1)] j; m j; m (5.96) jj j ijj h j ¡ j i ¡ h j i 2 2 J j; m = j; m J Jz(Jz 1) j; m =[j(j +1) m(m 1)] j; m j; m (5.97) jj ¡j ijj h j ¡ ¡ j i ¡ ¡ h j i Now, adding and subtracting a factor of jm from each parenthetical term on the right, these last expressions can be factored into

2 2 J+ j; m =(j m)(j + m +1) j; m (5.98) jj j ijj ¡ jjj ijj J j; m 2 =(j + m)(j m +1) j; m 2: (5.99) jj ¡j ijj ¡ jjj ijj For these quantities to remain positive de…nite, we must have

(j m)(j + m +1) 0 (5.100) ¡ ¸ and (j + m)(j m +1) 0: (5.101) ¡ ¸ For positive j; the …rst inequality requires that

j m and m (j +1) (5.102) ¸ ¸¡ and the second that m j and j +1 m: (5.103) ¸¡ ¸ All four inequalities are satis…ed if and only if

j m j; (5.104) ¸ ¸¡ Eigenstates and Eigenvalues of Angular Momentum Operators 177

which shows, as required, that the eigenvalue m of Jz must lie between j. § Having narrowed the range for the eigenvalues of Jz; we now show that the vector 2 J+ j; m vanishes if and only if m = j; and that otherwise J+ j; m is an eigenvector of J j i j i and Jz with angular momentum (j; m +1); i.e., it is associated with the same eigenvalue 2 j(j +1)of J ; but is associated with an eigenvalue of Jz increased by one, as a result of the action of J+. To show the …rst half of the statement, we note from (5.98), that

2 J+ j; m =(j m)(j + m +1) j; m : (5.105) jj j ijj ¡ jjj ijj

Given the bounds on m; it follows that J+ j; m vanishes if and only if m = j: To prove the j i second part, we use the commutation relation [Jz;J+]=J+ to write JzJ+ = J+Jz + J+ and thus JzJ+ j; m =(J+Jz + J+) j; m =(m +1)J+ j; m ; (5.106) j i j i j i showing that J+ j; m is an eigenvector of Jz with eigenvalue m +1: Also, because 2 j i [J ;J+]=0; 2 2 J J+ j; m = J+J j; m = j(j +1)J+ j; m (5.107) j i j i j i 2 showing that if m = j; then the vector J+ j; m is an eigenvector of J with eigenvalue j(j +1): 6 j i In a similar fashion, using (5.99) we …nd that

J j; m =(j + m)(j m +1) j; m 2: (5.108) jj ¡j ijj ¡ jjj ijj Given the bounds on m; this proves that the vector J j; m vanishes if and only if m = j; ¡j 2 i ¡ while the commutation relations [Jz;J ]= J and [J ;J ]=0imply that ¡ ¡ ¡ ¡

JzJ j; m =(J Jz Jz) j; m =(m 1)J j; m (5.109) ¡j i ¡ ¡ j i ¡ ¡j i J2J j; m = J J2 j; m = j(j +1)J j; m : (5.110) ¡j i ¡ j i ¡j i 2 Thus, when m = j; the vector J j; m is an eigenvector of J and Jz with angular momentum (j; m6 ¡1): ¡j i ¡ Thus, J+ is referred to as the raising operator, since it acts to increase the component of angular momentum along the z-axis by one unit and J is referred to as the lowering operator. Neither operator has any e¤ect on the total angular¡ momentum of the system, as represented by the quantum number j labeling the eigenvalues of J2: 2 We now proceed to restrict even further the spectra of J and Jz: We note, e.g., from our preceding analysis that, given any vector j; m of angular momentum (j; m) we can produce a sequence j i 2 3 J+ j; m ;J j; m ;J j; m ; (5.111) j i +j i +j i ¢¢¢ 2 of mutually orthogonal eigenvectors of J with eigenvalue j(j +1)and of Jz with eigenvalues m; (m +1); (m +2); : (5.112) ¢¢¢ This sequence must terminate, or else produce eigenvectors of Jz with eigenvalues violating the upper bound in Eq. (5.93). Termination occurs when J+ acts on the last nonzero n vector of the sequence, J+ j; m say, and takes it on to the null vector. But, as we have j i n shown, this can only occur if m + n = j; i.e., if J+ j; m is an eigenvector of angular momentum (j; m + n)=(j; j): Thus, there must existj an integeri n such that

n = j m: (5.113) ¡ 178 Angular Momentum and Rotations

This, by itself, does not require that m or j be integers, only that the values of m change by an integer amount, so that the di¤erence between m and j be an integer. But similar arguments can be made for the sequence

J j; m ;J2 j; m ;J3 j; m ; (5.114) ¡j i ¡j i ¡j i ¢¢¢ which will be a series of mutually orthogonal eigenvectors of J2 with eigenvalue j(j +1) and of Jz with eigenvalues m; (m 1); (m 2); : (5.115) ¡ ¡ ¢¢¢ Again, termination is required to now avoid producing eigenvectors of Jz with eigenvalues violating the lower bound in Eq. (5.93). Thus, the action of J on the last nonzero vector n0 ¡ of the sequence, J+ j; m say, is to take it onto the null vector. But this only occurs if jn0 i m n0 = j; i.e., if J j; m is an eigenvector of angular momentum (j; m n0)=(j; j): ¡ ¡ + j i ¡ ¡ Thus there exists an integer n0 such that

n0 = j + m: (5.116)

Adding these two relations, we deduce that there exists an integer N = n + n0 such that 2j = n + n0 = N,or N j = : (5.117) 2 Thus, j must be either an integer or a half-integer. If N is an even integer, then j is itself an integer and must be contained in the set j 0; 1; 2; : For this situation, the results of the proceeding analysis indicate that m 2fmust also¢¢¢g be an integer and, for a given integer value of j; the values m must take on each of the 2j +1 integer values m =0; 1; 2; j: In this case, j is said to be an integral value of angular momentum. §If N§is¢¢¢§ an odd integer, then j di¤ers from an integer by 1=2; i.e., it is contained in the set j 1=2; 3=2; 5=2; ; and is said to be half-integral (short for half-odd-integral). For a given2f half-integral value¢¢¢g of j; the values of m must then take on each of the 2j +1 half-odd-integer values m = 1=2; ; j. Thus,wehavededucedthevaluesof§ ¢¢¢ § j and m that are consistent with the commutation relations (5.79). In particular, the allowed values of j that can occur are the non-negative integers and the positive half-odd-integers. For each value of j; there are always (at least) 2j +1fold mutually-orthogonal eigenvectors

j; m m = j; j +1; ;j (5.118) fj ij ¡ ¡ ¢¢¢ g 2 2 of J and Jz corresponding to the same eigenvalue j(j +1) of J , but di¤erent eigenvalues m of Jz (the orthogonality of the di¤erent vectors in the set follows from the fact that they are eigenvectors of the Hermitian observable Jz corresponding to di¤erent eigenvalues.) In any given problem involving an angular momentum J~ it must be determined whichoftheallowedvaluesofj and how many subspaces for each such value actually occur. All of the integer values of angular momentum do, in fact, arise in the study of the orbital angular momentum of a single particle, or of a group of particles. Half-integral values of angular momentum, on the other hand, are invariably found to have their source in the half-integral angular momenta associated with the internal or spin degrees of freedom of particles that are anti-symmetric under exchange, i.e., fermions. Bosons, by contrast, are empirically found to have integer spins. This apparently universal relationship between the exchange symmetry of identical particles and their spin degrees of freedom has actually been derived under a rather broad set of assumptions using the techniques of quantum …eld theory. Orthonormalization of Angular Momentum Eigenstates 179

The total angular momentum of a system of particles will generally have contributions from both orbital and spin angular momenta and can be either integral or half-integral, depending upon the number and type of particles in the system. Thus, generally speaking, there do exist di¤erent systems in which the possible values of j and m deduced above are actually realized. In other words, there appear to be no superselection rules in nature that further restrict the allowed values of angular momentum from those allowed by the fundamental commutation relations.

5.7 Orthonormalization of Angular Momentum Eigenstates We have seen in the last section that, given any eigenvector j; m with angular momen- j i 2 tum (j; m); it is possible to construct a set of 2j +1common eigenvectors of J and Jz corresponding to the same value of j, but di¤erent values of m: Speci…cally, the vectors obtained by repeated application of J+ to j; m will produce a set of eigenvectors of Jz with eigenvalues m+1;m+2; ;j;while repeatedj i application of J to j; m will produce ¢¢¢ ¡ j i the remaining eigenvectors of Jz with eigenvalues m 1;m 2;; j: Unfortunately, even when the original angular momentum eigenstate¡ j; m ¡is suitably¢¢¢¡ normalized, the eigenvectors vectors obtained by application of the raisingj andi lowering operators to this state are not. In this section, therefore, we consider the construction of a basis of normalized angular momentum eigenstates. To this end, we restrict our use of the notation j; m so that it refers only to normalized states. We can then express the action of J+ onj suchi a normalized state in the form

J+ j; m = ¸m j; m +1 (5.119) j i j i where ¸m is a constant, and j; m+1 represents, according to our de…nition, a normalized j i state with angular momentum (j; m+1). We can determine the constant ¸m by considering the quantity 2 2 J+ j; m = j; m J J+ j; m = ¸m (5.120) jj j ijj h j ¡ j i j j which from the analysis following Eqs. (5.98) and (5.99), and the assumed normalization of 2 the state j; m ; reduces to ¸m = j(j+1) m(m+1): Choosing ¸m real and positive,this implies thej followingi relationj j ¡

J+ j; m = j(j +1) m(m +1) j; m +1 (5.121) j i ¡ j i 2 between normalized eigenvectors ofpJ and Jz di¤ering by one unit of angular momentum along the z axis. A similar analysis applied to the operator J leads to the relation ¡ J j; m = j(j +1) m(m 1) j; m 1 : (5.122) ¡j i ¡ ¡ j ¡ i These last two relations can alsop be written in the sometimes more convenient form

J+ j; m j; m +1 = j i (5.123) j i j(j +1) m(m +1) ¡ p J j; m j; m 1 = ¡j i : (5.124) j ¡ i j(j +1) m(m 1) ¡ ¡ or p J j; m j; m 1 = §j i : (5.125) j § i j(j +1) m(m 1) ¡ § 2 Now, if J and Jz do not comprisep a complete set of commuting observables for the space on which they are de…ned, then other commuting observables (e.g., the energy) will 180 Angular Momentum and Rotations be required to form a uniquely labeled basis of orthogonal eigenstates, i.e., to distinguish 2 between di¤erent eigenvectors of J and Jz having the same angular momentum (j; m): If we let ¿ denote the collection of quantum numbers (i.e., eigenvalues) associated with the 2 other observables needed along with J and Jz to form a complete set of observables, then the normalized basis vectors of such a representation can be written in the form ¿;j;m : Typically, many representations of this type are possible, since we can alwaysfj form linearig combinations of vectors with the same values of j and m to generate a new basis set. Thus, in general, basis vectors having the same values of ¿ and j; but di¤erent values of m need not obey the relationships derived above involving the raising and lowering operators. However, as we show below, it is always possible to construct a so-called standard representation, in which the relationships (5.123) and (5.124) are maintained. To construct such a standard representation, it su¢ces to work within each eigensubspace S(j) of J2 with …xed j; since states with di¤erent values of j are automatically orthogonal (since J2 is Hermitian). Within any such subspace S(j) of J2, there are always contained even smaller eigenspaces S(j; m) spanned by the vectors ¿;j;m of …xed j and …xed m: We focus in particular on the subspace S(j; j) containingfj eigenvectorsig of Jz for which m takes its highest value m = j, and denote by ¿;j;j a complete set of normalized basis vectors for this subspace, with the index ¿fjdistinguishingig between di¤erent orthogonal basis vectors with angular momentum (j; j). By assumption, then, for the states in this set, ¿;j;j ¿ 0;j;j = ±¿;¿ (5.126) h j i 0 For each member ¿;j;j of this set, we now construct the natural sequence of 2j +1 basis vectors by repeatedj applicationi of J ; i.e., using (5.124) we set ¡ J ¿;j;j J ¿;j;j ¿;j;j 1 = ¡j i = ¡j i (5.127) j ¡ i j(j +1) j(j 1) p2j ¡ ¡ and the remaining members of thep set according to the relation

J ¿;j;m ¿;j;m 1 = ¡j i ; (5.128) j ¡ i j(j +1) m(m 1) ¡ ¡ terminating the sequence with the vectorp¿;j; j : Since the members ¿;j;m of this set j ¡ i j i (with ¿ and j …xed and m = j; ;j) are eigenvectors of Jz corresponding to di¤erent eigenvalues, they are mutually¡ orthogonal¢¢¢ and, by construction, properly normalized. It is also straightforward to show that the vectors ¿;j;m generatedinthiswayfrom j i the basis vector ¿;j;j of S(j; j) are orthogonal to the vectors ¿ 0;j;m generated from j i j i a di¤erent basis vector ¿ 0;j;j of S(j; j). To see this, we consider the inner product j i ¿;j;m 1 ¿ 0;j;m 1 and, using the adjoint of (5.128), h ¡ j ¡ i

¿;j;m J+ ¿;j;m 1 = h j ; (5.129) h ¡ j j(j +1) m(m 1) ¡ ¡ we …nd that p

¿;j;m J+J ¿ 0;j;m ¿;j;m 1 ¿ 0;j;m 1 = h j ¡j i = ¿;j;m ¿ 0;j;m (5.130) h ¡ j ¡ i j(j +1) m(m 1) h j i ¡ ¡ where we have used (5.99) to evaluate the matrix element of J+J : This shows that, ¡ if ¿ 0;j;m and ¿;j;m are orthogonal, then so are the states generated from them by j i j i application of J : Since the basis states ¿;j;j and ¿ 0;j;j used to start each sequence are orthogonal,¡ by construction, so, it follows,j i arej any twoi sequences of basis vectors Orthonormalization of Angular Momentum Eigenstates 181

so produced, and so also are the subspaces S(¿;j) and S(¿ 0;j) spanned by those basis vectors. Proceeding in this way for all values of j a standard representation of basis vectors for the entire space is produced. Indeed, the entire space can be written as a direct sum of the subspaces S(¿;j) so formed for each j; i.e., we can write

S = S(j) S(j0) S(j00)+ © © ¢¢¢ with a similar decomposition

S(j)=S(¿;j) S(¿ 0;j) S(¿ 00;j)+ © © ¢¢¢ for the eigenspaces S(j) of j2. The basis vectors for this representation satisfy the obvious orthonormality and completeness relations

¿ 0;j0;m0 ¿;j;m = ±¿ ;¿ ±j ;j±m ;m (5.131) h j i 0 0 0 ¿;j;m ¿;j;m = 1: (5.132) ¿;j;mj ih j X The matrices representing the components of the angular momentum operators are easily computed in any standard representation. In particular, it is easily veri…ed that the matrix elements of J2 are given in any standard representation by the expression

2 ¿ 0;j0;m0 J ¿;j;m = j(j +1)±¿ ;¿ ±j ;j±m ;m (5.133) h j j i 0 0 0 while the components of J~ have the following matrix elements

¿ 0;j0;m0 Jz ¿;j;m = m±¿ ;¿ ±j ;j±m ;m (5.134) h j j i 0 0 0

¿ 0;j0;m0 J ¿;j;m = j(j +1) m(m 1)±¿ ;¿ ±j ;j±m ;m 1: (5.135) h j §j i ¡ § 0 0 0 § The matrices representing the Cartesianp components Jx and Jy can then be constructed from the matrices for J+ and J ; using relations (5.84), i.e. ¡ 1 ¿ 0;j0;m0 Jx ¿;j;m = ±¿ ;¿ ±j ;j j(j +1) m(m 1)±m ;m 1 h j j i 2 0 0 ¡ ¡ 0 ¡ hp + j(j +1) m(m +1)±m ;m+1 (5.136) ¡ 0 p i i ¿ 0;j0;m0 Jy ¿;j;m = ±¿ ;¿ ±j ;j j(j +1) m(m 1)±m ;m 1 h j j i 2 0 0 ¡ ¡ 0 ¡ hp j(j +1) m(m +1)±m ;m+1 (5.137) ¡ ¡ 0 i Clearly, the simplest possible subspacep of …xed total angular momentum j is one corresponding to j =0; which according to the results derived above must be of dimension 2j +1 = 1: Thus, the one basis vector ¿;0; 0 in such a space is a simultaneous eigenvector 2 j i of J and of Jz with eigenvalue j(j +1)=m =0. Such a state, as it turns out, is also an 2 eigenvector of Jx and Jy (indeed of J~ itself). The 1 1 matrices representing J ;Jz;J ; £ § Jx; and J in such a space are all identical to the null operator. The¡ next largest possible angular momentum subspace corresponds to the value j =1=2; which coincides with the 2j +1 = 2 dimensional spin space of electrons, protons, and neutrons, i.e., particles of spin 1/2. In general, the full Hilbert space of a single particle of spin s can be considered the direct product

S = Sspatial Sspin (5.138) 182 Angular Momentum and Rotations of the Hilbert space describing the particle’s motion through space (a space spanned, e.g.,bythepositionstates ~r ) and a …nite dimensional space describing its internal spin j i degrees of freedom. The spin space Ss of a particle of spin s is by de…nition a 2s +1 dimensional space having as its fundamental observables the components Sx;Sy; and Sz of a spin angular momentum vector S:~ In keeping with the analysis of Sec. (5.3), these operators characterize the way that the spin part of the state vector transforms under rotations and, as such, satisfy the standard angular momentum commutation relations, i.e.,

[Si;Sj]=i "ijkSk: (5.139) k X Thus, e.g., an ONB for the space of one such particle consists of the states ~r; ms ; which j i are eigenstates of the position operator R;~ the total square of the spin angular momentum 2 S ; and the component of spin Sz along the z-axis according to the relations

R~ ~r; ms = ~r ~r; ms (5.140) j i j i 2 S ~r; ms = s(s +1)~r;ms (5.141) j i j i Sz ~r; ms = ms ~r; ms : (5.142) j i j i (As is customary, in these last expressions the label s indicating the eigenvalue of S2 has been suppressed, since for a given class of particle s does not change.) The state vector Ã of such a particle, when expanded in such a basis, takes the form j i s s 3 3 Ã = d r ~r; ms ~r; ms Ã = d rÃ (~r) ~r; ms (5.143) j i j ih j i ms j i ms= s ms= s X¡ Z X¡ Z and thus has a “wave function” with 2s +1 components Ãms (~r). As one would expect from the de…nition of the direct product, operators from the spatial part of the space have no e¤ect on the spin part and vice versa. Thus, spin and spatial operators automatically commute with each other. In problems dealing only with the spin degrees of freedom, therefore, it is often convenient to simply ignore the part of the space associated with the spatial degrees of freedom (as the spin degrees of freedom are often generally ignored in exploring the basic features of quantum mechanics in real space). Thus, the spin space of a particle of spin s =1=2, is spanned by two basis vectors, often designated + and ; with j i j¡i 1 1 3 S2 = +1 = (5.144) j§i 2 2 j§i 4j§i µ ¶ and 1 Sz = : (5.145) j§i §2j§i The matrices representing the di¤erent components of S~ within a standard representation for such a space are readily computed from (5.133)-(5.137),

3 10 S2 = (5.146) 4 01 µ ¶ 1 01 1 0 i 1 10 Sx = Sy = ¡ Sz = (5.147) 2 10 2 i 0 2 0 1 µ ¶ µ ¶ µ ¡ ¶ Orbital Angular Momentum Revisited 183

01 00 S+ = S = (5.148) 00 ¡ 10 µ ¶ µ ¶ The Cartesian components Sx;Sy; and Sz are often expressed in terms of the so-called Pauli ¾ matrices

01 0 i 10 ¾ = ¾ = ¾ = (5.149) x 10 y i ¡0 z 0 1 µ ¶ µ ¶ µ ¡ ¶ 1 in terms of which Si = 2 ¾i.The¾ matrices have a number of interesting properties that follow from their relation to the angular momentum operators.

5.8 Orbital Angular Momentum Revisited As an additional example of the occurrance of angular momentum subspaces with integral values of j we consider again the orbital angular momentum of a single particle as de…ned ~ ~ ~ by the operator ` = R K with Cartesian components `i = j;k "ijkXjKk: In the position representation these take£ the form of di¤erential operators P @ `i = i "ijkxj : (5.150) ¡ @xk j;k X As it turns out, in standard spherical coordinates (r; µ; Á); where

x = r sin µ cos Áy= r sin µ sin Áz= r cos µ; (5.151) the components of ~` take a form which is independent of the radial variable r. Indeed, using the chain rule it is readily found that

@ @ ` = i sin Á +cosÁ cot µ (5.152) x @µ @Á µ ¶ @ @ `y = i cos Á +sinÁ cot µ (5.153) ¡ @µ @Á µ ¶ @ `z = i : (5.154) ¡ @Á

In keeping with our previous development, it is useful to construct from `x and `y the raising and lowering operators ` = `x i`y (5.155) § § which (5.152) and (5.153) reduce to

iÁ @ @ ` = e§ + i cot µ : (5.156) § §@µ @Á · ¸ From these it is also straightforward to construct the di¤erential operator

@2 @ 1 @2 `2 = +cotµ + (5.157) ¡ @µ2 @µ sin2 µ @Á2 · ¸ representing the total square of the angular momentum. 2 Now we are interested in …nding common eigenstates of ` and `z: Since these operators are independent of r; it su¢ces to consider only the angular dependence. It 184 Angular Momentum and Rotations is clear, in other words, that the eigenfunctions of these operators can be written in the form m Ãl;m(r; µ; Á)=f(r)Yl (µ;Á) (5.158) m where f(r) is any acceptable function of r and the functions Yl (µ; Á) are solutions to the eigenvalue equations

2 m m ` Yl (µ; Á)=l(l +1)Yl (µ; Á) m m `zYl (µ; Á)=mYl (µ; Á) : (5.159)

m Thus, it su¢ces to …nd the functions Yl (µ; Á) ; whichare,ofcourse,justthespherical harmonics. More formally, we can consider the position eigenstates ~r as de…ning direct product states j i ~r = r; µ; Á = r µ; Á ; (5.160) j i j i j i j i i.e., the space can be decomposed into a direct product of a part describing the radial part of the wave function and a part describing the angular dependence. The angular part represents the space of functions on the unit sphere, and is spanned by the “angular position eigenstates” µ; Á : In this space, an arbitrary function Â(µ; Á) on the unit sphere is associated with a ketj i

2¼ ¼ Â = d Â(µ; Á) µ; Á = dÁ dµ sin µÂ(µ; Á) µ; Á (5.161) j i j i j i Z Z0 Z0 where Â(µ; Á)= µ; Á Â and the integration is over all solid angle, d =sinµdµdÁ:The states µ; Á formh a completej i set of states for this space, and so j i d µ; Á µ; Á = 1: (5.162) j ih j Z The normalization of these states is slightly di¤erent than the usual Dirac normalization, however, because of the factor associated with the transformation from Cartesian to spherical coordinates. To determine the appropriate normalization we note that

2¼ ¼ Â(µ0;Á0)= µ0;Á0 Â = dÁ dµ sin µÂ(µ; Á) µ0;Á0 µ; Á (5.163) h j i h j i Z0 Z0 which leads to the identi…cation ±(Á Á0)±(µ µ0)=sinµ µ0;Á0 µ; Á ; or ¡ ¡ h j i 1 µ0;Á0 µ; Á = ±(Á Á0)±(µ µ0)=±(Á Á0)±(cos µ cos µ0): (5.164) h j i sin µ ¡ ¡ ¡ ¡ Clearly, the components of ~` are operators de…ned on this space and so we denote by l; m 2 j i the appropriate eigenstates of the Hermitian operators ` and `z within this space (this assumes that l; m are su¢cient to specify each eigenstate, which, of course, turns out to be true). By assumption, then, these states satisfy the eigenvalue equations

`2 l; m = l(l +1)l; m j i j i `z l; m = m l; m (5.165) j i j i and can be expanded in the angular “position representation”, i.e.,

l; m = ± µ; Á µ; Á l; m = ± Y m(µ; Á) µ; Á (5.166) j i j ih j i l j i Z Z Orbital Angular Momentum Revisited 185 where the functions Y m(µ; Á)= µ; Á l; m (5.167) l h j i are clearly the same as those introduced above, and will turn out to be the spherical harmonics. m To obtain the states l; m (or equivalently the functions Yl ) we proceed in three stages. First, we determine thej generali Á-dependence of the solution from the eigenvalue 2 equation for `z: Then, rather than solving the second order equation for ` directly, we determine the general form of the solution for the states l; l having the largest component of angular momentum along the z-axis consistent withj ai given value of l. Finally, we use the lowering operator ` to develop a general prescription for constructing arbitrary 2 ¡ eigenstates of ` and `z: The Á-dependence of the eigenfunctions in the position representation follows from the simple form taken by the operator `z in this representation. Indeed, using (5.154), it follows that m @ m m `zY (µ; Á)= i Y (µ; Á)=mY (µ; Á); (5.168) l ¡ @Á l l which has the general solution

m m imÁ Yl (µ; Á)=Fl (µ)e : (5.169) Single-valuedness of the wave function in this representation imposes the requirement that m m Yl (µ; Á)=Yl (µ;Á +2¼); which leads to the restriction m 0; 1; 2; : Thus, for the case of orbital angular momentum only integral values2f of m§(and§ therefore¢¢¢g l)can occur. (We have yet to show that all integral values of l do, in fact, occur, however.) From this result we now proceed to determine the eigenstates l; l ; as represented by the wave functions j i l l ilÁ Yl (µ; Á)=Fl (µ)e : (5.170) To this end, we recall the general result that any such state of maximal angular momentum along the z axis is taken by the raising operator onto the null vector, i.e., `+ l; l =0.In the position representation, using (5.156), this takes the form j i

iÁ @ @ l µ; Á `+ l; l = e + i cot µ Y (µ; Á) h j j i @µ @Á l · ¸ @ @ = eiÁ + i cot µ F l(µ)eilÁ =0: (5.171) @µ @Á l · ¸ l Performing the Á derivative reduces this to a …rst order equation for Fl (µ), i.e., dF l(µ) l = l cot µF l(µ) (5.172) dµ l

l dFl d (sin µ) l = l (5.173) Fl sin µ which integrates to give, up to an overall multiplicative constant, a single linearly independent solution l l Fl (µ)=cl sin µ (5.174) for each allowed value of l. Thus, all values of l consistent with the integer values of m deduced above give acceptable solutions. Up to normalization we have, therefore, for each l =0; 1; 2 ; the functions ¢¢¢ l l ilÁ Yl (µ; Á)=cl sin µe : (5.175) 186 Angular Momentum and Rotations

The appropriate normalization for these functions follows from the relation

l; m l; m =1 (5.176) h j i which in the position representation becomes

m m d l;m µ; Á µ; Á l; m = d [Y (µ; Á)]¤ Y 0 (µ; Á) h j ih j i l l0 Z Z 2¼ ¼ = dÁ dµ sin µ Y m(µ;Á) 2 =1 (5.177) j l j Z0 Z0 l l ilÁ Substituting in the function Yl (µ; Á)=cl sin µe ; the magnitude of the constants cl can be determined by iteration, with the result that

(2l +1)!! 1 (2l +1)! cl = = l ; (5.178) j j s 4¼ (2l)!! 2 l!r 4¼ where the double factorial notation is de…ned on the postive integers as follows:

n(n 2)(n 4) (2) if n an even integer n!! = n(n ¡ 2)(n ¡ 4) ¢¢¢(1) if n an odd integer (5.179) 8 ¡ ¡ ¢¢¢ < 1 if n =0

0 With the phase of cl; :(chosen so that Yl (0; 0) is real and positive) given by the relation l cl =( 1) cl we have the …nal form for the spherical harmonic of order (l;l); i.e. ¡ j j l l ( 1) (2l +1)! l ilÁ Yl (µ; Á)= ¡l sin µe : (5.180) 2 l! r 4¼ From this, the remaining spherical harmonics of the same order l can be generated through application of the lowering operator, e.g., through the relation

` l; m l; m 1 = ¡j i (5.181) j ¡ i l(l +1) m(m 1) ¡ ¡ which, in the position representation takesp the form

m 1 iÁ @ m Y ¡ (µ; Á)=e¡ m cot µ Y (µ; Á) : (5.182) l ¡@µ ¡ l · ¸ In fact, it straightforward to derive the following expression

(l + m)! l m l; m = [` ] ¡ l; l (5.183) j i s(2l)!(l + m)! ¡ j i relating an arbitrary state l;m to the state l; l which we have explicitly found. This last relation is straightforwardj toi prove by induction.j i We …rst assume that it holds for some value of m and then consider

` l; m 1 (l m)! l m+1 l; m 1 = ¡j i = ¡ [` ] ¡ l; l : j ¡ i l(l +1) m(m 1) l(l +1) m(m 1)s(2l)!(l + m)! ¡ j i ¡ ¡ ¡ ¡ (5.184) p p Orbital Angular Momentum Revisited 187

But, as we have noted before, l(l +1) m(m 1) = (l + m)(l m +1): Substituting this into the radical on the right and canceling¡ the¡ obvious terms¡ which arise we …nd that

[l (m 1)]! l (m 1) l; m 1 = ¡ ¡ [` ] ¡ ¡ l;l (5.185) j ¡ i (2l)!(l + m 1)! ¡ j i s ¡ which is of the same form as the expression we are trying to prove with m m 1: Thus, if it is true for any m it is true for all values of m less than the original. We! then¡ note that the expression is trivially true for m = l; which completes the proof. Thus, the spherical harmonic of order (l;m) can be expressed in terms of the one of order (l; l) in the form

l (m 1) m [l (m 1)]! i(l m)Á @ @ ¡ ¡ l Y (µ; Á)= ¡ ¡ e¡ ¡ + i cot µ Y (µ; Á): (5.186) l (2l)!(l + m 1)! ¡@µ @Á l s ¡ · ¸ It is not our intention to provide here a complete derivation of the properties of the spherical harmonics, but rather to show how they …t into the general scheme we have developed regarding angular momentum eigenstates in general. To round things out a bit we mention without proof a number of their useful properties. 1. Parity - The parity operator ¦ acts on the eigenstates of the position representation and inverts them through the origin, i.e., ¦ ~r = ~r : It is straightforward to show that in the position representation this takesj i thej¡ formi ¦Ã(~r)=Ã( ~r): In spherical coordinates it is also easily veri…ed that under the parity operation r¡ r; µ ¼ µ; and Á Á + ¼: Thus, for functions on the unit sphere, ¦f(µ; Á)=f!(¼ µ;! Á +¡¼): ! ¡ The parity operator commutes with the components of ~` and with `2: Indeed, the states l; m are eigenstates of parity and satisfy the eigenvalue equation j i ¦ l; m =( 1)l l; m (5.187) j i ¡ j i which implies for the spherical harmonics that ¦Y m(µ;Á)=Y m(¼ µ; Á + ¼)=( 1)l Y m(µ; Á): (5.188) l l ¡ ¡ l 2. Complex Conjugation - It is straightforward to show that m m m [Y (µ; Á)]¤ =( 1) Y ¡ (µ; Á): (5.189) l ¡ l This allows spherical harmonics with m<0 to be obtained very simply from those with m>0. 3. Relation to Legendre Functions - The spherical harmonics with m =0are directly related to the Legendre polynomials l l ( 1) d 2 l Pl(u)= ¡ 1 u (5.190) 2ll! dul ¡ through the relation ¡ ¢

0 (2l +1)! Yl (µ; Á)= Pl(cos µ): (5.191) r 4¼ The other spherical harmonics with m>0 are related to the associated Legendre functions dmP (u) P m(u)= (1 u2)m l ; (5.192) l ¡ dum through the relation p

m m (2l +1)!(l m)! m imÁ Yl (µ; Á)=( 1) ¡ Pl (cos µ)e : (5.193) ¡ s 4¼ (l + m)! 188 Angular Momentum and Rotations

5.9 Rotational Invariance As we have seen, the behavior exhibited by a quantum system under rotations can usually be connected to a property related to the angular momentum of the system. In this chapter we consider in some detail the consequences of rotational invariance,thatis,we explore just what is implied by the statement that a certain physical state or quantity is unchanged as a result of rotations imposed upon the system. To begin the discussion we introduce the important idea of rotationally invariant subspaces. 5.9.1 Irreducible Invariant Subspaces In our discussion of commuting or compatible observables we encountered the idea of global invariance and saw, e.g., that the eigenspaces Sa of an observable A are globally invariant with respect to any operator B that commutes with A: It is useful to extend this idea to apply to more than one operator at a time. We therefore introduce the idea of invariant subspaces. A subspace S0 of the state space S is said to be invariant with respect to theactionofasetG = R1;R2; of operators if, for every Ã in S0; the vectors f ¢¢¢g j i R1 Ã ;R2 Ã ; are all in S0 as well. S0 is than said to be an invariant subspace of the j i j i ¢¢¢ speci…ed set of operators. The basic idea here is that the operators Ri all respect the boundaries of the subspace S0; in the sense that they never take a state in S0 onto a state outside of S0. With this de…nition, we consider a quantum mechanical system with state space S; characterized by total angular momentum J:~ It is always possible to express the state space S as a direct sum of orthogonal eigenspaces associated with any observable. (Recall that a space can be decomposed into a direct sum of two or more orthogonal subspaces if any vector in the space can be written as a linear combination of vectors from each subspace.) In the present context we consider the decomposition

S = S(j) S(j0) S(j00)+ (5.194) © © ¢¢¢ of our original space S into eigenspaces S(j) of the operator J2.Noweachoneofthe spaces S(j) associated with a particular eigenvalue j(j+1) of J2 can, itself, be decomposed into a direct sum S(j)=S(¿;j) S(¿ 0;j) S(¿ 00;j)+ (5.195) © © ¢¢¢ of 2j +1dimensional subspaces S(¿;j) associated with a standard representation for the space S, i.e., the vectors in S(¿;j) comprise all linear combinations

Ã = Ãm ¿;j;m (5.196) j i m= j j i X¡ of the 2j +1 basis states ¿;j;m with …xed ¿ and j. We now show that each of these j i spaces S(¿;j) is invariant under the action of the operator components Ju = J~ u^ of ¢ the total angular momentum. This basically follows from the the way thatn such a standardo representation is constructed. We just need to show that the action of any component of J~ on such a vector takes it onto another vector in the same space, i.e., onto a linear ~ combination of the same basis vectors. But the action of the operator Ju = J u^ = i uiJi is completely determined by the action of the three Cartesian components¢ of the vector operator J~. Clearly, however, the vector P

Jz Ã = mÃm ¿;j;m (5.197) j i m= j j i X¡ Rotational Invariance 189

lies in the same subspace S(¿;j) as Ã : Moreover, Jx and Jy are simple linear combinations of J ; which take such a state ontoj i § j

J Ã = j(j +1) m(m 1)Ãm ¿;j;m 1 (5.198) § j i m= j ¡ § j § i X¡ p which is also in the subspace S(¿;j). Thus it is clear the the components of J~ cannot take an arbitrary state Ã in S(¿;j) outside the subspace. Hence S(¿;j) is an invariant subspace of the angularj momentumi operators. Indeed, it is precisely this invariance that leads to the fact the the matrices representing the components of J~ are block-diagonal in any standard representation. It is not di¢cult to see that this invariance with respect to the action of the operator Ju extends to any operator function F (Ju). In particular, it extends to the unitary rotation operators Uu^(®)=exp( i®Ju) which are simple exponential functions ¡ of the components of J~. Thus, we conclude that the spaces S(¿;j) are also invariant subspaces of the group of operators Uu^(®) : We say that each subspace S(¿;j) is an invariant subspace of the rotation groupf ,orthatg S(¿;j) is rotationally invariant. We note that the eigenspace S(j) is also an an invariant subspace of the rotation group, since any vector in it is a linear combination of vectors from the invariant subspaces S(¿;j) and so the action of Uu^(®) on an arbitrary vector in Sj is to take it onto another vector in S(j); with the di¤erent parts in each subspace S(¿;j) staying within that respective subspace. It is clear, however, that although S(j) is invariant with respect to the operators of the rotation group, it can often be reduced or decomposed into lower dimensional invariant parts S(¿;j). It is is reasonable in light of this decomposability exhibited by S(j); to ask whether the rotationally invariant subspaces S(¿;j) of which S(j) is formed are similarly decomposable. To answer this question we are led to the idea of irreducibility. An invariant subspace S0 of a group of operators G = R1;R2; is said to be irreducible with respect to G (or is an irreducible invariant subspacef of¢¢¢gG) if, for every non-zero vector Ã in S0,thevectors Ri Ã span S0.Conversely,S0 is reducible if j i f j ig there exists a nonzero vector Ã in S0 for which the vectors Ri Ã fail to span the space. j i f j ig Clearly, in the latter case the vectors spanned by the set Ri Ã form a subspace of S0 that is itself invariant with respect to G: f j ig We now answer the question we posed above, and show explicitly that any invariant subspace S(¿;j) associated with a standard representation for the state space S is, in fact, an irreducible invariant subspace of the rotation group Uu^(®) , i.e., S(¿;j) cannot be decomposed into smaller invariant subspaces. To prove thisf requiresg several steps. To begin, we let j

Ã = Ãm ¿;j;m (5.199) j i m= j j i X¡ again be an arbitrary (nonzero) vector in S(¿;j) and we formally denote by SR the subspace of S(¿;j) spanned by the vectors Uu^(®) Ã ; that is, SR is the subspace of all vectors that can be written as a linear combinationf j ofig vectors obtained through a rotation of the state Ã : It is clear that SR is contained in S(¿;j); since the latter is invariant j i under rotations; the vectors Uu^(®) Ã must all lie inside S(¿;j). We wish to show that f j ig in fact SR = S(¿;j): To do this we show that SR contains a basis for S(¿;j) and hence the two spaces are equivalent. To this end we note that the vectors Ju Ã are all contained in SR: This follows j i from the form of in…nitesimal rotations Uu^(±®)=1 i±®Ju; which imply that ¡ 1 1 Ju = [1 Uu^(±a)] = [Uu^(0) Uu^(±a)] : (5.200) i® ¡ i® ¡ 190 Angular Momentum and Rotations

Thus, 1 1 Ju Ã = [Uu^(0) Uu^(±a)] Ã = [Uu^(0) Ã Uu^(±a) Ã ] (5.201) j i i® ¡ j i i® j i¡ j i which is, indeed, a linear combination of the vectors Uu^(®) Ã , and thus is in the space f j ig SR spanned by such vectors. By a straightforward extension, it follows, that the vectors

J Ã =(Jx iJy) Ã §j i § j i 1 = [Ux(0) Ux(±a) iUy(0) iUy(±a)] Ã (5.202) i® ¡ § ¨ j i are also in SR: In fact, any vector of the form

q P (J ) (J+) Ã (5.203) ¡ j i will be expressable as a linear combination of various products of Ux(0);Uy(0);Ux(±®); and Uy(±®) acting on Ã . Since the product of any two such rotation operators is itself j i a rotation, the result will be a linear combination of the vectors Uu^(®) Ã ; and so will f j ig also lie in SR: We now note, that since Ã is a linear combination of the vectors ¿;j;m ,each j i j i of which is raised or annihilated by the operator J+; there exists an integer P<2j for P which (J+) Ã = ¸ ¿;j;j (in other words, we keep raising the components of Ã up and j i j i j i annhilating them till the component that initially had the smallest value of Jz is the only one left.) It follows, therefore, that the basis state ¿;j;j lies in the subspace SR.We are now home free, since we can now repeatedly applyj thei lowering operator, remaining within SR with each application, to deduce that all of the basis vectors ¿;j;m lie in the j i subspace SR: Thus SR is a subspace of S(¿;j) that contains a basis for S(¿;j): The only way this can happen is if SR = S(¿;j): It follows that the under the unitary transformation associated with a general rotation, the basis vectors ¿;j;m of S(¿;j) are transformed into new basis vectors for the same invariant subspace.fj Indeed,ig it is not hard to see, based upon our earlier description of the rotation process, that a rotation R that takes the unit vector z^ onto 2 a new direction z^0 will take the basis kets ¿;j;mz , which are eigenstates of J and j i Jz onto a new set of basis kets ¿;j;mz0 for the same space that are now eigenstates 2 j ~ i of J and the component Jz0 of J along the new direction. These new vectors can, of course, be expressed as linear combinations of the original ones. The picture that emerges is that, under rotations, the vectors ¿;j;m transform (irreducibly) into linear combinations of themselves. This transformationj is essentiallyi geometric in nature and is analogous to the way that normal basis vectors in R3 transform into linear combinations of one another. Indeed, by analogy, the coe¢cients of this linear transformation are identical in any subspace of a standard representation having the same value of j; since the basis vectors of such a representation have been constructed using the angular momentum operators in precisely the same fashion. This leads to the concept of rotation matrices, i.e., a set of standard matrices representing the rotation operators Uu^(®) in terms of their e¤ect on the vectors within any irreducible invariant subspace S(¿;j). Just as with the matrices representing the components of J~ within any irreducible subspace S(¿;j),the elements of the rotation matrices will depend upon j and m but are indepndent of ¿. Thus, e.g., a rotation UR of a basis ket ¿;j;m of S(¿;j) results in a linear combination j i Rotational Invariance 191 of such states of the form j

UR ¿;j;m = ¿;j;m0 ¿;j;m0 UR ¿;j;m j i j ih j j i m = j X0 ¡ j (j) = ¿;j;m0 R (5.204) j i m0;m m = j X0 ¡ where, as the notation suggests, the elements

(j) R = ¿;j;m0 UR ¿;j;m = j; m0 UR ;j;m (5.205) m0;m h j j i h j j i of the 2j +1dimensional rotation matrix are independent of ¿: In terms of the elements R(j) of the rotation matrices, the invariance of the subspaces S(¿;j) under rotations m0;m leads to the general relation

(j) ¿ 0;j0;m0 UR ¿;j;m = ±j ;j±¿ ;¿ R : (5.206) h j j i 0 0 m0;m These matrices are straightforward to compute for low dimensional subspaces, and general formulas have been developed for calculating the matrices for rotations associated with the Euler angles (®; ¯; °). For rotations about the z axis the matrices take a particulalrly simple form, since the rotation operator in this case is a simple function of the operator Jz of which the states j; m are eigenstates. Thus, e.g., j i (j) i®Jz im® R (z;®)= j; m0 e¡ j; m = e¡ ±m;m (5.207) m0m h j j i 0 In a subspace with j =1=2; for example, this takes the form

e i®=2 0 R^(1=2)(z;®)= ¡ ; (5.208) 0 e+i®=2 µ ¶ while in a space with j =1we have

i® e¡ 00 R^(1)(z;®)= 010: 0 00ei® 1 @ A The point is that, once the rotation matrices have been worked out for a given value of j, they can be used for a standard representation of any quantum mechanical system. Thus, e.g., we can deduce a transformation law associated with rotations of the spherical harmonics, i.e.,

m R [Y (µ; Á)] = µ; Á UR l; m l h j j i l = Y m0 (µ; Á)R(l) : (5.209) l m0;m m = l X0 ¡ For rotations about the z-axis, this takes the form

m i®`z Rz(®)[Yl (µ; Á)] = µ; Á e¡ l; m h im®j m j i im® m imÁ = e¡ Yl (µ; Á)=e¡ Fl (µ)e m im(Á ®) = Fl (µ)e ¡ = Y m(µ; Á ®) (5.210) l ¡ which is readily con…rmed from simple geometric arguments. 192 Angular Momentum and Rotations

5.9.2 Rotational Invariance of States We now consider a physical state of the system that is invariant under rotations, i.e., that has the property that UR Ã = Ã (5.211) j i j i for all rotations UR. This can be expressed in terms of the angular momentum of the system by noting that invariance under in…nitesimal rotations

Uu^(±®) Ã = Ã i±®Ju Ã = Ã ; j i j i¡ j i j i requires that Ju Ã =0 (5.212) j i for all u^, which also implies that J2 Ã =0: Thus, a state Ã is rotationally invariant if and only if it has zero angular momentum.j i j i 5.9.3 Rotational Invariance of Operators If an observable Q is rotationally invariant it is, by our earlier de…nition, a scalar with respect to rotations, and we can deduce the following:

1. [UR;Q]=0

2 2. [Ju;Q]=0=[J ;Q]

2 3. There exists an ONB of eigenstates states ¿;q;j;m common to Jz;J ; and Q. fj ig 4. The eigenvalues of q of Q within any irreducible subspace S(j) are (at least) 2j +1 fold degenerate.

This degeneracy, referred to as a rotational or essential degeneracy, is straight- foward to show. Suppose that q is an eigenstate of Q; so that Q q = q q : Then j i j i j i

QUR q = URQ q = qUR q : (5.213) j i j i j i

This shows that UR q is an eigenstate of Q with the same eigenvalue. Of course not all j i the states UR q are linearly independent. From this set of states we can form linear f j ig 2 combinations which are also eigenstates of J and Jz; and which can be partitioned into irreducible invariant subspaces of well de…ned j: The 2j +1 linearly independent basis states associated with each such irreducible subspace S(q; j) are then 2j+1 fold degenerate eigenstates of Q. As an important special case, suppose that the Hamiltonian of a quantum system is a scalar with respect to rotations. We can then conclude that

1. H is rotationally invariant.

2 2. [UR;H]=[Ju;H]=[J ;H]=0:

3. The components of J~ are constants of the motion, since d i Ju = [H; Ju] =0: (5.214) dth i ~h i

4. The equations of motion are invariant under rotations. Thus, if Ã(t) is a solution to j i d i~ H Ã(t) =0 (5.215) dt ¡ j i µ ¶ Addition of Angular Momenta 193

then d d UR i~ H Ã(t) = i~ H UR Ã(t) =0 (5.216) dt ¡ j i dt ¡ j i µ ¶ µ ¶ which shows that the rotated state UR Ã(t) is also a solution to the equations of motion. j i

2 5. There exists an ONB of eigenstates states E;¿;j;m common to Jz;J ; and H. fj ig 6. The eigenvalues E of H within any irreducible subspace are (at least) 2j +1 fold degenerate. For the case of the Hamiltonian, these degeneracies are referred to as multiplets. Thus, a nondegenerate subspace associated with a state of zero angular momentum is referred to as a singlet, a doubly-degenerate state associated with a two-fold degenerate j =1=2 state is a doublet, and a three-fold degenerate state associated with angular momentum j =1is referred to as a triplet.

5.10 Addition of Angular Momenta

Let S1 and S2 be two quantum mechanical state spaces associated with angular momentum J~1 an J~2; respectively. Let ¿ 1;j1;m1 denote the basis vectors of a standard fj ig representation for S1, which is decomposable into corresponding irreducible subspaces S1(¿ 1;j1), and denote by ¿ 2;j2;m2 the basis vectors of a standard representation for fj ig S2; decomposable into irreducible subspaces S2(¿ 2;j2). The combined quantum system formed from S1 and S2 is an element of the direct product space S = S1 S2: (5.217) The direct product states

¿ 1;j1;m1; ¿ 2;j2;m2 = ¿ 1;j1;m1 ¿ 2;j2;m2 (5.218) j i j ij i form an orthonormal basis for S. As we will see, however, these direct product states do not de…ne a standard representation for S. Indeed, for this combined space the total angular momentum vector is the sum

J~ = J~1 + J~2 (5.219) of those assocated with each “factor space”. What this means is that the rotation operators for the combined space are just the products of the rotation operators for each individual space

(1) (2) Uu^(®)=Uu^ (®)Uu^ (®) ~ ~ ~ ~ i®J1 u^ i®J2 u^ i®(J1+J2) u^ = e¡ ¢ e¡ ¢ = e¡ ¢ (5.220) ~ ~ ~ i®J u^ i®(J1+J2) u^ e¡ ¢ = e¡ ¢ (5.221) so that the generators of rotations for each factor space simply add. At this point, we make no speci…c identi…cation of the nature of the two subspaces involved. Accordingly, the results that we will derive will apply equally well to the description of two spinless particles (for which J~ = L~ = L~ 1 + L~ 2 is the total orbital angular momentum of the pair), to the description of a single particle with spin (for which J~ = L~ + S~ is the sum of the orbital and spin angular momenta of the particle), or even to a collection of particles (where J~ = L~ + S~ is again the sum of the orbital and spin angular momentum, but where ~ ~ now the latter represent the corresponding orbital and spin angular momenta L = ® L® and S~ = S~ for the entire collection of particles). ® ® P P 194 Angular Momentum and Rotations

In either of these cases the problem of interest is to construct a standard repre- 2 sentation ¿;j;m of common eigenstates of J and Jz associated with the total angular j i momentum vector J~ of the system as linear combinations of the direct product states ¿ 1;j1;m1; ¿ 2;j2;m2 . As it turns out, the latter are eigenstates of j i

Jz =(J~1 + J~2) z^ ¢ = J1z + J2z (5.222)

since they are individually eigenstates of J1z and J2z; i.e.,

Jz ¿ 1;j1;m1; ¿ 2;j2;m2 =(J1z + J2z) ¿ 1;j1;m1; ¿ 2;j2;m2 j i j i =(m1 + m2) ¿ 1;j1;m1; ¿ 2;j2;m2 j i = m ¿ 1;j1;m1; ¿ 2;j2;m2 (5.223) j i

where m = m1 + m2: The problem is that these direct product states are generally not eigenstates of

2 J =(J~1 + J~2) (J~1 + J~2) ¢ 2 2 = J + J +2J~1 J~2 (5.224) 1 2 ¢ 2 2 because, although they are eigenstates of J1 and J2 ; they are not eigenstates of

J~1 J~2 = J1iJ2i (5.225) ¢ i X

due to the presence in this latter expression of operator components of J~1 and J~2 perpendicular to the z axis. Using the language of invariant subspaces, another way of expressing the problem at hand is as follows: determine how the direct product space S = S1 S2 can be decomposed into its own irreducible invariant subspaces S(¿;j). This way of thinking about the problem actually leads to a simpli…cation. We note that since S1 and S2 can each be written as a direct sum

S1 = S1(¿ 1;j1) S2 = S2(¿ 2;j2) (5.226) ¿ ;j ¿ ;j X1 1 X2 2

of rotationally invariant subspaces, the direct product of S1 and S2 can also be writtten as a direct sum

S = S1 S2 = S1(¿ 1;j1) S2(¿ 2;j2) ¿ ;j ;¿ ;j 1 X1 2 2 = S(¿ 1;¿2;j1;j2) (5.227) ¿ ;j ;¿ ;j 1 X1 2 2

of direct product subspaces S(¿ 1;¿2;j1;j2)=S1(¿ 1;j1) S2(¿ 2;j2). Now, because S1(¿ 1;j1) and S2(¿ 2;j2) are rotationally invariant, so is their direct product, i.e., any vector ¿ 1j1m1; ¿ 2j2m2 in this space will be take by an arbi- j i trary rotation onto the direct product of two other vectors, one from S1(¿ 1;j1) and one from S2(¿ 2;j2); it will remain inside S(¿ 1;¿2;j1;j2): On the other hand, although S(¿ 1;¿2;j1;j2) is rotationally invariant there is no reason to expect it that it is also irreducible. However, in decomposing S into irreducible invariant subspaces, we can use the Addition of Angular Momenta 195

fact that we already have a natural decomposition of that space into smaller invariant subspaces. To completely reduce the space S we just need to break these smaller parts into even smaller irreducible parts. Within any such space S(¿ 1;¿2;j1;j2) the values of ¿ 1;¿2;j1; and j2 are …xed. Thus we can simplify our notation in accord with the simpler problem at hand which can be stated thusly: …nd the irreducible invariant subspaces of a direct product space S(¿ 1;¿2;j1;j2)=S1(¿ 1;j1) S2(¿ 2;j2) with …xed values of j1 and j2: While working in this subspace we suppress any constant labels, and so denote by

S(j1;j2)=S(¿ 1;¿2;j1;j2)=S1(¿ 1;j1) S2(¿ 2;j2) (5.228) the subspace of interest and by

m1;m2 = ¿ 1;j1;m1; ¿ 2;j2;m2 (5.229) j i j i the original direct product states within this subspace. These latter are eigenvectors of 2 2 J1 and J2 with eigenvalues j1(j1 +1) and j2(j2 +1); respectively, and of J1z and J2z with 2 eigenvalues m1 and m2. We will denote the sought-after common eigenstates of J and Jz in this subspace by the vectors

j; m = ¿ 1;j1;¿2;j2; j; m (5.230) j i j i

which are to be formed as linear combinations of the states m1;m2 . With this notation we now proceed to prove the mainj result,i referred to as

The addition theorem:The(2j1 + 1)(2j2 +1)dimensional space S(j1;j2) contains exactly one irreducible subspace S(j) for each value of j in the sequence j = j1 + j2;j1 + j2 1; ; j1 j2 : (5.231) ¡ ¢¢¢ j ¡ j In other words, the subspace S(j1;j2) can be reduced into a direct sum

S(j1;j2)=S(j1 + j2) S(j1 + j2 1) S( j1 j2 ) (5.232) © ¡ ©¢¢¢ j ¡ j of irreducible invariant subspaces of the rotation group, where each space S(j) is spanned by 2j +1basis vectors j; m . j i To prove this result we begin with a few general observations, and then follow up with what is essentially a proof-by-construction. First, we note that since the space S(j1;j2) contains states m1;m2 with j i

j1 m1 j1 and j2 m2 j2 (5.233) ¸ ¸¡ ¸ ¸¡

the corresponding eigenvalues m = m1 + m2 of Jz within this subspace can only take on values in the range j1 + j2 m (j1 + j2): (5.234) ¸ ¸¡ This implies that the eigenvalues of J2 must, themselves be labeled by values of j satisfying the bound j1 + j2 j: (5.235) ¸ Moreover, it is not hard to see that the sum of m1 and m2 will result in integral values of m if m1 and m2are both integral or both half-integral and will result in half-integral 196 Angular Momentum and Rotations values of m if one is integral and the other half-integral. Since the integral character of m1 and m2 is determined by the character of j1 and j2; we deduce that

integral if j1;j2 are both integral or both half-integral j = (5.236) 8 < half-integral otherwise. With: these preliminary observations out of the way, we now proceed to observe that in the subspace S(j1;j2) there is only one direct product state m1;m2 in which the j i value of m = m1 + m2 takes on its largest value of j1 + j2; namely that vector in which m1 and m2 both individually take on their largest values, j1 and j2. This fact, we assert, 2 implies that the vector, m1;m2 = j1;j2 must be also be an eigenvector of J and Jz j i j i with angular momentum (j; m)=(j1 + j2;j1 + j2): In other words, it is that vector of an irreducible subspace S(j) with j = j1 + j2 having the maximum possible component of angular momentum along the z axis consistent with that value of j.Toprovethis assertion, we note that if this were not the case, we could act on this vector with the raising operator J+ = J1+ + J2+ (5.237) and produce an eigenstate of Jz with eigenvalue m = j1 + j2 +1. This state would have to be in S(j1;j2) because the latter is invariant under the action of the components of J~. But no vector exists in this space with m larger than j1 + j2: Thus, when J+ acts on j1;j2 it must take it onto the null vector. The only states having this property are those jof thei form j; m with j = m, which proves the assertion. j i Since there is only one such state in S(j1;j2) with this value of m; moreover, there can be only one irreducible subspace S(j) with j = j1 + j2 (in general there would be one such vector starting the sequence of basis vectors for each such subspace). Thus we identify j1 + j2;j1 + j2 = j1;j2 ; j i j i where the left side of this expression indicates the j; m state, the right side indicates the j i original direct product state m1;m2 : The remaining basis states j; m in this irreducible j i j i space S(j) with j = j1 + j2 can now, in principle, be produced by repeated application of the lowering operator J = J1 + J2 : (5.238) ¡ ¡ ¡ For example, we note that, in the j; m representation, j i

J j1 + j2;j1 + j2 = ¡j i

(j1 + j2)(j1 + j2 +1) (j1 + j2)(j1 + j2 1) j1 + j2;j1 + j2 1 ¡ ¡ j ¡ i = p2(j1 + j2) j1 + j2;j1 + j2 1 : (5.239) j ¡ i p But J = J1 +J2 ; so this same expression can be written in the m1;m2 representation, after some¡ manipulation,¡ ¡ as j i

(J1 + J2 ) j1;j2 = 2j1 j1 1;j2 + 2j2 j1;j2 1 : (5.240) ¡ ¡ j i j ¡ i j ¡ i Equating these last two results then givesp p

j2 j1 j1 + j2;j1 + j2 1 = j1;j2 1 + j1 1;j2 : (5.241) j ¡ i sj1 + j2 j ¡ i sj1 + j2 j ¡ i Addition of Angular Momenta 197

This procedure can obviously be repeated for the remaining basis vectors with this value of j. We now proced to essentially repeat the argument, by noticing that there are exactly two direct product states m1;m2 in which the value of m = m1 + m2 takes the j i next largest value possible, i.e., m = j1 + j2 1; namely, the states ¡ m1;m2 = j1;j2 1 j i j ¡ i m1;m2 = j1 1;j2 : (5.242) j i j ¡ i From these two orthogonal states we can produce any eigenvectors of Jz in S(j1;j2) having eigenvalue m = j1 + j2 1: In particular, we can form the state (5.241), which 2 ¡ is an eigenvector of J with j = j1 + j2: But we can also produce from these two direct product states a vector orthogonal to (5.241), e.g., the vector

j2 j1 j1;j2 1 j1 1;j2 : (5.243) sj1 + j2 j ¡ i¡sj1 + j2 j ¡ i Analogous to our previous argument we argue that this latter state must be an eigenstate 2 of J and Jz with angular momentum (j; m)=(j1 + j2 1;j1 + j2 1): In other words, ¡ ¡ it is that vector of an irreducible subspace S(j) with j = j1 + j2 1 having the maximum possible component of angular momentum along the z axis consistent¡ with that value of j. To prove this assertion, assume it were not the case. We could then act both on this vector and on (5.241) with the raising operator and produce in S(j1;j2) two orthogonal eigenstates of Jz with eigenvalue m = j1 + j2 (since,aswehaveseentheraisingand lowering operators preserve the orthogonality of such sequences). But there is only one such state with this value of m; and it is obtained by applying the raising operator to (5.241). Thus, application of J+ to (5.243) must take it onto the null vector, and hence it must be a state of the type asserted. Since there are no other orthogonal states of this type that can be constructed, we deduce that there is exactly one irreducible invariant subspace S(j) with j = j1 + j2 1; and we identify (5.243) with the state heading the sequence of basis vectors for that¡ space. As before, the remaining basis vectors j; m for j i this value of j can then be generated by applying the lowering operator J = J1 + J2 to (5.243). ¡ ¡ ¡ The next steps, we hope, are clear: repeat this procedure until we run out of vectors. The basic idea is that as we move down through each value of j we always encounter just enough linearly-independent direct product states with a given value of m to form by linear combination the basis vectors associated with those irreducible spaces already generated with higher values of j; as well as one additional vector which is to be constructed orthogonal to those from the other irreducible spaces. This remaining state forms the beginning vector for a new sequence of 2j +1basis vectors associated with the present value of j. Thus, e.g., for small enough n; we …nd that there are exactly n+1 direct product states m1;m2 in which the value of m = m1 + m2 takes the value m = j1 + j2 n; namely,j the statesi with ¡ m1 = m2 = j1 nj2 ¡ j1 n +1 j2 1 : (5.244) . ¡ . ¡ . . j1 j2 n ¡ If, at this stage, there has been exactly one irreducible invariant subspace S(j) for all values of j greater than j1 + j2 n; than we can form from these n +1 vectors those ¡ 198 Angular Momentum and Rotations n vectors having this m value that have already been obtained using J from spaces with higher values of j. We can then form exactly one additional vector¡ orthogonal to these (e.g., by the Gram-Schmidt procedure) which cannot be associated with any of the subspaces S(j) already constructed and so, by elimination, must be associated with the one existing subspace S(j) having j = j1 + j2 n: Thisstatemust,moreover,bethat state of this space in which m = j, (if it were¡ not we could use the raising operator to produce n +1orthogonal states with m having the next higher value which exceeds the number of orthogonal states of this type). Thus, the remaining basis vectors of this space can be constructed using the lowering operator on this state. We have inductively shown, therefore, that if there is exactly one irreducible invariant subspace S(j) for all values of j greater than j1 + j2 n; then there is exactly one such subspace for j = j1 + j2 n. This argument proceeds¡ until we reach a value of n for which there are not n +1basis¡ vectors with the required value of m. From the table above, we see that this occurs when the values of m1 or m2 exceed there natural lower bounds of j1 and j2: Conversely, the induction proof holds for all values of n such that ¡ ¡

j1 n j1 and j2 n j2 (5.245) ¡ ¸¡ ¡ ¸¡ or

2j1 n and 2j2 n (5.246) ¸ ¸ With j = j1 + j2 n; this implies that ¡

j j2 j1 and j j1 j2 (5.247) ¸ ¡ ¸ ¡ or more simply

j j1 j2 : (5.248) ¸j ¡ j Thus, the arguments above prove the basic statement of the addition theorem, namely that there exists in S(j1;j2) exactly one space S(j) for values of j starting at j1 + j2 and stepping down one unit at a time to the value of j1 j2 : Moreover, the method of proof contains an outline of the basic procedure usedj to¡ actuallyj construct the subspaces of interest. Of course there remains the logical possibility that there exist other irreducible subspaces in S(j1;j2) that are simply not accessible using the procedure outlined. It is straightforward to show that this is not the case, however, by simply counting the number of vectors produced through the procedure outlined. Indeed, for each value of j = j1 + j2; ; j1 j2 the procedure outlined above generates 2j +1 basis vectors. The total number¢¢¢ j of¡ suchj basis vectors is then represented by the readily computable sum

j1+j2 (2j +1)=(2j1 + 1)(2j2 +1) (5.249)

j= j1 j2 jX¡ j showing that they are su¢cient in number to generate a space of dimension equal to the original. Thus, the states j; m = ¿ 1;j1;¿2;j2; j; m formed in this way comprise an or- j i j i thonormal basis for the subspace S(j1;j2): Implicitly, therefore, there is within this subspace a unitary transformation between the original direct product states m1;m2 = j i ¿ 1;j1;¿2;j2; m1;m2 and the basis states j; m assocated with the total angular momen- j i j i tum J~. By an appropriate choice of phase, the expansion coe¢cients (i.e., the matrix elements of the unitary transformation between these two sets) can be chosen independent of ¿ and ¿ 0 and dependent only on the values j; m; j1;j2;m1; and m2: Thus, e.g., the Addition of Angular Momenta 199 new basis states can be written as linear combinations of the direct product states in the usual way, i.e.,

j1 j2 j; m = j1;j2;m1;m2 j1;j2;m1;m2 j; m (5.250) j i j ih j i m1= j1 m2= j2 X¡ X¡ wherewehaveincludedthequantumnumbersj1 and j2 in the expansion to explicitly indicate the subspaces that are being combined. Similarly, the direct product states can be written as linear combinations of the new basis states j; m ; i.e., j i j1+j2 j j1;j2;m1;m2 = j; m j; m j1;j2;m1;m2 : (5.251) j i j ih j i j= j1 j2 m= j jX¡ j X¡ These expansions are completely determined once we know the corresponding expansion coe¢cients j1;j2;m1;m2 j; m ; which are referred to as Clebsch-Gordon coe¢cients (or CG coe¢cients).h Di¤erentj authorsi denote these expansion coe¢cients in di¤erent ways, e.g., j1;j2;m1;m2 C = j; m j1;j2;m1;m2 : (5.252) j;m h j i It is straightforward, using the procedure outlined above, to generate the CG coe¢cients for given values of j1;j2; and j. They obey certain properties that follow from their de…nition and from the way in which they are constructed. We enumerate some of these properties below. 1. Restrictions on j and m - It is clear from the proof of the addition theorem detailed above that the CG coe¢cient j1;j2;m1;m2 j; m must vanish unless the two states in the innner product have theh same z componentj i of total angular momentum. In addition, we must have the value of total j on the right lie within the range produced by the angular momenta j1 and j2.Thuswehavetherestriction

j1;j2;m1;m2 j; m =0 unless m = m1 + m2 h j i j1;j2;m1;m2 j; m =0 unless j1 + j2 j j1 j2 : (5.253) h j i ¸ ¸j ¡ j The restriction on j is referred to as the triangle inequality, since it is equivalent to the condition that the positive numbers j; j1; and j2 be able to represent the lengths of the three sides of some triangle. As such, it is easily shown to apply to any permutation of these three numbers, i.e., its validity also implies that j1 + j ¸ j2 j1 j and that j + j2 j1 j j2 . ¸j ¡ j ¸ ¸j ¡ j 2. Phase convention - From the process outlined above, the only ambiguity involved in constructing the states j; m from the direct product states m1;m2 is at the point where we construct thej maximallyi aligned vector j; j for eachj subspacei S(j): This vector can always be constructed orthogonal to thej statesi with the same value of m associated with higher values of j; but the phase of the state so constructed can, in principle, take any value. To unambiguously specify the CG coe¢cients, therefore, this phase must be unambiguously speci…ed. This is done by de…ning the phase of this state relative to a particular direct product state. In particular, we de…ne the CG coe¢cients so that the coe¢cient

j1;j2;j1;j j1 j; j = j; j j1;j2;j1;j j1 0 (5.254) h ¡ j i h j ¡ i¸ is real and positive. Since the remaining states j; m are constructed from j; j using the lowering operator this choice makes all ofj thei CG coe¢cients real j i

j1;j2;m1;m2 j; m = j; m j1;j2;m1;m2 ; (5.255) h j i h j i 200 Angular Momentum and Rotations

although not necessarily positive (one can show, for example, that the sign of j1 m1 j1;j2;m1;m2 j; j is ( 1) ¡ .) h j i ¡ 3. Orthogonality and completeness relations - Being eigenstates of Hermitian operators the two sets of states j; m and m1;m2 each form an ONB for the fj ig fj ig subspace S(j1;j2): Orthonormality implies that

j1j2m1m2 j1j2m0 m0 = ±m ;m ±m ;m (5.256) h j 1 2i 1 10 2 20

j; m j0;m0 = ±j;j ±m;m (5.257) h j i 0 0 While completeness of each set within this subspace implies that

j1+j2 j j; m j; m =1 within S(j1;j2) (5.258) j ih j j= j1 j2 m= j jX¡ j X¡

j1 j2 j1;j2;m1;m2 j1;j2;m1;m2 =1 within S(j1;j2) (5.259) j ih j m1= j1 m2= j2 X¡ X¡ inserting the completeness relations into the orthonormality relations gives corresponding orthonormality conditions for the CG coe¢cients, i.e.,

j1+j2 j j1j2m1m2 j; m j; m j1j2m0 m0 = ±m ;m (5.260) h j ih j 1 2i 1 10 j= j1 j2 m= j jX¡ j X¡

j1 j2 j; m j1;j2;m1;m2 j1;j2;m1;m2 j0;m0 = ±j;j ±m;m : (5.261) h j ih j i 0 0 m1= j1 m2= j2 X¡ X¡ 4. Recursion relation -Thestates j; m in each irreducible subspace S(j) are formed from the state j; j by applicationj ofi the lowering operator. It is possible, as a result, to use thej loweringi operator to obtain recursion relations for the Clebsch- Gordon coe¢cients associated with …xed values of j; j1; and j2. To develop these relations we consider the matrix element of J between the states j; m and the § j i states j1;j2;m1;m2 ; i.e., we consider j i

j; m J j1;j2;m1;m2 = j; m (J1 + J2 ) j1;j2;m1;m2 (5.262) h j §j i h j § § j i On the left hand side of this expression we let J act on the bra j; m .Sincethis is the adjoint of J j; m the role of the raising and§ lowering operatorsh j is reversed, i.e., ¨j i j; m J = j(j +1) m(m 1) j; m 1 : (5.263) h j § ¡ ¨ h ¨ j Substituting this in above andp letting J1 and J2 act to the right we obtain the relations § § j(j +1) m(m 1) j; m 1 j1;j2;m1;m2 ¡ ¨ h ¨ j i p = j1(j1 +1) m1(m1 1) j; m j1;j2;m1 1;m2 ¡ § h j § i +p j2(j2 +1) m2(m2 1) j; m j1;j2;m1;m2 1 : (5.264) ¡ § h j § i p These relations allow all CG coe¢cients for …xed j; j1; and j2; to be obtained from a single one, e.g., j1;j2;j1;j j1 j; j . h ¡ j i Addition of Angular Momenta 201

5. Clebsch-Gordon series - As a …nal property of the CG coe¢cients we derive a relation that follows from the fact that the space S(j1;j2) is formed from the direct product of irreducible invariant subspaces S1(j1) and S2(j2). In particular, we know that in the space S1(j1) the basis vectors j1;m1 transform under rotations into linear combinations of the themselves accordingj toi the relation

j1 (1) (j1) UR j1m1 = j1;m10 Rm ;m ; (5.265) j i j i 10 1 m = j1 10X¡ where R(j1) is the rotation matrix associated with an irreducible invariant sub- m10 ;m1 space with angular momentum j1: Similarly, for the states j2;m2 of S2(j2) we have the relation j i j2 (2) (j2) UR j2m2 = j2;m20 Rm ;m : (5.266) j i j i 20 2 m = j2 20X¡ It follows that in the combined space the direct product states j1;j2;m1;m2 transform under rotations as follows j i

(1) (2) UR j1;j2;m1;m2 = U U j1;j2;m1;m2 j i R R j i (j1) (j2) = j1;j2;m10 ;m20 Rm ;m Rm ;m : (5.267) j i 10 1 20 2 m ;m X10 20

On the other hand, we can also express the states j1;j2;m1;m2 in terms of the states j; m ; i.e., j i j i

UR j1;j2;m1;m2 = UR j; m j; m j1;j2;m1;m2 : (5.268) j i j;m j ih j i X But the states j; m are the basis vectors of an irreducible invariant subspace S(j) of the combinedj subspace,i and so tranform accordingly,

(j) UR j; m = j; m0 R : (5.269) j i j i m0;m m X0 Thus, we deduce that

(j) UR j1;j2;m1;m2 = j; m0 R j; m j1;j2;m1;m2 (5.270) j i j i m0;mh j i j;m;m X 0 (j) = j1;j2;m10 ;m20 j1;j2;m10 ;m20 j; m0 R j; m j1;j2;m1;m2 j ih j i m0;mh j i m ;m j;m;m X10 20 X 0 (5.271) where in the second line we have transformed the states j; m0 back to the direct product representation. Comparing coe¢cients in (5.267)j and (5.271),i we deduce a relation between matrix elements of the rotation matrices for di¤erent values of j; namely,

(j1) (j2) (j) Rm ;m Rm ;m = j1;j2;m10 ;m20 j; m0 Rm ;m j; m j1;j2;m1;m2 (5.272) 10 1 20 2 h j i 0 h j i j;m;m X 0 which is referred to as the Clebsch-Gordon series. Note that in this last expression the CG coe¢cients actually allow the sum over m and m0 to both collapse to a single 202 Angular Momentum and Rotations

term with m = m1 + m2 and m0 = m10 + m20 ; making it equivalent to

j1+j2 (j1) (j2) (j) Rm ;m Rm ;m = j1;j2;m10 ;m20 j; m10 +m20 Rm +m ;m +m j; m1+m2 j1;j2;m1;m2 10 1 20 2 h j i 10 20 1 2 h j i j= j1 j2 jX¡ j (5.273)

5.11 Reducible and Irreducible Tensor Operators We have seen that it is possible to classify observables of a system in terms of their transformation properties. Thus, scalar observables are, by de…nition, invariant under rotations. The components of a vector observable, on the other hand, transform into well de…ned linear combinations of one another under an arbitrary rotation. As it turns out, these two examples constitute special cases of a more general classi…cation scheme involving the concept of tensor operators. By de…nition, a collection of n operators Q1;Q2; ;Qn comprise an n-component rotational tensor operator Q if they transform underf rotations¢¢¢ g into linear combinations of each other, i.e., if for each rotation R there are a set of coe¢cients Dji(R) such that

+ R [Qi]=URQiUr = QjDji(R): (5.274) j X It is straightforward to show that under these circumstances the matrices D(R) with matrix elements Dji(R) form a representation for the rotation group. As an example, we note that the Cartesian components Vx;Vy;Vz of a vector f g operator V~ form the components of a 3-component tensor V. Indeed, under an arbitrary rotation, the operator Vu = V~ u^ is transformed into ¢

R [Vu]=Vu = V~ u^0 (5.275) 0 ¢ where u^0 = ARu^ indicates the direction obtained by performing the rotation R on the vector u^.Thus,

ui0 = Aijuj: (5.276) j X If u^ corresponds to the Cartesian unit vector x^k; then uj = ±j;k and ui0 = Aik; so that

~ R [Vk]=V x^k0 = ViAik (5.277) ¢ i X which shows that the components of V~ transform into linear combinations of one another under rotations, with coe¢cients given by the 3 3 rotation matrices AR. As a second example, if V~ and W~ are vector£ operators and Q is a scalar operator, then the operators Q; Vx;Vy;Vz;Wx;Wy;Wz form a seven component tensor, since under an arbitrary rotationf R they are taken ontog

Q Q + 0 Vj + 0 Wj (5.278) ! j ¢ j ¢ X X Vi VjAji + 0 Wj +0 Q (5.279) ! j j ¢ ¢ X X Wi WjAji + 0 Vj +0 Q (5.280) ! j j ¢ ¢ X X Reducible and Irreducible Tensor Operators 203 which satis…es the de…nition. Clearly in this case, however, the set of seven components can be partitioned into 3 separate sets of operators Q ; Vx;Vy;Vz ; and Wx;Wy;Wz ; which independently transform into linear combinationsf g f of one another,g i.e.,f the tensorg can be reduced into two 3-component tensors and one tensor having just one component (obviously any scalar operator constitutes a one-component tensor). Note also that in this circumstance the matrices D(R) governing the transformation of these operators is block diagonal, i.e., it has the form 1 0 1 AR 0 1 D(R)=B C : (5.281) B C B @ A C B C B AR C B 0 1 C B C B C @ @ A A Thus this representation of the rotation group is really a combination of three separate representations, one of which is 1 dimensional and two of which are 3 dimensional. This leads us to the idea of irreducible tensors. A tensor operator T is said to be reducible if its components T1;T2; ;Tn , or any set of linear combinations thereof, can be partitioned into tensorsf having a¢¢¢ smallerg number of components. If a tensor cannot be so partitioned it is said to be irreducible. For example, if V~ is a vector operator, the set of operators Vx;Vy do not form f g a two component tensor, because a rotation of Vx about the y axis through ¼=2 takes it onto Vz; which cannot be expressed as a linear combination of Vx and Vy: Thus, a vector operator cannot be reduced into smaller subsets of operators. It follows that all vector operators are irreducible. It is interesting to note that the language that we are using here to describe the components of tensor operators is clearly very similar to that describing the behavior of the basis vectors associated with rotationally invariant subspaces of a quantum mechanical Hilbert space. Indeed, in a certain sense the basic reducibility problem has alread been solved in the context of combining angular momenta. We are led quite naturally, therefore, to introduce a very useful class of irreducible tensors referred to as spherical tensors. By de…nition, a collection of 2j +1 operators T m m = j; ;j form the com- f j j ¡ ¢¢¢ g ponents of an irreducible spherical tensor Tj of rank j if they transform under rotations into linear combinations of one another in the same way as the basis vectors j; m of an irreducible invariant subspace S(j): Speci…cally, this means that under a rotationj iR; the m operator Tj is taken onto

m m + m0 (j) R T = URT U = T R (5.282) j j R j m0;m m £ ¤ X0 where R(j) represents the rotation matrix associated with an eigenspace of J2 with this m0;m value of j. Since the basis vectors j; m transform irreducibly, it is not hard to see that the components of spherical tensorsj of thisi sort do so as well. 0 It is not hard to see that, according to this de…nition, a scalar observable Q = Q0 is an irreducible spherical tensor of rank zero, i.e., its transformation law

0 0 R Q0 = Q0 (5.283) is the same as that of the single basis£ vector¤ 0; 0 associated with a subspace of zero angular momentum, for which j i R [ 0; 0 ]= 0; 0 : (5.284) j i j i 204 Angular Momentum and Rotations

Similary, a vector operator V~ , which de…nes an irreducible three component tensor, can be viewed as a spherical tensor V1 of rank one. By de…nition, the spherical tensor components V m m =1; 0; 1 of a vector operator V~ are given as the following linear combinations f 1 j ¡ g

1 (Vx + iVy) 0 1 Vx iVy V = V = Vz V ¡ = ¡ (5.285) 1 ¡ p2 1 1 p2 of its Cartesian components. In this representation we see that the raising and lowering operators can be expressed in terms of the spherical tensor components of the angular momentum operator J~ through the relation

1 J = p2J1§ : (5.286) § ¨ To see that the spherical components of a vector de…ne an irreducible tensor of unit rank we must show that they transform appropriately. To this end it su¢ces to demonstrate the transformation properties for any vector operator, since the transformation law will clearly be the same for all vector operators (as it is for the Cartesian components). Consider, then, in the space of a single particle the vector operator R^ which has the e¤ect in the position representation of multiplying the wavefunction at ~r by the radial unit vector r;^ i.e.,

~r R^ ~r =^r ~r = ~r (5.287) j i j i ~r j i j j ~rÃ(~r) ~r R^ Ã =^rÃ(~r)= : (5.288) h j j i ~r j j In the position representation the Cartesian components of this operator can be written in spherical coordinates (r; µ; Á) in the usual way, i.e., x R^ = =cosÁ sin µ (5.289) x r y R^ = =sinÁ sin µ (5.290) y r z R^ = =cosµ (5.291) z r In this same representation the spherical components of this vector operator take the form 1 1 R^1 = (cos Á sin µ + i sin Á sin µ)= eiÁ sin µ (5.292) 1 ¡p2 ¡p2 ^0 R1 =cosµ (5.293) 1 1 1 iÁ R^¡ = (cos Á sin µ i sin Á sin µ)= e¡ sin µ (5.294) 1 p2 ¡ p2 Aside from an overall constant, these are equivalent to the spherical harmonics of order one, i.e.,

4¼ R^1 = Y 1(µ; Á) (5.295) 1 3 1 r 4¼ R^0 = Y 0(µ; Á) (5.296) 1 3 1 r 1 4¼ 1 R^¡ = Y ¡ (µ; Á) (5.297) 1 3 1 r Reducible and Irreducible Tensor Operators 205 which, as we have seen, transform as the basis vectors of an irreducible subspace with j =1

1 R [Y m]= Y m0 R(1) (5.298) 1 1 m0;m m = 1 X0 ¡ from which it follows that the spherical components of R^ transform in the same way, i.e.,

1 R R^m = R^m0 R(1) (5.299) 1 1 m0;m m = 1 h i X0 ¡

Thus, this vector operator (and hence all vector operators) de…ne a spherical tensor of rank one. As a natural extension of this, it is not hard to see that the spherical harmonics of a given order l de…ne the components of a tensor operator in (e.g.) the position representation. Thus, we can de…ne an irreducible tensor operator Yl with 2l +1components Y m m = l; ;l which have the following e¤ect in the position representation f l j ¡ ¢¢¢ g

~r Y m Ã = Y m(µ; Á)Ã(r; µ; Á): (5.300) h j l j i l

We note in passing that the components of this tensor operator arise quite naturally in the multipole expansion of electrostatic and magnetostatic …elds.

The product of two tensors is, itself, generally a tensor. For example, if Tj1 and Qj2 represent spherical tensors of rank j1 and j2; respectively, then the set of products T m1 Qm2 form a (2j + 1)(2j +1)component tensor. In general, however, such j1 j2 1 2 a tensor is reducible into tensors of smaller rank. Indeed, since the components T m1 ; j1 m2 © ª Q of each tensor transform as the basis vectors j1;m1 ; j2;m2 of an irreducible sub- j2 j i j i space S(j1);S(j2) the process of reducing the product of two spherical tensors into irreducible components is essentially identical to the process of reducing a direct product space S(j ;j ) into its irreducible components. Speci…cally, from the components T m1 Qm2 1 2 j1 j2 we can form, for each j = j1 + j2; ; j1 j2 an irreducible tensor Wj with components ¢¢¢ j ¡ j © ª

j1 j2 m m1 m2 W = T Q j1;j2;m1;m2 j; m m = j; ;j: (5.301) j j1 j2 h j i ¡ ¢¢¢ m1= j1 m2= j2 X¡ X¡

To show that these 2j +1 components comprise a spherical tensor of rank j we must show that they satisfy the approriate transformation law. Consider

m m + m1 m2 + R W = URW U = URT Q U j1;j2;m1;m2 j; m j j R j1 j2 R h j i m1;m2 £ ¤ X = U T m1 U +U Qm2 U + j ;j ;m ;m j; m R j1 R R j2 R 1 2 1 2 m ;m h j i X1 2 m10 m20 (j1) (j2) = Tj Qj Rm ;m Rm ;m j1;j2;m1;m2 j; m (5.302) 1 2 10 1 20 2 h j i m1;m2 m ;m X X10 20 206 Angular Momentum and Rotations

Using the Clebsch-Gordon series this last expression can be written

m m10 m20 (j0) R Wj = T Q j1;j2;m10 ;m20 j0;m0 R j1 j2 h j i m0;m00 m1;m2 m ;m j ;m ;m £ ¤ X X10 20 0 X0 00

j0;m00 j1j2;m1;m2 j1;j2;m1;m2 j; m £h j ih j i

m10 m20 (j) = T Q j1;j2;m10 ;m20 j0;m0 R j0;m00 j; m j1 j2 h j i m0;m00 h j i m ;m j;m ;m X10 20 X0 00 m10 m20 (j) = T Q j1;j2;m10 ;m20 j; m0 R j1 j2 h j i m0;m m m ;m X0 X10 20 = W m0 R(j) (5.303) j m0;m m X0 m which shows that Wj does indeed transform as the mth component of a tensor of rank j; and where we have used the orthonormality and completeness relations associated with the C-G coe¢cients.

5.12 Tensor Commutation Relations Just as scalar and vector observables obey characteristic commutation relations

[Ji;Q]=0 (5.304)

[Ji;Vj]=i "ijkVk (5.305) ijk X with the components of angular momentum, so do the components of a general spherical tensor operator. As with scalars and vectors, these commutation relations follow from the way that these operators transform under in…nitesimal rotations. Recall that under an in…nitesimal rotation Uu^(±®)=1 i±®Ju; an arbitrary operator Q is transformed into ¡ Q0 = Q i±® [Ju;Q] (5.306) ¡ Thus, the mth component of the spherical tensor Tj is transformed by Uu^(±®) into

m m m Ru^(±®)[T ]=T i±® Ju;T (5.307) j j ¡ j m On the other hand, by de…nition, under any rotation£Tj is transformed¤ into

j m m0 (j) Ru^(±®)[T ]= T R (^u; ±®): (5.308) j j m0;m m = j X0 ¡ But (j) R (^u; ±®)= j; m0 Uu^(±®) j; m m0;m h j j i = j; m0 1 i±®Ju j; m h j ¡ j i = ±m;m i±® j; m0 Ju j; m (5.309) 0 ¡ h j j i which implies that

j m m m0 Ru^(±®)[T ]=T i±® T j; m0 Ju j; m : (5.310) j j ¡ j h j j i m = j X0 ¡ The Wigner Eckart Theorem 207

Comparing these we deduce the commutation relations

j m m0 Ju;T = T j; m0 Ju j; m : (5.311) j j h j j i m = j £ ¤ X0 ¡ As special cases of this we have

m m Jz;Tj = mTj :

and £ ¤ m m 1 J ;Tj = j(j +1) m(m 1) Tj § : (5.312) § ¡ § £ ¤ p 5.13 The Wigner Eckart Theorem We now prove and important result regarding the matrix elements of tensor operators between basis states ¿;j;m of any standard representation. This result, known as the Wigner-Eckart theorem,j illustrates,i in a certain sense, the constraints put on the components of tensor operators by the transformation laws that they satisfy. Speci…cally, we will show that the matrix elements of the components of a spherical tensor operator TJ between basis states ¿;j;m aregivenbytheproduct j i M ¿;j;m T ¿ 0;j0;m0 = j; m J; j0;M;m0 ¿;j T ¿ 0j0 (5.313) h j J j i h j ih jj jj i of the Clebsch Gordon coe¢cient j; m J; j0;M;m0 and a quantity ¿;j T ¿ 0j0 that h j i h jj jj i is independent of m; M; and m0; referred to as the reduced matrix element. Thus, the “orientational” dependence of the matrix element is completely determined by geometrical considerations. This result is not entirely surprising, given that the two quantities on the M right hand side of the matrix element, i.e., the T ¿ 0;j0;m0 transform under rotations like J j i a direct product ket of the form J; M j0;m0 ; while the quantity on the left transforms j ij i as a bra of total angular momentum (j; m). The reduced matrix element ¿;j T ¿ 0j0 h jj jj i characterizes the extent to which the given tensor operator TJ mixes the two subspaces S(¿;j) and S(¿ 0;j0); and is generally di¤erent for each tensor operator. To prove the Wigner-Eckart theorem we will simply show that the matrix elements of interest obey the same recursion relations as the CG coe¢cients. To this end, we use the simplifying notation

j1;j2;m1;m2 C = j; m j2;j1;m1;m2 (5.314) j;m h j i for the CG coe¢cients and denote the matrix elements of interest in a similar fashion, i.e., j1;j2;m1;m2 m2 T = ¿;j;m T ¿ 0;j2;m2 : (5.315) j;m h j j1 j i This latter quantity is implicitly a function of the labels ¿;¿0 , but we will suppress this dependence until it is needed. We then recall that the CG coe¢cients obey a recursion relation that is generated by consideration of the matrix elements

j; m J j2;j1;m1;m2 = j; m (J1 + J2 ) j2;j1;m1;m2 (5.316) h j §j i h j § § j i which leads to the relation

j1;j2;m1;m2 j1;j2;m1 1;m2 j(j +1) m(m 1)Cj;m 1 = j1(j1 +1) m1(m1 1) Cj;m § ¡ ¨ ¨ ¡ § j1;j2;m1;m2 1 p p+ j2(j2 +1) m2(m2 1)C (5.317)§ ¡ § j;m p 208 Angular Momentum and Rotations

To obtain a similar relation for the matrix elements of Tj1 we consider an “analogous” matrix element

m1 m1 ¿;j;m J Tj ¿ 0;j2;m2 = j(j +1) m(m 1) ¿;j;m 1 Tj ¿ 0;j2;m2 h j § 1 j i ¡ ¨ h ¨ j 1 j i j1;j2;m1;m2 = pj(j +1) m(m 1) Tj;m 1 : (5.318) ¡ ¨ ¨ We can evaluate this in a second wayp by using the commutation relations satis…ed by J and T m1 ; i.e., we can write § j1

m1 m1 m1 J Tj = J ;Tj + Tj J § 1 § 1 1 § m1 1 m1 = £ j1(j1 +1)¤ m1(m1 1) Tj § + Tj J (5.319) ¡ § 1 1 § which allows us to express thep matrix element above in the form

m1 m1 1 ¿;j;m J Tj ¿ 0;j2;m2 = j1(j1 +1) m1(m1 1) ¿;j;m Tj § ¿ 0;j2;m2 h j § 1 j i ¡ § h j 1 j i + ¿;j;m T m1 J ¿ ;j ;m (5.320) p j1 0 2 2 h j §j i which reduces to

m1 j1;j2;m1 1;m2 ¿;j;m J Tj ¿ 0;j2;m2 = j1(j1 +1) m1(m1 1) Tj;m § h j § 1 j i ¡ § j1;j2;m1;m2 1 p+ j2(j2 +1) m2(m2 1) T § (5.321) ¡ § j;m Comparingthetwoexpressionsfor ¿;j;mp J T m1 ¿ ;j ;m we deduce the recursion j1 0 2 2 relation h j § j i

j1;j2;m1;m2 j1;j2;m1 1;m2 j(j +1) m(m 1) Tj;m 1 = j1(j1 +1) m1(m1 1) Tj;m § ¡ ¨ ¨ ¡ § j1;j2;m1;m2 1 p p+ j2(j2 +1) m2(m2 1) T (5.322)§ ¡ § j;m

p j1;j2;m1;m2 which is precisely the same as that obeyed by the Clebsch-Gordon coe¢cients Cj;m : The two sets of number, for given values of j; j1; and j2; must be proportional to one another. Introducing the reduced matrix element ¿;j T ¿ 0j0 as the constant of proportionality, we deduce that h jj jj i

j1;j2;m1;m2 j1;j2;m1;m2 T = ¿;j T ¿ 0j0 C (5.323) j;m h jj jj i j;m which becomes after a little rearrangement

M ¿;j;m T ¿ 0;j0;m0 = j; m J; j0;M;m0 ¿;j T ¿ 0j0 : (5.324) h j J j i h j ih jj jj i This theorem is very useful because it leads automatically to certain selection rules. Indeed, because of the CG coe¢cient on the right hand side we see that the matrix M element of TJ between two states of this type vanishes unless

¢m = m m0 = M (5.325) ¡

j0 + J j j0 J : (5.326) ¸ ¸j ¡ j 0 Thus, for example we see that the matrix elements of a scalar operator Q0 vanish unless ¢m =0and ¢j = j j0 =0: Thus, scalar operators cannot change the angular momentum of any states that¡ they act upon. (They are often said to carry no angular momentum, in contrast to tensor operators of higher rank, which can and do change the The Wigner Eckart Theorem 209 angular momentum of the states that they act upon.) Thus the matrix elements for scalar operators take the form

0 ¿;j;m Q ¿ 0;j0;m0 = Q¿;¿ ±j;j ±m;m : (5.327) h j 0j i 0 0 0 In particular, it follows that within any irreducible subspace S(¿;j) the matrix represent- 0 ing any scalar Q0 is just a constant Q¿ times the identity matrix for that space (con…rming the rotational invariance of scalar observables within any such subspace), i.e.,

0 ¿;j;m Q ¿;j;m0 = Q¿ ±m;m : (5.328) h j 0j i 0 Application of the Wigner-Eckart theorem to vector operators V;~ leads to con- m sideration of the spherical components V1 m =0; 1 of such an operator. The corresponding matrix elements satisfy the relationf j § g

M ¿;j;m V ¿ 0;j0;m0 = j; m 1;j0;M;m0 ¿;j V ¿ 0j0 ; (5.329) h j 1 j i h j ih jj jj i and vanish unless ¢m = M 0; 1 : (5.330) 2f § g Similarly, application of the triangle inequality to vector operators leads to the selection rule ¢j =0; 1: (5.331) § Thus, vector operators act as though they impart or take away angular momentum j =1. The matrix elements of a vector operator within any given irreducible space are proportional to those of any other vector operator, such as the angular momentum operator J;~ whose spherical components satisfy

M ¿;j;m J ¿ 0;j0;m0 = j; m 1;j0;M;m0 ¿;j J ¿ 0j0 ; (5.332) h j 1 j i h j ih jj jj i from which it follows that

M M ¿;j;m V ¿;j;m0 = ®(¿;j) ¿;j;m J ¿;j;m0 (5.333) h j 1 j i h j 1 j i where ®(¿;j)= ¿;j V ¿;j = ¿;j J ¿;j is a constant. Thus, within any subspace h jj jj i h jj jj i S¿ (j) all vector operators are proportional, we can write

V~ = ®J~ within S¿ (j): (5.334) It is a straight forward exercise to compute the constant of proportionality in terms of the scalar observable J~ V;~ the result being what is referred to as the projection theorem, i.e., ¢ J~ V~ V~ = h ¢ i J~ within S¿ (j); (5.335) j(j +1) where the mean value J~ V~ ; being a scalar with respect to rotation can be taken with h ¢ i respect to any state in the subspace S¿ (j): In a similar manner one …nds that the nonzero matrix elements within any irreducible subspace are proportional, i.e., for two nonzero tensor operators TJ and WJ of the same rank, it follows that, provided ¿;j W ¿j =0; h jj jj i6 M M ¿;j;m T ¿ 0;j0;m0 = ® ¿;j;m W ¿ 0;j0;m0 (5.336) h j J j i h j J j i where ® = ¿;j T ¿ 0j0 = ¿;j W ¿ 0;j0 : Thus, the orientational dependence of the (2j + h jj jj i h jj jj i 1)(2J + 1)(2j0 +1) matrix elements is completely determined by the transformational properties of the states and the tensors involved. 210 Angular Momentum and Rotations Chapter 4 BOUND STATES OF A CENTRAL POTENTIAL

4.1 General Considerations As an application of some of these ideas we brieﬂy investigate the properties of a particle of mass m subject to a force deriving from a spherically symmetric potential V (r). Obviously, in many of the cases where this problem is of interest, the mass m referred to is the reduced mass m m m = 1 2 (4.1) m1 + m2 of an interacting two-particle system and the radial coordinate r corresponds to the magnitude of the relative position vector ~r = ~r2 ~r1, although these details need not concern us here. The corresponding quantum system− is governed by a Hamiltonian operator P 2 H = + V (R) (4.2) 2m which in the position representation takes the usual form

2 ~2 ∇ + V (r), (4.3) − 2m and our goal is information about the bound state solutions ψ to the energy eigenvalue equation | i (H ε) ψ =0, (4.4) − | i whereweassumethatV (r) 0 as r , so that bound state solutions are identiﬁed as those square normalizeable→ solutions→∞

ψ ψ = d3r ψ(~r) 2 =1 (4.5) h | i | | Z for which ε 0. The≤ spherical symmetry of V (r) ( and thus of H) suggest the use of spherical coordinates for which the identity 1 ∂2 1 ∂2 ∂ 1 ∂2 2 = r + +cotθ + (4.6) ∇ r ∂r2 r2 ∂θ2 ∂θ sin2 θ ∂φ2 · ¸ holds for all points except at r =0, which is a singular point of the transformation. This last expression can be written in the form

2 2 2 2 2 2 2 ~ ∂ ~ L P = ~ = r + (4.7) − ∇ − r ∂r2 r2 where ∂2 ∂ 1 ∂2 L2 = +cotθ + (4.8) − ∂θ2 ∂θ sin2 θ ∂φ2 · ¸ 124 Bound States of a Central Potential is the operator, in this representation, associated with the square of the (dimensionless) orbital angular momentum L~ = R~ K.~ Thus, as in classical mechanics, the kinetic energy × H0 separates into a radial part and a rotational part

P 2 2L2 H = r + ~ , (4.9) 0 2m 2mr2 where 2 2 2 2 1 ∂ ~ 1 ∂ Pr = ~ r = r . (4.10) − r ∂r2 i r ∂r · ¸ This last form suggests that the operator

1 ∂ P = ~ r (4.11) r i r ∂r can be viewed as the radial component of the momentum. This is a legitimate inference, and indeed Pr as deﬁned above does correspond to the Hermitian part 1 Pr = P~ Rˆ + Rˆ P~ . (4.12) 2 · · h i of the operator P~ Rˆ. Some care must be taken, however, because (it can be shown that) · Pr is only Hermitian on the space of wave functions ψ(~r) such that limr 0 rψ(~r)=0, 1 → while the eigenfunctions of Pr all diverge at r =0as r− . Thus, Pr is an example of a Hermitian operator that is not an observable. None of this is too important for the task at hand, however. Indeed, with the aforementioned formulae we can write the energy eigenvalue equation of interest in the form

2 1 ∂2 2L2 ~ r + ~ + V (r) ε ψ(~r)=0 (4.13) −2m r ∂r2 2mr2 − ½ ¾ which is valid at all points except r =0. 2 We now make the observation that L and Lz = i∂/∂φ (or any other component − of L~ ) commute with the each other and with any function of r or Pr.Itisobvious, therefore, that they commute with H.Thus,because

2 2 H, L = L ,Lz =[H, Lz]=0 (4.14) we know that there exists a basis£ of¤ eigenstates£ ¤ common to the three operators H, L2, and Lz. These states, which we will denote by n, l, m are characterized by their associated eigenvalues {| i}

H n, l, m = εn,l n, l, m (4.15) | i | i L2 n, l, m = l(l +1)n, l, m (4.16) | i | i Lz n, l, m = m n, l, m (4.17) | i | i and we denote by ψ (~r)= ~r n, l, m (4.18) n,l,m h | i the wavefunctions in the position representation associated with these states. Note that the eigenvalues of H are labeled by the principle quantum number n and by the total angular momentum quantum number `, but not by the azimuthal quantum number m, thus reﬂecting the necessary rotational degeneracy associated with the scalar operator H. General Considerations 125

2 It is also clear from our earlier discussions that the eigenfunctions of L and Lz can be written in the general form

m ψn,l,m(~r)=Fn,l(r)Yl (θ, φ) (4.19) involving the spherical harmonics, with ` =0, 1, 2, , and m = `, , +`. Substitution of this assumed form into the energy eigenvalue equation··· results− in the··· following ordinary diﬀerential equation

~2 1 d2 `(` +1)~2 rFn,l + + V (r) εn,l Fn,l =0 (4.20) −2m r dr2 2mr2 − µ ¶ for the radial functions Fn,l(r). This equation is independent of the eigenvalue m of Lz, confirming our labeling of the eigenfunctions based upon the expected rotational degeneracy of the system. Thus, we can anticipate that each energy eigenvalue εn,l will be 2` +1 fold degenerate due to the rotational invariance of H. Since the differential equation does, generally, depend upon the quantum number `, we can expect a different series of eigenvalues εn,l for each value of l. Additional accidental degeneracies may arise, however, (and in fact do arise when V (r)=kr2, which corresponds to the 3D harmonic oscillator, and when V (r)= k/r, which corresponds to the Coulomb potential.) As a further simplification− it is customary to introduce a new function

φn,l(r)=rFn,l(r) (4.21) which obeys the equation

~2 `(` +1)~2 φ00 + + V (r) εn,l φ =0. (4.22) −2m n,l 2mr2 − n,l · ¸ This latter equation is of the precise form

~2 φ00 +[Veff (r) εn,l] φ =0. −2m n,l − n,l obeyed by a particle moving in a 1-dimensional eﬀective potential

`(` +1) 2 V (r)= ~ + V (r) (4.23) eff 2mr2 consisting of the so-called “centrigugal barrier” `(`+1)~2/2mr2 in addition to the central potential experienced by the particle. Note that the wavefunction is deﬁned only for r>0. Moreover, the normalization condition

2 2 r dr dΩ ψn,l,m =1 (4.24) Z ¯ ¯ leads, along with the standard normalization¯ condition¯ for the spherical harmonics

dΩ Y m 2 =1 (4.25) | ` | Z to a normalization condition

∞ 2 2 ∞ 2 dr r Fn,l(r) = dr φ (r) =1 (4.26) | | n,l Z0 Z0 ¯ ¯ ¯ ¯ 126 Bound States of a Central Potential

for the functions φn,l which make them correspond to a particle whose wave function is restricted to the positive real axis. As we will see, the boundary condition obeyed by these functions for most cases of interest corresponds to one in which φn,l 0 as r 0, as though there were an inﬁnite barrier at the origin conﬁning to the particle→ to the→ region r>0. To see this, we consider the general properties of the solution to Eq.(4.22) for small values of r. First, we rewrite the equation in the form

`(` +1) φ00 + v(r)+kn,l φ =0 (4.27) n,l − r2 n,l · ¸ where 2mV (r) 2 2m εn,l v(r)= kn,l = | | (4.28) ~2 ~2 and we are assuming the εn,l < 0, as is appropriate to bound state solutions. We further assume the following:

1. V (r) is bounded except near r =0. 2. V (r) M/r as r 0 for some positive constant M. In other words, the magnitude |of V diverges| ≤ at the→ origin no more quickly than the Coulomb potential. 3. With these constraints, we now assume a solution near the origin which is of the form of a power law in r, i.e., we assume that

φ (r) Crs+1 r 0 (4.29) n,l ∼ → for some positive constant s. This assumed form corresponds to the actual radial s wave function having the behavior Fn,l(r) Cr as r 0. ∼ → If we now substitute these limiting forms into the energy eigenvalue equation we obtain the following algebraic relation

s 1 s 1 s+1 2 s+1 s(s +1)r − `(` +1)r − v(r)r + k r =0. (4.30) − − − n,l Now as r 0, the last two terms are at most of order rs and become negligible compared → s 1 to terms of order r − . Thus, for this form to be consistent at small r we must have s(s +1)=`(` +1), which has two solutions

s = `s= (` +1). (4.31) − In principle, this gives us two linearly independent solutions to the second order diﬀerential equation. The solution with s = ` gives a solution for which, at small r,

`+1 ` φ (r) Cr Fn,l(r) Cr (4.32) n,l ∼ ∼ while the solution with s = (` +1)gives a solution which for small r has the limiting behavior − C C φ (r) Fn,l(r) . (4.33) n,l ∼ r` ∼ r`+1 Of these, the solutions of the second type can be rejected on physical grounds. First, if `>0, then the solution of the form φ(r) C/r` is not normalizeable ∼ a dr C 2 ε 0. (4.34) | | r2` →∞ → Zε Hydrogenic Atoms: The Coulomb Problem 127

In the l =0case the situation is a little diﬀerent. In fact, if l =0, the kinetic energy operator acting on any function proportional to 1/r gives a delta function, i.e., for r near zero, 2 2 1 Fn,0(r) =4πδ(~r). (4.35) ∇ ∼∇ r · ¸ The presence of this delta function prevents the energy eigenvalue equation from being satisﬁed at the origin, since the potential only has at most algebraic singularities at that point. Thus, the second “solution” for l =0is not actually a valid solution to the eigenvalue equation. It is a spurious solution arising from the singular nature of the transformation to spherical coordinates at r =0. Thus, in the end we conclude that under these conditions we obtain exactly one regular radial function

`+1 ` φ Cr Fn,l Cr r 0 (4.36) n,l ∼ ∼ → for each value ` =0, 1, 2, of the orbital angular momentum. Note that this limiting form implies the following··· “boundary condition”

lim φn,l(r)=0 (4.37) r 0 → for the eﬀective one-dimensional problem, making the wave function act as though there were an inﬁnite potential barrier at r =0. We have gone about as far as we can go without actually specifying the potential. We thus turn to what is certainly the most important application of these ideas to atomic systems, namely, the Coulomb problem associated with a particle moving in a Coulombic potential well.

4.2 Hydrogenic Atoms: The Coulomb Problem We now assume that potential of interest is of the form

1 Zq2 V (r)= (4.38) −4πε0 r associated with the Coulomb interaction between a nucleus of charge +Zq and a single electron of charge q. We ignore the spin of the contituents, and rewrite the potential as − Ze2 V (r)= (4.39) − r 2 2 by introducing e = q /4πε0. From dimensional arguments it is possible to generate estimates of the energy and length scales associated with bound states of this system. This is a useful exercise since it allows us to express the equations of interest in dimensionless form. These scales of interest arise from the fact that the mean energy

E = T + V (4.40) h i h i h i of a bound state depends upon both its kinetic energy

P 2 T = h i (4.41) h i 2m and its potential energy Ze2 V (4.42) h i ∼− a 128 Bound States of a Central Potential where a r is a typical distance of the electron from the origin, i.e., its spread a ∆r about the∼ nucleus.h i In any stationary bound state the mean position of the particle∼ must itself be stationary, which requires that P~ =0, and so, using the uncertainty principle we have the estimate h i

2 ∆P = P 2 ~ = ~ T ~ . (4.43) h i ∼ ∆r a h i ∼ 2ma2 p Thus, an order of magnitude estimate gives

2 Ze2 E ~ . (4.44) h i ∼ 2ma2 − a To estimate the ground state energy, therefore, we minimize this with respect to the parameter a.Setting∂ E /∂a =0we ﬁnd that h i 1 2 a a = ~ = 0 (4.45) Z me2 Z where ~2 a0 = 0.51angstrom (4.46) me2 ∼ turns out to be the Bohr radius. Putting this into our expression for the energy gives the estimate 2 2 Z ~ 2 2 E = 2 = Z ε0 Z 13.6eV (4.47) −2ma0 − ∼− × where ~2 ε0 = 2 13.6eV 2ma0 ∼ In addition, we can obtain an estimate for a typical velocity for the problem through the relation 2 p ~ e 6 v = = Z = Zv0 Z 2.2 10 m/s. (4.48) m ∼ ma ~ ∼ × × In atomic physics it is useful to introduce a set of units in which e = ~ = me =1, so that a0 =1,v0 =1, and E0 =1/2. We shall not do that in the present treatment, but we will use these length and energy scales to appropriately transform the problem. For example, the radial equation for this problem takes the form

~2 Ze2 l(l +1)~2 φ00 + + En,l φ =0. (4.49) 2m n,l r − 2mr2 n,l µ ¶ Introducing a dimensionless position variable ρ = r/a thiscanbewrittenintheform

~2 d2φ Ze2 1 l(l +1)~2 1 + + En,l φ =0. (4.50) 2ma2 dρ2 2a ρ − 2ma2 ρ2 µ ¶ 2 2 2 Dividing this through by E0 = Z ε0 = ~ /2ma this reduces to the form d2φ 2 l(l +1) + λ2 φ =0 (4.51) dρ2 ρ − ρ2 − n,l µ ¶ where 2 En,l λn,l = 2 0 (4.52) −Z ε0 ≥ Hydrogenic Atoms: The Coulomb Problem 129 is a positive dimensionless quantity for the bound states that we seek. From this equation we now determine the asymptotic behavior of the function φ(ρ) for large values of ρ. Indeed, for ρ >> 1 (which corresponds to r>>a) we can neglect 1 2 terms of order ρ− and ρ− in the eigenvalue equation, which then has the asymptotic form 2 d φ 2 = λ φ (ρ ) λ = λn,l. (4.53) dρ2 →∞ λρ This has two independent solutions φ Ae± , of which the diverging solution must be rejected as being non-normalizable. From∼ the well-behaved solution we introduce a new substitution, setting λρ u(ρ)=un,l(ρ)=φn,l(ρ)e (4.54) which makes u(ρ) a slowly varying function compared to the exponential. To determine the function u(ρ) we evaluate dφ λρ = e− [u0 λu] (4.55) dρ − 2 d φ λρ 2 = e− u00 2λu0 + λ u (4.56) dρ2 − in terms of which the radial equation can£ be re-expressed as¤

2 l(l +1) u00 2λu0 + u =0. (4.57) − ρ − ρ2 · ¸ The classical method of solving such an equation is to assume a power series solution

∞ j s ∞ k u(ρ)= cjρ = ρ bkρ (4.58) j=0 k=0 X X where in the second form we explicitly assume the ﬁrst coeﬃcient b0 is explicitly not equal to zero, and so have pulled out the leading power in the expansion. Indeed, we know that l+1 for small r the function φn,l(r) goes to zero as r . It follows that in the same limit the function u(ρ) has the leading behavior

λρ l+1 l+2 un,l(ρ) e φ Aρ + O(ρ ). (4.59) ∼ n,l ∼ Thus we write l+1 ∞ k ∞ k+l+1 u(ρ)=ρ bkρ = bkρ . (4.60) k=0 k=0 X X Calculating the derivatives of this expression term-by-term and substituting into the differential equation for u(ρ) wecancombinetermsinpowersofρ to determine a recursion relation for the coeﬃcients:

k(k + l +1)bk =2[(k + l)λ 1] bk 1 (4.61) − −

This relation allows all coefficients to be expressed in terms of a single coefficient b0. We observe that b 2kλ 2λ lim k+1 0. (4.62) k b ∼ k2 ∼ k → →∞ k Thus, the series converges for all fixed ρ. However, in general, the function u(ρ) to which it converges will itself diverge exponentially as a function of ρ as e2λρ. This leads to unacceptable solutions of the form φ eλρ. To prevent this from happening the n,l ∼ 130 Bound States of a Central Potential power series has to terminate, so that it reduces to a polynomial function. Thus, we conclude that acceptable solutions to the eigenvalue equation have the property that the power series terminates: there exists some value of k 1 for which the corresponding coefficients vanish, i.e., for which ≥

bk+n =0 n =0, 1, 2, . (4.63) ··· For this to occur, we must have

bk =2[(k + l) λ 1] bk 1 =0. (4.64) − − This leads to the condition that (k + l)λ =1, hence that the allowed values of λ satisfy the equation 1 λ = k =1, 2, (4.65) k + l ··· Turning this back into an equation for the energies, we ﬁnd that

2 En,l = λn,lE0 = λn,lZ ε0 (4.66) − − where 1 1 λn,l = = k =1, 2, ,n=(l +1), (l +2), . (4.67) k + l n ··· ··· Thus, the spectrum of energies takes the form

2 Z ε0 En,l = n =(l +1), (l +2), . (4.68) − n2 ··· Note that for ﬁxed n, several l values produce the same energy. This is an accidental degeneracy that is not required by the rotational invariance of the problem, and reﬂects other symmetries to the hydrogen atom problem that are not apparent in the present treatment. Expressed in the more standard way, the spectrum can be written

E0 En = l =0, 1, 2, ,n 1. (4.69) − n2 ··· −

Having determined the energies, and the corresponding values of λn,l, the coeﬃcients of the power series expansion are then determined by recursion. Since the series terminates at k = n l, the acceptable solutions take the form of polynomials − n l 1 n − − k+l 1 k un,l(ρ)= bkρ − = ckρ (4.70) k=0 k=l+1 X X of order n in ρ; the functions

n 1 − λn,lr/a0 k φn,l = e− ck(r/a0) (4.71) k=l X involve polynomials of order n 1 in r and the total wavefunctions take the form − n 1 k − r r/na0 m ψn,l,m = e− CkYl (θ, φ) (4.72) a0 k=l X µ ¶ and have energies E0 En = n =1, 2 . (4.73) − n2 ··· The 3-D Isotropic Oscillator 131

For each energy level En there are (2l +1)-fold degenerate angular momentum multiplets corresponding to l =0, 1, (n 1). The degeneracy of the nth level, therefore, can be calculated as ··· − n 1 − 2 gn = (2l +1)=n . (4.74) l=0 X In spectrocopic notation the bound states n, l, m for angular momentum states with l =0, 1, 2, 3, 4, 5, are indicated with a letter| s, p,i d, f, g, h, , respectively. Thus, e.g., ··· ···

ψ2p 2, 1,m (4.75) m → | i ψ3d 3, 2,m (4.76) m → | i and so on.

4.3 The 3-D Isotropic Oscillator The Hamiltonian P 2 1 H = + mω2R2 (4.77) 2m 2 3 P 2 1 = α + mω2X2 (4.78) 2m 2 α α=1 X · ¸ is separable in Cartesian coordinates, and has eigenstates

ψ (~r)= ~r n = ~r nx,ny,nz = ψ (x)ψ (y)ψ (z) n h | i h | i nx ny nz that are the products of 1D oscillator wave functions

1/4 π− q2 /2 ψ (x)= e− x H (q ) nx n nx x √2 x nx! where Hnx (q) is the Hermite polynomial, qx = βx, and β = mω/~. The spectrum of the isotropic oscillator Hamiltonian are the energies p 3 En =(n + )~ω n = nx + ny + nz 0, 1, 2, . 2 ∈ { ···} which are generally degenerate due to the many diﬀerent ways a positive integer n can be represented as the sum of three other positive integers. We note for future reference that the product form of the eigenfunctions implies that

β2r ψ(~r)=e− fn(~r) where fn(~r) is of polynomial order in r. Because H is clearly a scalar with respect to rotations (being a function of P~ P~ · and R~ R~) we know that there also exists an basis of eigenstates · ψ (~r)= ~r n, l, m = Fn,l(r)Yl,m(θ, φ) n,l,m h | i 2 common to the operators H, L , and Lz. Indeed, we know that the function φn(r)= rFn,l(r) satisiﬁes the equation

2 2 ~ `(` +1)~ 1 2 2 φ00 + + mω r εn,l φ =0. (4.79) −2m n,l 2mr2 2 − n,l · ¸ 132 Bound States of a Central Potential

As before, we simplify by multiplying through by 2m/~2 to obtain (surpressing indices) − `(` +1) 4 2 φ00 + β r ν φ =0, (4.80) − r2 − · ¸ 2 4 2 2 2 where ν =2mεn,l/~ and β = m ω /~ . As in the Coulomb problem we first determine 2 the asymptotic behavior of the solution for r>>β− , in which limit the differential equation above simplifies to 4 2 φ00 β r φ =0, (4.81) − which has two solutoins, which at large r are dominated by the behavior exp( β2r2/2). The decaying solution corresponds to the behavior already noted in the Cartesian± form, while the growing one is clearly unacceptable. We thus introduce a new function u(r) such that β2r2 φ(r)=u(r)e− in terms of which β2r2 2 φ0(r)=e− [u0 β ru] − β2r2 2 4 2 2 φ00 = e− u00 2β ru0 + β (r β )u . − − The equation for the functoin u(r) is then£ found to take the form ¤

2 2 `(` +1) u00 2β ru0 β + ν u =0. − − r2 − · ¸ We assume a power series solution consistent with the small r behavior deduced earlier, i.e.,

∞ k+`+1 u(r)= bkr b0 =0. 6 k=0 X Substitution into the differential equation for u leads, eventually, to a series of equations. The first equation of interest comes from setting the coefficients of r`+2 equal to zero and requires for its solution that b1 =0.For k>1 the resulting equations reduce to a recursion relation of the form 2 (k +2)(k +2` +3)bk+2 = (2k +2` +3)β ν bk. − From this we deduce that all coefficeints bk £with k odd are identically¤ zero. From the large k behavior of the even terms we deduce that if the series does not terminate it will converge to a function that for large r behaves as exp(β2r2), which leads again to the unacceptable solution. Thus, as in the Coulomb problem, the acceptable solutions must terminate, i.e., there exists an even integer k such that bk =0, but 0=bk+2 = bk+4 = . Thiscanonlyhappenif,forsomeevenintegerk, 6 ··· ν =2β2 (k + ` +3/2) k =0, 2, 4, ··· With the definition of ν and β this implies that

εn,l = ~ω (n +3/2) n = k + `k=0, 2, 4, ··· Clearly, for a given value of n, the allowed ` values of orbital angular momentum satisfy the relation εn,l = ~ω (n +3/2) ` = n k − Thus, more than one value of ` is generally allowed for each value of n. For n even the allowed values of ` include ` =0, 2, ,n, while for odd n the allowed values include ` = 1, 3, ,n. There is, again, an accidental··· degeneracy in this system above that required by the··· rotational invariance of the Hamiltonian. It is a straightforward exercise to show that the degeneracy of the nth level of the isotropic oscillator is (n +1)(n +2)/2. Chapter 8 TIME DEPENDENT PERTURBATIONS: TRANSITION THEORY

8.1 General Considerations The methods of the last chapter have as their goal expressions for the exact energy eigenstates of a system in terms of those of a closely related system to which a constant perturbation has been applied. In the present chapter we consider a related problem, namely, that of determining the rate at which transitions occur between energy eigenstates of a quantum system of interest as a result of a time-dependent, usually externally applied, perturbation. Indeed, it is often the case that the only way of experimentally determining the structure of the energy eigenstates of a quantum mechanical system is by perturbing it in some way. We know, e.g., that if a system is in an eigenstate of the Hamiltonian, then it will remain in that state for all time. By applying perturbations, however, we can induce transitions between di¤erent eigenstates of the unperturbed Hamiltonian. By probing the rate at which such transitions occur, and the energies absorbed or emitted by the system in the process, we can infer information about the states involved. The calculation of transition rates for such situations, and a number of others of practical interest are addressed in this chapter. To begin, we consider a system described by time-independent Hamiltonian H0 to which a time-dependent perturbation V^ (t) is applied. Thus, while the perturbation is acting, the total system Hamiltonian can be written

H(t)=H0 + V^ (t): (8.1)

It will be implicitly assumed unless otherwise stated in what follows that the perturbation V^ (t) is small compared to the unperturbed Hamiltonian H0; if we want to study the eigenstates of H0 we do not want to change those eigenstates drastically by applying a strong perturbation. In fact, we will often write the perturbation of interest in the form

V^ (t)=¸V (t) (8.2) where ¸ is a smallness parameter that we can use to tune the strength of V^ . We will denote by n a complete ONB of eigenstates of H0 with unperturbed energies "n; so that,byassumptionfj ig

H0 n = "n n n n =1 n n0 = ±n;n0 : (8.3) j i j i n j ih j h j i X Our general goal is to calculate the amplitude (or probability) to …nd the system in a given …nal state Ã at time t if it was known to be in some other particular state Ã j f i j ii at time t = t0: Implicit in this statement is the idea that we are going to let the system evolve from Ã until time t and then make a measurement of an observable A of which j ii Ã is an eigenstate (e.g., we might be measuring the operator Pf = Ã Ã ). A little j f i j f ih f j less generally, if the system was initially in the unperturbed eigenstate ni of H0 at t0; j i 240 Time Dependent Perturbations: Transition Theory we wish to …nd the amplitude that it will be left in (or will be found to have made a transition to) the eigenstate nf at time t>t0; where now the measurement will be that of the unperturbed Hamiltonianj i itself. We note in passing that if we could solve the full Schrödinger equation d i~ Ãi(t) = H(t) Ãi(t) (8.4) dtj i j i for the initital condition Ãi(t0) = Ãi of interest, the solution to the general problem would be immediate. Thej correspondingi j i transition amplitude would then just be the inner product Ti f (t)= Ãf Ãi(t) and the transition probability would be ! h j i 2 2 Wi f = Ti f = Ãf Ãi(t) = Ãi(t) Ãf Ãf Ãi(t) : (8.5) ! j ! j h j i h j ih j i ¯ ¯ It is useful, in what follows, to develop¯ our techniques¯ for solving time-dependent problems of this kind in terms of the evolution operator U(t; t0), or propagator, which evolves the system over time Ã(t) = U(t; t0) Ã(t0) : j i j i We recall a few general features of the evolution operator

1. It is Unitary, i.e., + 1 U (t; t0)=U ¡ (t; t0)=U(t0;t): (8.6)

2. It obeys a simple composition rule

U(t; t0)=U(t; t0)U(t0;t0): (8.7)

3. It is smoothly connected to the identity operator

lim U(t; t0)=1: (8.8) t t ! 0 4. It obeys an operator form of the Schrödinger equation d i U(t; t )=H(t)U(t; t ): (8.9) ~dt 0 0

5. If H(t)=H0 is independent of time, then the evolution operator takes a particularly simple form, i.e., iH0(t t0)=~ U = U0(t; t0)=e¡ ¡ : (8.10)

By comparison with what we have written above, the transition amplitude Ti f can be expressed as the matrix element of the evolution operator between the initial and! …nal states, i.e., Ti f = Ãf Ãi(t) = Ãf U(t; t0) Ãi ; (8.11) ! h j i h j j i or, if we are interested in transitions between eigenstates of H0; we have

2 Tn m = m U(t; t0) n Wn m = m U(t; t0) n (8.12) ! h j j i ! jh j j ij Typically, of course, it is the presence of the perturbation V^ (t) that renders the full Schrödinger equation intractable. Indeed, when ¸ =0; each eigenstate of H0 evolves so as to acquire an oscillating phase

i!n(t t0) U0(t; t0) n = e¡ ¡ n (8.13) j i j i General Considerations 241 but no transitions between di¤erent eigenstates occur:

Wnm = ±nm: It is the perturbation V^ (t) that allows the system to evolve into a mixture of unperturbed states, an evolution that is viewed as inducing transitions between them. Our goal, then, is to develop a general expansion for the full evolution operator U(t; t0) in powers of the perturbation, or equivalently, in powers of the small parameter ¸. To this end, it is useful to observe that the unperturbed evolution of the system is not the goal of our calculation, involving as it does all of the unperturbed eigenenergies of the system. Indeed, that problem is assumed to have been completely solved. It would be convenient, therefore, to transform to a set of variables that evolve, in a certain sense, along with the unperturbed system, so that we can focus on the relatively slow part of the evolution induced by the weak externally applied perturbation, without worrying about all the rapid oscillation of the phase factors associated with the evolution occuring under H0. The idea here is similar to tranforming to a rotating coordinate system to ease the solution of simple mechanical problems. In the present context, we expect that in the presence of a small perturbation the unperturbed evolution changes from the form given above into a mixture of di¤erent states, which we can generally write in the form

i"m(t t0)=~ U(t; t0) n = Ám(t)e¡ ¡ m (8.14) j i m j i X where for small enough ¸ the expansion coe¢cients Ám(t) are, it is too be hoped, slowly varying relative to the rapidly oscillating phase factors associated with the unperturbed evolution. As suggested above, we can formally eliminate this fast evolution generated by H0 by working in the so-called “interaction picture”. Recall that our axioms of quantum mechanics were developed in the Schrodinger picture in which the state of the system evolves in time

Ã (t) = U(t; t0) Ã(t0) (8.15) j Sch i j i while fundamental observables of the system are associated with time-independent Her- mitian operators A = ASch. By contrast, it is possible to develop a di¤erent formulation of quantum mechanics, the so-called Heisenberg picture, in which the state of the system

+ Ã (t) = Ã(t0) = U (t; t0) Ã (t) (8.16) j H i j i j Sch i remains …xed in time, but observables are associated with time-evolving operators

+ AH (t)=U (t; t0)ASchU(t; t0): (8.17) The kets and operators of one picture are related to those of the other through the unitary transformation induced by the evolution operator U(t; t0) and its adjoint, and preserve the mean values, and hence predictions, of quantum mechanics in the process. In this same spirit, it is possible to develop a formulation in which some of the time evolution is associated with the kets of the system and some of it associated with the operators of interest. An interaction picture of this sort can be de…ned for any system in which the Hamiltonian can be written in the form H = H0 + V (t); with the state vector of this picture + Ã (t) = U (t; t0) Ã (t) (8.18) j I i 0 j sch i being de…ned relative to that of the Schrödinger picture through the inverse of the unitary transformation U0(t; t0)=exp[ iH0(t t0)=~] (8.19) ¡ ¡ 242 Time Dependent Perturbations: Transition Theory which governs the system in the absence of the perturbation. This form for the state vector suggests that the inverse (or adjoint) operator U + acts on the fully-evolving state vector of the Schrödinger picture to “back out” or undo the fast evolution associated with the unperturbed part of the Hamiltonian. In a similar fashion, the operators

+ AI (t)=U0 (t; t0)ASch(t)U0(t; t0); (8.20) of the interaction picture are related to those of the Schrodinger picture through the same corresponding unitary transformation, but as applied to operators (we have included a time dependence in the Schrodinger operator ASch(t) on the right to take into account any intrinisic time dependence exhibited by such operators, as occurs, e.g., with a sinusoidally applied perturbing …eld). Naturally, we can de…ne an evolution operator UI (t; t0) for the interaction picture that evolves the state vector Ã (t) in time, according to the relation j I i Ã (t) = UI (t; t0) Ã (t0) (8.21) j I i j I i Using the de…nitions given above we deduce that

+ UI (t; t0)=U0 (t; t0)U(t; t0) (8.22) or, multiplying this last equation through by U0(t; t0); we obtain a result for the full evolution operator U(t; t0)=U0(t; t0)UI (t; t0): (8.23) in terms of the evolution operators U0 and UI : To obtain information about transitions between the unperturbed eigenstates of H0; then, we need the transition amplitudes

i!m(t t0) Tn m = m U(t; t0) n = m U0(t; t0)UI (t; t0) n = e¡ ¡ m UI (t; t0) n (8.24) ! h j j i h j j i h j j i and transition probabilities

2 2 Wn m = Tn m = m UI (t; t0) n : (8.25) ! j ! j jh j j ij We see, therefore, that the evolution operator of the interaction picture does indeed contain all information about transitions induced between the unperturbed eigenstates. The evolution equation obeyed by UI (t; t0) is also straightforward to obtain. By taking derivatives of U(t; t0) we establish (with t0 …xed) that dU dU dU = 0 U + U I : (8.26) dt dt I 0 dt But clearly dU i i = ¡ [H0 + V^ (t)]U = ¡ [H0 + V^ (t)]U0UI (8.27) dt ~ ~ and dU0 i = ¡ H0U0: (8.28) dt ~ From these last three equations we deduce that dU i I = U +V^ (t)U U (8.29) ~ dt 0 0 I which we can write as dU i I = V^ (t)U : (8.30) ~ dt I I General Considerations 243

Thus, the evolution operator in the interaction picture evolves under a Schrödinger equation that is governed by a Hamiltonian

^ + ^ VI (t)=U0 V (t)U0 (8.31) that only includes the perturbing part of the Hamiltonian (the interaction), as represented in this picture. Since UI (t; t0) shares the limiting behavior

+ lim UI (t; t0) = lim U0 (t; t0)U(t; t0)=1 (8.32) t t0 t t0 ! ! of any evolution operator, it obeys the integral equation that we derived earlier for evolution operators governed by a time-dependent Hamiltonian, i.e.,

i t UI (t; t0)=1 dt0 V^I (t0)UI (t0;t0) (8.33) ¡ ~ t Z 0 andhencecanbeexpandedinthesamewayinpowersoftheperturbation,i.e.,

1 (k) UI (t; t0)= UI (t; t0) (8.34) k=0 X where

k t t2 (k) 1 ^ ^ ^ UI (t; t0)= dtk dt1 VI (tk)VI (tk 1) VI (t1) i~ t ¢¢¢ t ¡ ¢¢¢ µ ¶ Z 0 Z 0 k 1 t t2 = dtk dt1 U0(t; tk)V^ (tk)U0(tk;tk 1)V^ (tk 1) i~ t ¢¢¢ t ¡ ¡ ¢¢¢ µ ¶ Z 0 Z 0 V^ (t2)U0(t2;t1)V^ (t1) (8.35) ¢¢¢ Combining this with (8.23) it is possible to deduce a similar expansion

1 (k) U(t; t0)= U (t; t0) (8.36) k=0 X

k t t2 (k) 1 U (t; t0)= dtk dt1 U0(t; tk)V^ (tk)U0(tk;tk 1)V^ (tk 1) i~ t ¢¢¢ t ¡ ¡ ¢¢¢ µ ¶ Z 0 Z 0 V^ (t2)U0(t2;t1)V^ (t1)U0(t1;t0) (8.37) ¢¢¢ for the full evolution operator. Note the structure of this is of a sum (integral) over all processes whereby the system evolves under H0 without perturbation from t0 to t1,at which time it is acted upon by the perturbation V^ (t1); then evolves without perturbation from t1 to t2; at which time it is acted upon by the perturbation V^ (t2); and so on. The kth order contribution arises from all those those processes in which the system is scattered (or perturbed) exactly k times between t0 and t; with the particular times at which those perturbations could have acted being integrated over. It is this structure that forms the basis for diagrammatic representations for the perturbation process, such as those introduced in the context of electrodynamics by Feynman. If the perturbation is small enough, this formal expansion for the propagator of the system can be trunctated after the …rst order term, and as such allows us to address in a perturbative sense the problem originally posed. For example, if the system is at 244 Time Dependent Perturbations: Transition Theory

t = t0 initially in an eigenstate Ã(t0) = n of the unperturbed Hamiltonian, the results of the above expansion reveal thatj thei statej i of the system at time t will be given by the expansion

Ã(t) = Ãm(t) m (8.38) j i m j i X where

Ã (t)= m U(t; t0) Ã(t0) = m U(t; t0) n m h j j i h j j i i!m(t t0) = m U0(t; t0)UI (t; 0) n = e¡ ¡ m UI (t; t0) n : (8.39) h j j i h j j i Truncating the expression for UI at …rst order i t UI (t; t0)=1 dt0 V^I (t0): (8.40) ¡ ~ t Z 0 and inserting the result into the expression for Ãm; keeping the lowest-order non-zero result for each coe¢cient, we obtain a basic equation of time-dependent perturbation theory: i t i!m(t t0) i!mn(t0 t0) Ãm(t)=e¡ ¡ ±n;m dt0 Vmn(t0)e ¡ (8.41) ¡ ~ t · Z 0 ¸ where !mn = !m !n is the Bohr frequency associated with the transition between levels n and m. Clearly¡ in this last expression, the …rst term, involving the Kronecker delta function is associated with the amplitude for the system to be found in the initial state, while the remaining terms give the desired (…rst-order) transition amplitudes

i t i!m(t t0) i!mn(t0 t0) Tn m = e¡ ¡ dt0 Vmn(t0)e ¡ (8.42) ! ¡~ t Z 0 from which follow the corresponding transition probabilities

t 2 2 i!mnt0 Wn m = ~¡ dt0 Vmn(t0)e : (8.43) ! ¯Zt0 ¯ ¯ ¯ ¯ ¯ For a perturbation that starts in the far¯ distant past and dissapears¯ in the far distant future, these results reduce to a particulalry simple form, in which the total transition probability can be written

2 1 2¼ 2 2 i!mnt0 ~ Wn m = ~¡ dt0 Vmn(t0)e = Vmn(!mn) (8.44) ! ~2 ¯Z¡1 ¯ ¯ ¯ ¯ ¯ where ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 1 i!mnt0 V~mn(!mn)= dt0 Vmn(t0)e (8.45) p2¼ Z¡1 is simply the Fourier transform of the perturbing matrix element connecting the two states involved in the transition, evaluated at a frequency !mn corresponding to the energy di¤erence between the two states involved. As an example, we consider a 1D harmonic oscillator P 2 1 H = + m!2X2 (8.46) 0 2m 2 which is initially (at t = ) in its ground state when a perturbing electric …eld pulse is applied of the form ¡1 V^ (t)= f(t)X: (8.47) ¡ Periodic Perturbations: Fermi’s Golden Rule 245

In this expression, f(t)=eE(t) represents the spatially uniform, but time-dependent force exerted by the …eld on the charged harmonically-bound particle. We might, e.g., example take a pulse envelope t2=¿ 2 f(t)=f0e¡ (8.48) with a Gaussian shape that peaks at a time that for convenience we have set equal to t =0. Our goal is to …nd the the probability that the particle is left by this pulse in the nth excited state. Provided the pulse strength is su¢ciently low, the transition probability canthenbewritten 2¼ 2 W0 n = V~n;0(n!) (8.49) ! ~2 ¯ ¯ where ¯ ¯ ¯ ¯ Xn;0 1 i!t0 V~n;0(!)= dt0 f(t0)e (8.50) p2¼ Z¡1 and ~ Xn;0 = n X 0 = ±n;1: (8.51) h j j i r2mw Clearly, the …rst-order transition amplitude vanishes except for the …rst excited state, i.e., n =1. For the Gaussian pulse, evaluation of the Fourier integral leads to the result that long after the pulse has passed through, the probability for the charge to be excited to the n =1state is 2 2 f0 ¼¿ 2 2 W0 1 = exp ! ¿ =2 : (8.52) ! 2m~! ¡ Note the transition probability becomes exponentially¡ small¢ as the duration of the pulse (as measured by the parameter ¿) increases, and that there is a maximum in the transition probability as a function of ¿. For this perturbative result to be valid, the strength f0 of the …eld must be small enough that the transition probability W0 1 is small compared to unity. !

8.2 Periodic Perturbations: Fermi’s Golden Rule An important class of problems involve perturbations that are harmonic in time, and expressible, therefore, in the form

i!t + i!t V^ (t)= Ve¡ + V e µ(t): (8.53)

Here, µ(t) is the Heaviside step function£ that describes¤ the initial application of the perturbation at t =0: Such a perturbation could describe, e.g., an electromagnetic wave applied to the system at t =0; with a wavelength much large than the system size. We consider here the situation in which the perturbation is simply left on and calculate, after all the transients of the system have died down, the steady-state transition rate

dWn m ¡n m = lim ! (8.54) ! t dt !1 which gives the number of transitions induced per unit time by the applied perturbation betweenaninitialstate n and a …nal state m . Using our …rst order result (8.43), the transition probability forj thisi situation can bej writteni in the form

t 2 1 i+t i t Wn m(t)= 2 Vmne + Vnm¤ e ¡ dt (8.55) ! ~ 0 ¯Z ¯ ¯ £ ¤ ¯ ¯ ¯ ¯ ¯ 246 Time Dependent Perturbations: Transition Theory in which we have de…ned the quantities

+ = !m !n ! (8.56) ¡ ¡ and = !m !n + !: ¡ ¡ Performing the integrals, we …nd

2 i+t i t 1 Vmn e 1 Vnm¤ e ¡ 1 Wn m(t)= 2 ¡ + ¡ ! ~ ¯ 2i (+=2) 2i ( =2) ¯ ¯ ¡ ¢ ¡ ¡ ¢¯ 2 ¯ i+t i t ¯ 1 ¯Vmne sin (+t=2) Vnm¤ e ¡ sin¯ ( t=2) ¡ = 2 ¯ + ¯ (8.57) ~ +=2 =2 ¯ ¡ ¯ ¯ ¯ ¯ ¯ which reduce to ¯ ¯ 2 2 1 2 sin (+t=2) 2 sin ( t=2) ¡ Wn m(t)= 2 Vnm 2 + Vnm¤ 2 ! ~ j j (+=2) j j ( =2) ½ ¡ ¾ 2 sin (+t=2) sin (+t=2) + Re ei!mntV V : (8.58) 2 mn nm¤ ( =2) ( =2) ~ ½ + + ¾ To put this in a form useful for exploring the long time limit, we now multiply and divide the …rst two terms by 2¼t and the last term by ¼2 to obtain

2 2 2 2¼ Vnm t 1 sin (+t=2) 1 sin ( t=2) ¡ Wn m(t)= j 2 j 2 + 2 ! ~ ¼ +t=2 ¼ t=2 ½ ¡ ¾ 2 2¼ VmnVnm¤ 1 sin (+t=2) 1 sin ( t=2) ¡ + j 2 j : (8.59) ~ ¼ (+=2) ¼ ( =2) · ¸· ¡ ¸ This form is convenient, because in the long time limit, the transient oscillations in the bracketed functions tend to die away, and they approach Dirac ±-functions as T . Speci…cally, it is straightforward to establish the following representations of the!1 Dirac ±-function 1 sin2 (!T=2) ±(!) = lim ±1(T;!) = lim (8.60) T T ¼ !2T=2 !1 !1 1 sin (!T=2) ±(!) = lim ±2(T;!) = lim (8.61) T T ¼ !=2 !1 !1 by showing that the functions ±(T;!) have, as T ; the appropriate limiting behavior (going to for ! =0; and going to 0 for ! =0!1; respectively), and that their integrals both approach1 unity for T : This allows6 us to write, for times t much greater than typical evolution times of the!1 unperturbed system

2 2 2¼ Vnm t 2¼ VmnVnm¤ Wn m(t)= j j ±(+)+±( ) + j j±(+)±( ): (8.62) ! ~2 f ¡ g ~2 ¡

Clearly, the product ±(+)±( )=±(!m !n + !)±(!m !n !) vanishes, since the ±-functions have di¤erent arguments.¡ This¡ leaves the …rst¡ two terms,¡ one of which must always vanish. If the …nal state has an energy greater than the initial, so that !m = !n+!, then the corresponding transition probability

2 2¼ Vnm t Wn m(t)= j j ±(!m !n !) (8.63) ! ~2 ¡ ¡ Periodic Perturbations: Fermi’s Golden Rule 247 describes the resonant absorbtion of a quantum ¢" = ~! of energy; if the …nal state has a lower energy than the initial one, so that !m = !n !; the transition probability ¡ 2 2¼ Vnm t Wn m(t)= j j ±(!m !n + !) (8.64) ! ~2 ¡ describes the stimulated emission of a quantum ¢" = ~! of energy. The …nal form of the transition rate for these processes can then be written

2 2 2¼ Vnm 2¼ Vnm ¡n m = j j ±(!m !n !)= j j ±("m "n ~!): (8.65) ! ~2 ¡ § ~ ¡ § This is the simple form of what is referred to as Fermi’s golden rule.Sincethe±-functions makes the transition rate formally in…nite or zero, this expression has meaning only when there is a distribution of …nal states having the right energy. Indeed, if we formally sum the transition rate ¡n m over all possible …nal states m, we can write the total transition probability in the form!

2¼ 2 ¡n = ¡n m = Vm;n ±("m "n ~!) ! m m ~ j j ¡ § X X 2¼ 2 = d" ±(" "n ~!) Vmn ±(" "m): (8.66) ~ ¡ § m j j ¡ Z X If Vmn is approximately a constant over those states of the right energy to which transitions can occur, the integral simpli…es and we end up with the second form of Fermi’s golden rule 2¼ 2 2¼ 2 ¡n = Vmn ½("n ~!)= Vmn ½("f ); (8.67) ~ j j § ~ j j which involves the so-called density of states

½(")= ±(" "m) (8.68) m ¡ X evaluated at the …nal energy "f = "n ~! to which the transition can occur. Note that the density of states (or state distribution§ function) so de…ned has the property that

"2 ½(")d" = N("1;"2) (8.69) Z"1 gives the number of states of the system with energies lying between "1 and "2: Typically, situations in which Fermi’s golden rule applies are those where the …nal set of states is part of a continuum (e.g., when a photon is given o¤ or absorbed, so that there are a continuum of possible directions associated with the incoming or outgoing photon), and thus the density of states function ½(") is to be considered a continuous function of the …nal energy. As an example of the application of Fermi’s golden rule, and to see how densities of states of the sort typically encountered are constructed, we consider a ground state hydrogen atom, with a single bound electron described by the wave function

1=2 3 r=a0 Ã0(r)= ¼a0 ¡ e¡ ; (8.70) to which a harmonic perturbing potential¡ ¢

V^ (~r; t)=V0 cos(~k0 ~r !t); (8.71) ¢ ¡ 248 Time Dependent Perturbations: Transition Theory

is applied, in which V0 is a constant having units of energy, and ~k0 = k0z:^ (The form is clearly suggestive of an electromagnetic perturbation of some sort.) Assume that the perturbation causes ionizing transitions in which the initially bound electron ends up in a “free particle state” with …nal wavevector ~k: We are interested in calculating the ~ “di¤erential ionization rate” d¡0(µ; Á)=d for transitions to free-particle k-states passing through an in…nitesimal solid angle d centered along some particular direction (µ; Á): To proceed, we note that the perturbation can be written in the form

i!t + i!t V^ (t)=Ve¡ + V e (8.72) where 1 ~ V = V eik0 ~r: (8.73) 2 0 ¢ From Fermi’s golden rule, irreversible transitions in which a quantum ~! is absorbed (stimulated absorption) can only occur to states with …nal energies "f = "i +~! = ~! "0: 2 2¡ This …nal energy is assumed to be associated with the …nal kinetic energy "f = ~ k =2m of the ionized electron, which requires the …nal wavevector to have magnitude

4 2 2m (~! "0) 2m (~! me =2~ ) k = kf = 2¡ = ¡ 2 : (8.74) r ~ r ~ The Fermi golden rule rate for transitions to a plane wave state of wavevector ~k having this magnitude can be written

2¼ 2 2¼ 2 2m¼ 2 ¡0 ~k = V~k;0 ±("k "f )= V~k;0 ±("k ~! +"0)= 3 V~k;0 ±(k kf ) (8.75) ! ~ ¡ ~ ¡ ~ k ¡ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ wherewehaveusedtheresult¯ ¯ ¯ ¯ ¯ ¯ 2 ~ 2 2 m ± k k = ±(k kf ): (8.76) 2m ¡ f 2k ¡ · ¸ ~ ¡ ¢ Note that this last ±-function involves only the magnitude of the wavevector. The transition rate d¡0(µ; Á) into all k-states passing through an in…nitesimal solid angle d along (µ; Á) is obtained by summing over all such …nal states, i.e.,

2m¼ 2 d¡0(µ; Á)= V~ ±(k kf ): (8.77) ~3k k;0 ¡ ¯ ¯ ~k d ¯ ¯ X2 ¯ ¯ where the sum really is a symbolic way of writing an integral over all those wavevectors passing through the solid angle d at (µ; Á): Working in the spherical coordinate representation in k-space this can be written in the form

1 2 ~ ±(k kf )= dk k d ½(k)±(k kf ) (8.78) ¡ 0 ¡ ~k d Z X2 where ½(~k)=½(k;µ; Á) isthedensityofplanewavestateswithwavevector~k; i.e., the number of states per unit volume of k-space. To obtain this quantity, it is convenient in problems of this sort to take the entire system to be contained in a large box of edge L; with normalized plane wave states

~ 3=2 i~k r ~r k = Á~ (~r)=L¡ e ¢ (8.79) h j i k Periodic Perturbations: Fermi’s Golden Rule 249 that satisfy periodic boundary conditions at the edges of the box. The allowed wavevectors in this situation are then of the form 2¼ ~k = (n ^{ + n |^+ n k^) (8.80) L x y z where nx;ny; and nz are integers. The points in k-space thus form a regular cubic lattice with edge length 2¼=L; so there is exactly one state in every k-space unit cell volume of (2¼=L)3: The resulting density of states in k space

L 3 ½(~k)= (8.81) 2¼ µ ¶ is uniform, therefore, independent of ~k. Thus, the density of “ionized” states along d takes the form

3 1 2 ~ L 2 ±(k0 k)= dk k d ½(k)±(k kf )= k d: (8.82) ¡ 0 ¡ 2¼ k0 d µ ¶ X2 Z

Puttingthisintotheexpressiongivenaboveford¡0(µ; Á), and dividing through by d; we obtain the following expression for the “di¤erential ionization rate”

2 3 2 d¡0(µ; Á) 2m¼k ~ mL k = V~ ½(k)d = V~ (8.83) d ~3 k;0 4¼2~3 k;0 ¯ ¯ ¯ ¯ ¯ ~ ¯ ¯ ¯ where it is understood at this point that¯ k ¯= kf as given above.¯ This¯ quantity gives the number of transitions per unit time per unit¯ ¯ solid angle along the speci…ed direction. To ¯ ¯ complete the calculation we need to evaluate¯ ¯ the matrix element

V ~ ~ ~ 0 3 ik ~r ik0 ~r V~ = k V Ã = d re¡ ¢ e ¢ Ã (r) k;0 h j j 0i 2L3=2 0 Z V V 0 3 i(~k ~k0) ~r 0 = d re¡ ¡ ¢ Ã (r)= Ã~ (~k ~k0) (8.84) 2L3=2 0 2L3=2 0 ¡ Z where, after a little hard work we …nd that 1 8 3 i~q ~r r=a0 3 Ã~ (~q)= d re¡ ¢ e¡ = ¼a (8.85) 0 3 0 2 2 2 ¼a (1 + a0q ) 0 Z q Combining these results wep obtain, …nally:

d¡ (µ; Á) 16mV 2a2 ka 0 = 0 0 0 d ¼ 3 2 ~ ~ 2 4 ~ (1 + a0 k k0 ) 2 2 j ¡ j 16mV0 a0 ka0 = 3 2 2 2 4 (8.86) ¼~ (1 + a0(k 2kk0 cos µ + k0)) ¡ which is symmetric about the z-axis (independent of Á) and has a maximim along the z direction associated with the wavevector ~k0 that characterizes the perturbation (suggest- ing the absorption of momentum from the plane wave perturbation). Note that although we adopted the “box convention” for determining the density of states, corresponding factors in the normalization of the …nal plane wave state led to a cancellation of any terms involving the size L of the box. We are free at this point to take L without a¤ecting the …nal answer. !1 250 Time Dependent Perturbations: Transition Theory

8.3 Perturbations that Turn On We now consider another class of problems, one in which the measurement question that is asked is slightly di¤erent. Consider a system subject to a time dependent perturbation

H(t)=H0 + V^ (t) (8.87) in which the perturbation begins to be applied to the system at some …xed instant of time (say t =0), but takes a certain amount of time to develop. (The current has to build up in the external circuits, for example). To describe this situation, we write the perturbation in the form V^ (t)=V0¸(t) (8.88) where the function ¸(t) describes the smooth increase in the strength of the perturbation V^ to its …nal value V0. The function ¸(t) is unspeci…ed, but is assumed to have the general features 0 for t<0 ¸(t)= (8.89) 8 < 1 for t>T where T is a measure of the time that it takes for the perturbation to build up to full : strength. We note that except for the interval T>t>0; while the Hamiltonian is actually changing, the system is described by time-independent Hamiltonia: H0 initially, and H0 + V0 afterwards. During these initial and …nal intervals the evolution is readily described by the corresponding eigenvectors and eigenvalues of these two di¤erent opera- (0) (0) tors. Borrowing from the notation we introduced previously, we denote by n and ²n j i the eigenvectors and eigenvalues of H0 and by n and "n the corresponding quantities for j i the …nal Hamiltonian H = H0 + V0. Then, by assumption, (0) (0) (0) H0 n = " n j i n j i (H0 + V0) n = "n n (8.90) j i j i We then ask the following question. If the system is known to be in an eigenstate (0) n of H0 at t =0; what is the amplitude for it to be in the eigenstate n0 of the …nal j i j i Hamiltonian H = H0 + V0 after the perturbation has fully turned on? This is clearly a relevant question, since information about the admixture of …nal eigenstates allows us to predict the subsequent evolution for t>T. So the basic question is, what happens to the system as the perturbation is increasing to its …nal form? The general answer to this question is complicated, but becomes very simple in two limiting cases: (1) a perturbation that is applied in…nitely fast, and (2) a perturbation that is applied in…nitely slowly. The …rst, referred to as a sudden perturbation occurswhenthechangeinthe Hamiltonian occurs much more rapidly then the system (either before or after the change) can respond. In this limit, the function ¸(t)=µ(t) is essentially a Heaviside step function. The opposite limit, that in which the turn-on time T is much longer than typical evolution times of the system describes what is referred to as an adiabatic perturbation. As a useful thought-experiment that provides a mental mnemonic for remember- ing what happens in these two cases, consider what happens when a marble is placed in the bottom (i.e., ground state) of a bowl, which is then raised slowly to some predeter- mined height. If the raised bowl is then suddenly lowered, the marble will be left hanging in air, in the “ground state” of the raised bowl, not the lowered one. It does not have time, under these circumstances to respond to the changing conditions (Hamiltonian) until long after the bowl is in the lowered position. When, on the other hand, the bowl is lowered very slowly, the marble stays in the “instantaneous ground state” of the bowl for each elevation, ultimately sitting in the bottom of the bowl in the …nal lowered position. These features also characterize the behavior of quantum mechanical systems. Perturbations that Turn On 251

8.3.1 Sudden Perturbations In keeping wsith the thought experiment just described, it is possible to show quite generally that the state vector Ã(t) of a system subject to an instantaneous change in its Hamiltonian undegoes no changej i itself as a result of the instantaneous change in the H. In such a circumstance, the Schrödinger equation can be written (in the so-called sudden approximation) in the form

d i~ H0 Ã(t) =0 t<0 dt ¡ j i µ ¶ d i~ H0 V0 Ã(t) =0 t>0 (8.91) dt ¡ ¡ j i µ ¶ To understand what happens to the state vector during this change, we formally integrate across the discontinuity in H(t) at t =0; as follows:

i d Ã(t) = ¡ H0 Ã(t) + µ(t)V^0 Ã(t) dt (8.92) j i ~ j i j i h i Ã+ i +" i " d Ã(t) = ¡ H0 Ã(t) dt V0 Ã(t) dt (8.93) Ã j i ~ " j i ¡ ~ 0 j i Z ¡ Z¡ Z Thus, we …nd that, for in…nitesimal " i i i Ã+ Ã = "H0 Ã+ + "H0 Ã "V0 Ã+ (8.94) j i¡j ¡i ¡~ j i ~ j ¡i¡~ j i

The right hand side is proportional to "; so provided that the strength of V^0 is …nite,

lim Ã+ Ã =0: (8.95) " 0 ¡ ! j i¡j i Hence Ã(t) is continuous across any …nite discontinuity in H. Thus in this limit, if the j i (0) system is initially in an eigenstate n of H0; it will still be in that state immediately after the change in the Hamiltonianj hasi occurred. The transition amplitude to …nd it, at that instant, in the eigenstate n0 of H0 + V^0 is just the inner product between the eigenstates of these two di¤erent Hamiltonia,j i i.e., :

2 (0) (0) Tn n = n0 n Wn n = n0 n (8.96) ! 0 h j i ! 0 h j i ¯ ¯ ¯ ¯ As an interesting example of this class of problem, consider¯ the beta¯ decay of the tritium atom, which is an isotope of hydrogen with a nucleus consisting of 2 neutrons and 1 proton, so Z =1. Suppose the single bound electron of this atom, which sees an electric potential identical to that of hydrogen, is initially in its ground state, when the tritium nucleus to which it is bound undergoes beta decay, a process in which the nucleus ejects an electron with high kinetic energy ( 17 KeV), leaving behind a Helium nucleus with 2 protons and a neutron. As a result of» the quick ejection of the “nuclear” electron, the bound atomic electron sees the potential in which its moving change very quickly from

e2 Vi = (8.97) ¡ r to 2e2 Vf = : (8.98) ¡ r 252 Time Dependent Perturbations: Transition Theory

Thus, immediately after the beta decay the electron is in the ground state of Hydrogen, 1 Ã (~r; Z =1)= r¹ Ã = exp ( r=a ) (8.99) 1;s 3 0 h j i ¼a0 ¡ but is moving in a potential corresponding to singlyp ionized Helium (He+). It is, therefore, in a linear combination of Helium ion ground and excited eigenstates. What is the probability amplitude that an energy measurement will …nd the electron in, say, the Ã2;s state of the Helium ion? It is just the inner product between the Ã1s ground state of Hydrogen (with Z =1)and the corresponding Ã2sstate (with Z =2)fortheHeion.For ahydrogenicatomwithZ =2;

r 1 r=a0 Ã2s = Ã2;0;0 = 1 e¡ (8.100) ¼a3 ¡ a0 0 µ ¶ so the relevant transition amplitude is p

r 3 4 1 2 2r=a0 1 T1s 2s = d rÃ2¤s(2;r)Ã1s(1;r)= 3 dr r 1 e¡ = ; ! a ¡ a0 ¡2 Z 0 Z0 µ ¶ 1 W1s 2s = (8.101) ! 4 There is, therefore, a 25% chance of it ending up in this state. Such transitions can be detected when the electron emits a photon and decays back to the ground state of the He ion. Obviously the emission spectrum for this process can be calculated by …nding the corresponding transition probabilities for the remaining excited states of the He+ ion. 8.3.2 The Adiabatic Theorem Perturbations that reach their full strength very slowly obey the so-called adiabatic theo- (0) rem: if the system is initially in an eigenstate n of H0 before the perturbation starts to change, then provided the change in H occursj slowlyi enough, it will adiabatically follow the change in the Hamiltonian, staying in an instantaneous eigenstate of H(t) while the change is taking place. Afterwards, therefore, it will be found in the corresponding eigenstate n of the …nal Hamiltonian H = H0 + V0: Toj seei this we present a “perturbative proof” of the adiabatic theorem, by focusing on an interval of time over which the Hamiltonian changes by a very small amount. Now, by assumption, the Hamiltonian H(t) of the system is evolving very slowly in time and may ultimately change by a great amount. Suppose, however, that there exists an instant during this evolution when the system happens to be in an instantaneous eigenstate n of H(t). Let us rede…ne our time scale and denote this instant of time as t =0; andj seti H0 = H(0): At some time T later, the Hamiltonian will have evolved into a new operator H(T )=H0 + V;^ where the change in H; represented by the operator V^ = H(T ) H0; is ¡ assumed small, in the perturbative sense, compared to H0. We are interested in exploring how the evolution of the system during this time interval depends upon the total time T for this change in the Hamiltonian to take place. As already discussed, we assume that the Hamiltonian varies in the intervening time interval T>t>0 in such a way that V (t)=H(t) H0 = ¸(t)V;^ where the function ¸(t) starts at t =0with the value ¸(0) = 0 and increases¡ monotonically to the …nal value ¸ =1when t = T: To allow for a parameterization of the speed with which the change in H occurs, we assume that the function ¸ can be reexpressed in the form ¸ = ¸(t=T )=¸(s); with the properties that ¸(0) = 0 and ¸(1) = 1: This allows us to smoothly decrease the rate at which the change in H is being made simply by increasing the time T over which the change occurs. For convenience, we also make the assumption that ¸(s) is a monotonically increasing Perturbations that Turn On 253 function of s = t=T for s between 0 and 1. Under these circumstances, for su¢ciently small perturbations V^ ,thestateattheendofthisintervaloftimewillbegiventoan excellent approximation by the results of …rst order time-dependent peturbation theory:

(0) (0) U(T;0) n = Ãm(T ) m (8.102) j i m j i X with

i!nT Ãn(T )=e¡ T i i!mT i!mnt Ãm(T )= e¡ dt Vm;n(t)e ¡~ 0 Z T iVmn i!mT i!mnt = e¡ dt ¸(t=T )e m = n: (8.103) ¡ ~ 0 6 Z Performing an integration by parts, and using the limiting values of the function ¸(t=T ) over this interval, leads then to the result

V e i!mT ei!mnT V e i!mT ei!mnT T d¸(t=T ) mn ¡ mn ¡ i!mnt Ãm(t)= + dt e : (8.104) ¡ !mn !mn dt Z0 Now in the limit that the time T over which this change takes place becomes very large, the second integral becomes as small as we like. This follows from the fact that d¸(t=T ) 1 d¸(s) = ¸0(t=T )= : (8.105) dt T ds ¯s=t=T ¯ ¯ Thus the integral of interest is bounded in magnitude by¯ the relation

T T T d¸(t=T ) i! t 1 i! t 1 dt e mn dt ¸0(t=T )e mn = dt ¸0(t=T ) dt · T T ¯Z0 ¯ Z0 Z0 ¯ ¯ ¯ ¯ ¯ ¯ ¸(1) ¸(0)¯ 1 ¯ ¯ ¯ = ¡ = : (8.106) ¯ ¯ T T where in evaluating the last integral we have used the assumed monotonicity of ¸. Hence the second term in the previous integration by parts is of order 1=T and becomes negligible relative to the …rst as T . In this limit, then, the …rst term gives for m = n the result !1 6 V e i!mT ei!mnt V e i!nt Ã (T )= mn ¡ = mn ¡ ; m (0) (0) (8.107) ¡ ~!mn ¡ "m "n ¡ where we have used the de…nition of !mn in terms of the corresponding eigenvalues of H0: Thus, to this order we can write V !nt (0) i!nt mn (0) Ã(t) = e¡ n + e¡ m j i j i (0) (0) j i m=n "m "n X6 ¡ !nt = e¡ n (8.108) j i where V n = n(0) + mn m(0) (8.109) j i j i (0) (0) j i m=n "m "n X6 ¡ is the perturbative result for the exact eigenstate of H(T )=H0 + V^ expressed as an expansion in eigenstates of H0 = H(0): Thus, if the system begins the time interval in 254 Time Dependent Perturbations: Transition Theory an eigenstate n(0) of H(0), it ends in an eigenstate n of H(T ): We can now repeat the process, presumably,j i by rede…ning the time such thatj i t = T corresponds to a new time variable t0 =0; rede…ne H0 as H(T )=H(t0 =0); and proceed in the same way as above. In this way, after many such (long) time intervals, the system has remained in the corresponding eigenstate of the evolving Hamiltonian, which can ultimately change by a very great amount. Provided that the change occurs su¢ciently slowly, however, the state of the system will adiabatically “follow” the slowly-evolving Hamiltonian. Thus, the amplitude to …nd the system in an eigenstate of the …nal Hamiltonian is unity, provided it started in the corresponding eigenstate of the initial Hamiltoninan. If the change that occurs in the Hamiltonian is not in…nitely slow, however, there will be transitions induced to other eigenstates of H(T ). In the case of a pair of energy levels that are made to cross as a result of a time dependent perturbation it is possible to determine the probability of transitions being induced between di¤erent corresponding levels. The resulting analysis of such “Landau-Zener” transitions is presented in an appendix. he details are a bit complicated and rely on properties of the parabolic cylinder functions. The end result, however, is the surpsrisingly simple expression

¼V 2 W =exp (8.110) ¡~ d"=dt µ j j¶ for the transition probability between a pair of levels whose time-dependent energies cross at a rate d"=dt and which are connected by a constant matrix element V: Note that as the time rate of change of the perturbation goes to zero, the transition probability becomes exponentially small, and can, consistent with the adiabatic theorem, be neglected provided

2 d"=dt << ¼V =~: (8.111)

8.4 Appendix: Landau-Zener Transitions Consider a pair of energy levels connected by a constant matrix element V .Ifthe(diagonal) energies of the original states remain constant, then the probability amplitude to be found in either one will oscillate in time with a frequency proportional to V and with an amplitude that depends upon the magnitude of the energy di¤erence between them. For widely separated levels very little amplitude is ever transferred from one state to the other. Even when the levels are degenerate, the transfer is complete but temporary, since the amplitude repeatedly oscillates entirely back to the original state. Consider, however, a time-dependent perturbation that causes two widely separated levels connected by a constant matrix element to temporarily become close, or even degenerate in energy, and then to separate. In this situation an irreversible transition can occur as a result of the strong transfer that takes place during the limited time that the levels are nearly degenerate, since some fraction of the amplitude will generally get “stranded” in each state as the levels become widely separated again in energy. Processes of this type are referred to as Landau-Zener transitions since they were originally studied independently by those two authors in the context of electronic transitions in molecular systems during collisions. The basic idea has a wider applicability and has more recently been applied to understand optically induced transitions between Stark-split states of atomic systems within the so-called dressed atom picture of Cohen-Tanoudji, et al. To understand the essence of the Landau-Zener transition we consider two states Á and Á ; subject to a time-dependent Hamiltonian H(t) for which j 1i j 2i

H(t) Á1 = ~!1(t) Á1 + ~v Á2 j i j i j i H(t) Á2 = ~!2(t) Á2 + ~v Á1 j i j i j i Appendix: Landau-Zener Transitions 255

where, for simplicity, !1(t) and !2(t) are taken to be linear functions of time such that !2(t)= !1(t)=®t=2; with ®>0: Thus, !2 is negative for negative times and positive ¡ for positive times, while !1 has the opposite behavior. At very large negative times the levels are widely separated with a positive energy splitting

! = !1 !2 = ®t ¡ ¡ indicating that !1 >!2 for t<0. These “bare” energy levels come together and cross at t =0,with!2 becoming larger than !1 for t>0. The exact instantaneous eigenenergies and eigenstates Ã+ and Ã are easily determined by diagonalizing the 2 2 matrix associated with Hj (t)i; the twoj ¡i roots to the secular equation £

1 E (t)= ~ v2 + ®2t2 § 4 § r are indicated schematically below along with the bare energies.

0 -4 -2 2t 4 -1

-2

Clearly, at large negative times !2 corresponds to the lower branch E and !1 to the upper ¡ branch E+: The situation becomes reversed at large positive times, where !2 corresponds to E+ and !1 to E : Thus, up to a phase factor, ¡ lim Ã (t) = lim Ã (t)=Á lim Ã (t) = lim Ã (t)=Á t + t 2 t t + 1 !1 !¡1 ¡ !1 ¡ !¡1 In the neighborhood of t =0; the exact eigenstates are nearly equal symmetric and antisymmmetric combinations of Á1 and Á2 ; and the two branches associated with the exact eigenergies exhibit the classicj “avoidedi j crossing”i behavior, never coming any closer together in energy than 2V =2~v. Suppose that initially, as t , the system is in the !¡1 ground state Á2 = Ã ( ) ; i.e., on the lower branch E (t). Then, according to the adiabatic theorem,j i providedj ¡ ¡1Hi(t) is varied slowly enough (®¡ 1), the system will remain on this lower branch at each instant as the system adiabatically¿ evolves. At large positive times, therefore, the system will (up to a phase) be in the state Á1 = Ã (+ ) with unit probability. On the other hand, transitions between the upperj i andj lower¡ 1 branchesi may occur if the variation is not su¢ciently slow. To analyze this process, we consider the following expansion

Substitution into the Schrödinger equation d i~ Ã(t) = H(t) Ã(t) dtj i j i yields the following set of …rst order di¤erential equations

t t dC1 i !dt0 dC2 i !dt0 i = ve 0 C2 i = ve¡ 0 C1: dt R dt R for the expansion coe¢cients. We seek solutions to these equations corresponding to the boundary conditions C1( ) =0 C2( ) =1; j ¡1 j j ¡1 j in which the system is initially in an eigenstate assocated with the lower branch E of the energy spectrum, and we are interested in the probability that at large positive¡ times, well after the levels have separated and are no longer strongly-interacting, the system has made a transition from the lower branch E to the upper branch E+: In this ¡ regime E+ corresponds to the state Á2 : Thus, the transition probability arising from the nonadiabaticity of the perturbationj is giveni by

2 2 P = C2(+ ) =1 C1(+ ) : j 1 j ¡j 1 j To proceed, we take another derivative and substitute back in to obtain the following pair of second order di¤erential equations

2 2 d C1 dC1 2 d C2 dC2 2 i! + v C1 =0 + i! + v C2 =0; dt2 ¡ dt dt2 dt The substitutions i t i t C1 = U1 exp !dt0 C2 = U2 exp !dt0 2 ¡2 µ Z0 ¶ µ Z0 ¶ along with the relation d!=dt = ® reduce these to ¡ 2 2 2 2 2 2 d U1 2 i® ® t d U2 2 i® ® t + v + U1 =0 + v + + U2 =0: dt2 ¡ 2 4 dt2 2 4 µ ¶ µ ¶ A …nal pair of substitutions

1=2 i¼=4 2 z = ® e¡ tn= iv =® = i° where ° = v2=® is positive and real, put these into the standard di¤erential equations

2 2 d U1 1 2 1 d U1 1 2 z n U1 = z + a1 U1 =0 dz2 ¡ 4 ¡ ¡ 2 dz2 ¡ 4 µ ¶ µ ¶ 2 2 2 d U2 1 2 1 d U2 z z n + U2 = + a2 U2 =0 dz2 ¡ 4 ¡ 2 dz2 ¡ 4 µ ¶ µ ¶ 1 obeyed by the parabolic cylinder functions U(a; z), where here a1 = n 2 and a2 = 1 ¡ ¡ n + 2 : ¡ The solution to the …rst of these equations having the right properties as t is the parabolic cylinder function !§1

U1(z)=AU( a1; iz)=AU(a; iz); ¡ ¡ ¡ Appendix: Landau-Zener Transitions 257

1 where a = n + 2 and the constant A must be determined from the initial conditions and the asymptotic properties of the functions U(a; x).Ast the argument of the i¼=4 1=2 i¼=4 !¡1 function can be written iz ie¡ ® t = e R; with R real and positive, ¡ ! j j !1 along which path¤

i¼=4 1 iR2 n 1 i¼(n+1)=4 U1 = AU(a; iz) AU(a; Re ) Ae¡ 4 R¡ ¡ e¡ : ¡ » » n i° ln R This clearly goes to zero as R as 1=R (note that R¡ = e¡ oscillates with unit magnitude as R increases because!1n is strictly imaginary). Thus this solution automatically satis…es the initial condition C1 ( ) = U1( ) =0: To determine the value of A we use the other initial conditionj ¡1 j j ¡1 j

1 dC1 1= C2( ) = lim j ¡1 j v t dt !¡1 ¯ ¯ ¯ ¯ where we have used the original di¤erential equation to¯ express¯ C in terms of the deriv- ¯ ¯ 2 ative of C1: Now using the relation between C1 and U1 the boundary condition for U1 becomes 1 i! dU1 1 = lim v¡ U1 + t ¡ 2 dt !¡1 ¯ ¯ ¯ ¯ It turns out that as t the …rst term in¯ the brackets has¯ precisely the same asymptotic behavior !¡1 ¯ ¯

i! i®t 1 1=2 1 iR2 n 1 i¼(n+1)=4 U2 = U1 iA® Re¡ 4 R¡ ¡ e¡ ¡ 2 ¡ 2 »¡2 iA 1=2 1 iR2 n i¼(n+1)=4 ® e¡ 4 R¡ e¡ ; »¡2 as the second term i¼=4 dU1 1=2 dU(a; Re ) 1=2 d 1 iR2 n 1 i¼(n+1)=4 = A® A® e¡ 4 R¡ ¡ e dt dR » dR iA 1=2 1 iR2 n i¼(n+1)=4 h i ® e¡ 4 R¡ e¡ : »¡2 Thus the boundary condition becomes

1 1=2 1 iR2 n i¼(n+1)=4 1=2 ¼°=4 1 = lim v¡ iA® e¡ 4 R¡ e¡ = A °¡ e ; R ¡ j j !1 ¯ ¯ from which we deduce that¯ ¯ ¯ 1=2 ¼°=4 ¯ A = ° e¡ : j j At large positive times the argument of the parabolic cylinder function can be written i¼=4 1=2 i3¼=4 iz ie¡ ® t = Re¡ . To determine the asymptotic properties in this ¡situationweusetheidentity!¡ ¯ ¯ y ¯ ¯ i¼=4 1 i ¼ ( a+ 1 ) i¼=4 i ¼ ( a+ 1 ) i3¼=4 p2¼U a; Re¡ =¡( + a) e¡ 2 ¡ 2 U(a; Re )+e 2 ¡ 2 U(a; Re¡ ) ¡ 2 From³ Abramowitz and´ Stegun, p.689,n Eq. 19.8.1 we have for x >> a when arg x <¼=2; that o ¤ j j j j j j x2=4 a 1=2 U(a; x) e¡ x¡ ¡ »

i¼=4 yHere we use, with x = Re¡ the expression from Abramowitz and Stegun, p. 687, Eq. 19.4.6, which gives

1 i¼( 1 a+ 1 ) i¼( 1 a+ 1 ) p2¼U (a; x)=¡( a) e¡ 2 4 U( a; ix)+e 2 4 U( a; ix) § 2 ¡ ¡ § ¡ ¨ n o 258 Time Dependent Perturbations: Transition Theory which implies that as t + ! 1

i3¼=4 i¼(n+1) i¼=4 p2¼ i¼n=2 i¼=4 U1 = AU(a; Re¡ )=A e U(a; Re )+ e U a; Re¡ " ¡(n +1) ¡ # ³ ´ i3¼(n+1)=4 1 iR2 n 1 p2¼ i¼n=2 1 iR2 n i¼n=4 Ap2¼ i¼n=2 1 iR2 n i¼n=4 A e e¡ 4 R¡ ¡ + e e 4 R e¡ e e 4 R e¡ » " ¡(n +1) # » ¡(n +1)

and so

p2¼° ¼°=2 U1 (+ ) = e¡ : j 1 j ¡(n +1)

The square of this gives the amplitude for the system to remain on the lower branch E ; i.e., ¡

2 i¼=4 2 2¼° ¼° C1 ( ) = lim U1 Re¡ = e¡ R + ¡(1+i°)¡(1 i°) j 1 j ! 1 j j ¼° ³ ´ ¼° ¡ =2e¡ sinh ¼° =1 e¡ ¡

and so the corresponding transition probability to the upper branch is given by the Landau-Zener formula

2 P =1 C1 ( ) ¡j 1 j 2 ¼° 2 2 ¼V = e¡ =exp ¼V =~ ® =exp : ¡ ¡~2 d!=dt µ j j¶ ¡ ¢ Chapter 9 SCATTERING THEORY

9.1 General Considerations In this chapter we consider a situation of considerable experimental and theoretical interest, namely, the scattering of particles o¤ of a medium containing some type of scattering centers, such as atoms, molecules, or nuclei. The basic experimental situation of interest is indicated in the …gure below.

An incident beam of particles impinges upon a target, which maybe a cell containing atoms or molecules in a gas, a thin metallic foil, or a beam of particles moving at right angles to the incident beam. As a result of interactions between the particles in the initial beam and those in the target, some of the particles in the beam are de‡ected and emerge from the target traveling along a direction (µ; Á) with respect to the original beam direction, while some are left unscattered and emerge out the other side having undergone no de‡ection (or undergo “forward scattering”). The number of particles de‡ected along a given direction are then counted in a detector of some sort. The kinds of interactions and the analyis of general scattering situations of this type can be quite complicated. We will focus in the following discussion on the scattering of incident particles by scattering centers in the target uner the following conditions:

1. The incident beam is composed of idealized spinless, structureless, point particles.

2. The interaction of the particles with the scattering centers is assumed to be elastic so that the energy of the scattered particle is …xed, the internal structure of the scatterer (if any) and, thus, the potential seen by the scattered particle does not change during the scattering event.

3. There is no multiple scattering, so that each incident particle interacts with at most one scattering center, a condition that can be obtained with su¢ciently thin or dilute targets. 264 Scattering Theory

4. The scattering potential V (~r1;~r2)=V ( ~r1 ~r2 ) between the incident particle and the scattering center is a central potential,j so¡ wej can work in the relative coordinate and reduced mass of the system.

Under these conditions, the picture of interest reduces to that depicted below, in with an incident particle characterized by a plane wave of wavevector ~k = kz^ along a direction that we take parallel to the z-axis, and scattered particles emerging through a in…nitesimal solid angle d along some direction (µ; Á). We may characterize the incident beam by its (assumed uniform) current density J~i along the z axis. Classically J~i = n~vi where n = dN=dV is the particle number density characterizing the beam. The incident particle current through a speci…ed surface S is then the surface integral

dNS I = = J~i dS:~ dt ¢ ZS The scattered particle current dIS into a far away detector subtending solid angle d along (µ; Á) is found to be proportional to (i) the magnitude of the incident ‡ux density Ji, and (ii) the magnitude of the solid angle d subtended by the detector. We write d¾(µ; Á) dI = J d S d i where the constant of proportionality d¾(µ; Á)=d is referred to as the di¤erential cross section for elastic scattering in the direction (µ; Á). This quantity contains all information experimentally available regarding the interaction between scattered particles and the scattering center. We also de…ne the total cross section ¾ = ¾tot in terms of the total scattered current IS through a detecting sphere centered on the scattering center: d¾(µ;Á) I = dI = J d = J ¾ S S i d i tot Z Z so that the total cross section is simply the integral over all solid angle d¾(µ; Á) ¾ = ±: tot d Zsphere

Figure 1 General Considerations 265

of the di¤erential cross section. We note that d¾=d and ¾tot both have units of area

Is d¾ 1 dIS ¾tot = = Ji d Ji d and physically represent the e¤ective cross sectional area of the target atom “seen” by the incident particle. As such it contains, in principle, information about the relative sizes of the particles involved in the collision as well as the e¤ective range of the interaction potential V (r) without which there would be no scattering. Cross sections are often 24 2 measured in “barns”, where by de…nition 1 barn = 10¡ cm , which corresponds to the 12 cross sectional area of an object with a linear extent on the order of 10¡ cm. Thus, given that the cross section is the primary observable of a scattering experiment, the main theoretical task reduces to the following: given the scattering potential V (r); calculate the di¤erential and total scattering cross sections d¾=d and ¾tot as a function of the energy or wavevector of the incident particle. This problem can be addressed in a number of di¤erent ways. Perhaps the simplest conceptual approach would be as follows:

1. Consider a particle in an initial state at t = corresponding to a wave packet at z = centered in momentum about ~k = k¡1z:^ ¡1 2. Evolve the wavepacket accordiing to the full Schrödinger equation d i~ Ã(t) = H Ã(t) dtj i j i to large positive times t + : ! 1 3. Evaluate the probability current through d along (µ; Á):

Such an approach leads to a study of the so-called S-matrix

S = lim U(t; t)=U( ; ): t !1 ¡ ¡1 1 Rather than proceed along this route, we make a few simplifying observations. First,we note that the essential scattering process is time-independent, and can yield steady-state scattering currents, with Ji and JS independent of time. Secondly, for elastic scattering the particle energy is …xed and well de…ned, and it seems a shame to throw this away by forming a wavepacket of the type described. Finally, we note that the evolution of the system is completely governed by the positive energy solutions to the energy eigenvalue equation (H ") Ã =0: ¡ j i This last observation leads us to ask whether or not there generally exist stationary solutions to the energy eigenvalue equation that have asymptotic properties corresponding to the experimental situation of interest. The answer, in general, is yes and the solutions of interest are referred to as stationary scattering states of the associated potential V (r): To understand these states it is useful to consider the 1D analogy of a free particle incident upon a potential barrier, as indicated in the diagram. For this situation, there exist solutions in which the wave function to the left of the barrier is a linear combination of a right-going (incident) and left-going (scattered) wave, while the wave function to the right of the barrier contains a part that corresponds to the transmitted or “forward scattered” part of the wave. We note that experimentally, the wave function in the barrier region is inaccessible, and the only information that we can obtain is by measuring the relative magnitudes of the forward and backward scattered waves. 266 Scattering Theory

Figure 2

Similar arguments in 3-dimensions lead us to seek solutions to the eigenvalue equation of the form ikz Á²(~r)=e + ÁS(~r) in which the …rst part on the right clearly corresponds to the incident part of the beam and the second term corresponds to the scattered part. The subscript " indicates the energy of the incoming and outgoing particle, which is related to the wavevector of the incoming particle through the standard relation " = ~2k2=2m: We expect that at large distances from the scattering center, where the potential vanishes, the scattered part of the wave takes the form of an outward propagating wave. Hence, as r ; we anticipate !1 that ÁS has the asymptotic behavior

f(µ;Á)eikr Á (~r) r : S » r !1 Note that this form satis…es the eigenvalue equation for large r beyond the range of the 2 potential, where the Hamiltonian reduces simply to the kinetic energy H H0 = P =2m: Thus we seek solutions to the eigenvalue equation !

(H0 + V ) Á = " Á j ²i j "i which have the asymptotic form

eikr Á (~r) eikz + f(µ; Á) " » r where the quantity f(µ; Á) which determines the angular distribution of the scattered part of the wave is referred to, appropriately as the scattering amplitude in the (µ; Á) direction or, alternatively, as the scattering length since it is readily determined by dimensional analysis that f(µ; Á) has units of length. The obvious question that arises at this point is the following: what is the relation between the scattering length f(µ; Á) and the scattering cross section d¾(µ; Á)=d? To anwer this question we note that, by de…nition, the current into the detector can be written

dIS = ¾(µ; Á)Jid: General Considerations 267

But we can also write this in terms of the scattered current density J~S = J~S(r; µ; Á); i.e.,

2 dIs = J~S dS~ = r JSd: ¢ Thus, classically, d¾(µ; Á) J = r2J (r; µ; Á): i d s Now for quantum systems these conditions will be obeyed by the corresponding mean values taken with respect to the stationary state of interest, so that

d¾(µ; Á) Js(r; µ; Á) = r2 h i: d Ji h i Thus, we need operators corresponding to the current density. Classically, for a single particle at ~r, p~ J~(~r0)=n(~r0)~v = ±(~r ~r0)~v = ±(~r ~r0) : ¡ ¡ m In going to quantum mechanics we replace ~r and ~p by operators and symmeterize to ensure Hermiticity. Thus, 1 J(~r0)= [±(R~ r0)P~ + P±~ (R~ ~r0)]: 2m ¡ ¡

After a short calculation, the mean value of J~(~r0) in the state Á is found to be j i 1 ~ Á J~(~r0) Á = Re Á¤(~r0) ~ Á(~r0) : h j j i m i r · ¸ Using this, the incident ‡ux, associated with the plane wave part of the eigenstate, can be written 1 ikz ~ ikz ~k Ji = Re e¡ ~ e = : h i m i r m ¯ · ¸¯ ¯ ¯ By contrast, the scattered ‡ux is¯ then given by the expression¯ ¯ ¯ 1 ~ J~S = J~(r; µ; Á) = Re Á¤ (~r) ~ Á (~r) h i hj i m S i r S · ¸ which is most conveniently expressed in spherical coordinates, for which @ 1 @ 1 @ r = µ = Á = r @r r r @µ r r sin µ @Á

Using these along with the assumed asymptotic form for ÁS we …nd that

~k 1 2 JS r = f(µ; Á) h i m r2 j j ~ 1 1 @f JS µ = Re f ¤ h i m r3 i @µ · ¸ ~ 1 1 @f JS Á = Re f ¤ h i m r3 sin µ i @Á · ¸ which shows that asymptotically the angular components of current density become negligible compared to the radial component. Thus, from the radial component of the current density we deduce that

d¾(µ; Á) Js(r; µ; Á) r = r2 h i = f(µ; Á) 2 : d Ji j j h i 268 Scattering Theory

Thus, there is a very simple relation d¾(µ; Á) = f(µ; Á) 2 d j j between the scattering length and the cross sections that are experimentally accessible. We now turn to the problem of actually solving the energy eigenvalue equation to …nd the stationary scattering solutions, and thereby to determine the scattering length f(µ; Á) for a given potential V (r):

9.2 An Integral Equation for the Scattering Eigenfunctions We seek solutions to the eigenvalue equation

(H ") Á =0 ¡ j "i H = H0 + V 2 2 in which H0 = ~ K =2m and we assume that 1 V (r) 0 as r (at least as fast as r¡ ) ! !1 Since we are describing a situation where we are sending in incident particles with well de…ned kinetic energy, we expect solutions for all positive energys " 0; so we need just …nd the corresponding eigenvectors Á , which have the form ¸ j "i Á = Á + Á j "i j 0i j Si ~ in which Á = k0 is the incident part, which is an eigenstate of H0 with energy " = j 0i j i ~2k2=2m i~k0 ~r ikz Á (~r)= ~r Á = e ¢ = e 0 h j 0i and ÁS is the scattered part, which should dissapear as the potential V (r) goes to zero. To proceed,j i we rewrite the eigenvalue equation

(H0 + V ") Á =0 ¡ j "i in the form (" H0) Á = V Á ¡ j "i j "i and substitute in the assumed form of the solution to obtain

(" H0)[Á + Á ]=V Á ¡ j 0i j Si j "i or (" H0) Á = V Á : ¡ j Si j "i In this last form, only the scattered part of the state appears on the left hand side. We note at this point that if the operator (" H0) possessed an inverse, we could apply it to the left hand side to obtain a formal expression¡ for Á : The problem with this idea j Si is that for ">0 the operator (" H0) has a large degenerate subspace with eigenvalue ¡ 0; since H0 generally has a degenerate subspace for any positive energy ".Thus,strictly speaking, det(" H0)=0and the inverse is not de…ned. To overcome¡ this di¢culty we employ a little analytic continuation, and de…ne the resolvent operator G(z); de…ned for all non-real z; as the operator inverse of z H0; i.e., ¡ 1 G(z)=(z H0)¡ ¡ An Integral Equation for the Scattering Eigenfunctions 269

since H0 has no non-real eigenvalues, this operator exists for all non-real values of the complex parameter z. We then de…ne the (causal) Green’s function operator G+(") as the limit, if it exists, of G(z) as z " + i´; where ´ =0+ is a positive in…nitesimal: ! 1 1 G+(") = lim (" + i´ H0)¡ ("+ H0)¡ : ´ 0+ ¡ ´ ¡ ! where "+ = " + i´: Thus, if the limit exists, then

lim G(" + i´)(" H0)=G+(")(" H0)=1 ´ 0+ ¡ ¡ ! and we can write G+(")(" H0) Á = G+(") Á ¡ j Si j "i or, more simply, Á = G+(") Á j Si j "i Adding the incident part of the state Á0 to both sides, we then obtain the so-called Lipmann-Schwinger equation j i

Á = Á + G+V Á j "i j 0i j "i which is itself a representation independent form of what is often referred to as the integral scattering equation. The latter follows from the Lipmann-Schwinger equation by expressing it in the position representation. Multiplying on the left by the bra ~r we obtain h j ~r Á = ~r Á + ~r G+V Á h j "i h j 0i h j j "i or, inserting a complete set of position states,

ikz 3 Á"(~r)=e + d ~r0 G+(~r;~r0)V (~r0)Á"(~r0): Z

Thus, we obtain an integral equation for the scattering eigenfunction Á"(~r) which has, we hope, the correct asymptotic behavior. To make this useful,we need to (i) evaluate the matrix elements G+(~r;~r0)= ~r G+(") ~r0 ; and (ii) actually solve the integral equation, at least in the asymptotic regime.h j j i 9.2.1 Evaluation Of The Green’s Function To evaluate the matrix elements of the Green’s function it is most convenient to begin the calculation in the wavevector representation in which the operator (" H0) is diagonal. Note that in k-space ¡ ~k ("+ H0) ~k0 =("+ "k) ±(~k ~k0) h j ¡ j i ¡ ¡ where 2 2 "k = ~ k =2m so that 3 ("+ H0)= d q ~q ("+ "q) ~q ¡ j i ¡ h j Z and hence, as is easily veri…ed by direct multiplication,

1 3 ~q ~q ("+ H0)¡ = d q j ih j ¡ ("+ "q) Z ¡ 270 Scattering Theory

If we set 2m"+ + k+ = 2 = k + i´ (´ =0 ) r ~ where the positivity of ´ in this last equation follows from the positive imaginary part of "+; then we can write 2m 3 ~q ~q G+(")= 2 d q j2 ih j2 ~ k+ q Z ¡ so that

2m 3 ~r ~q ~q ~r0 G+(~r;~r0)= 2 d q h j2 ih j 2 i ~ k+ q Z ¡ 3 i~q (~r ~r0) 2m d q e ¢ ¡ = 2 3 2 2 ~ (2¼) k+ q Z ¡ 2¼ ¼ 2 i~qR cos µ 2m 1 dq q e = 2 dÁ dµ sin µ 3 2 2 ~ 0 0 0 (2¼) k+ q Z Z Z ¡ ~ ~ where R = ~r ~r0 and R = R : The angular integrations are readily evaluated and give ¡ ¯ ¯ ¯ ¯ m¯ ¯ 1 q sin(qR) m 1 q sin(qR) G+(~r;~r0)= 2 2 dq 2 2 = 2 2 dq 2 2 ~ ¼ R 0 k+ q 2~ ¼ R k+ q Z ¡ Z¡1 ¡ where we have used the fact that the integrand is an even function of q. Splitting sin(qR) into exponentials and setting q0 = q in the second we …nd that ¡ iqR m 1 1 qe G+(~r;~r0)= 2 dq 2 ¼~ R 2¼i k+2 q Z¡1 ¡ This integral can be evaluated by contour integration in the complex q-plane using Cauchy’s theorem,which states that for a function f(z) that is analytic in and on a closed contour ¡ in the complex z-plane enclosing the point z = a,

1 f(z)dz = f(a): 2¼i z a I¡ ¡ In our case we choose a closed path in which q runs from Qto +Q andthencircles back around on a semicircle in the upper half plane, which is¡ ultimately taken to occur at Q = : Since the contribution from the integrand vanishes along this latter part, the integralj j over1 the closed contour coincides with the one of interest. To proceed, we note that

2 2 k q =(k+ q)(k+ + q) + ¡ ¡ which generates simple poles at q = k+ = k + i´ and q = k+ = k i´ ¡ ¡ ¡ only the …rst of which is enclosed by our contour. Setting

qeiqR f(q)= k+ + q An Integral Equation for the Scattering Eigenfunctions 271

Figure 3 we …nd that 1 f(q)dq 1 f(q)dq eik+R = = f(k+)= 2¼i k+ q ¡2¼i q k+ ¡ ¡ 2 I¡ ¡ I¡ ¡ Combing this with our previous formula, and taking the limit k+ k we obtain the Green’s function of interest, i.e., !

ik ~r ~r0 me j ¡ j G+(~r;~r0)= 2 = G+(~r ~r0): ¡2¼~ ~r ~r0 ¡ j ¡ j Putting this into our integral scattering equation gives the result

ik ~r ~r0 ikz m 3 e j ¡ j Á"(~r)=e 2 d r0 V (~r0)Á"(~r0): ¡ 2¼~ ~r ~r0 Z j ¡ j Before proceeding to solve this integral equation we should, perhaps, check to see that it gives solutions with the correct asymptotic behavior. To this end, we note that the integrand has contributions primarily from those regions where r0 is small, i.e., where the potential is signi…cant. At the detector, however, the magnitude of r is very large, and V (r) is negligible. Where the integrand is signi…cant, therefore, we have ~r >> ~r0 : In this limit we can write j j j j

2 2 ~r ~r0 = (~r ~r ) (~r ~r )= r 2~r ~r + r j ¡ j ¡ 0 ¢ ¡ 0 ¡ ¢ 0 0 2 2 pr 1 2~r ~r0=r = r 1p ~r ~r0=r ' ¡ ¢ ¡ ¢ r r^ ~r0 ' p¡ ¢ ¡ ¢ where r^ = ~r=r is a unit vector along the direction (µ; Á) associated with the detector. Hence in this limit we can write

ik ~r ~r0 ikr e ¡ e ikr^ ~r0 j j e¡ ¢ ~r ~r ' r j ¡ 0j 272 Scattering Theory and so our integral equation provides a solution of the form

ikr ikz m e 3 ikr^ ~r0 Á"(~r) e d r0 e¡ ¢ V (~r0)Á"(~r0) ' ¡ 2¼~2 r Z eikr eikz + f(µ; Á) ' r where m 3 ikr^ ~r0 f(µ; Á)= d r0 e¡ ¢ V (~r0)Á"(~r0) ¡2¼~2 Z is independent of r; but depends only upon r^ =^r(µ; Á); as it should. Thus, a solution to this equation should indeed have the correct asymptotic properties associated with the stationary scattering states of interest.

9.3 The Born Expansion

Now that we have an explicit representation for the Green’s function G+(") we can attempt a solution to the Lipmann-Schwinger equation

Á = Á + G+V Á : j "i j 0i j "i The traditional method of solving this kind of equation, or its integral equation equivalent is by iteration. Any approximation to Á" can be substituted into the right-hand side of the equation to generate a new approximationj i . Moreover, we can formally write

Á = Á + G+V [ Á + G+V Á ] j "i j 0i j 0i j "i = Á + G+V Á + G+VG+V Á j 0i j 0i j "i Proceeding in this way generates the so-called Born expansion

Á = Á + G+V Á + G+VG+V Á + j "i j 0i j 0i j 0i ¢¢¢ 1 n = (G+V ) Á : j 0i k=0 X The Born expansion gives a an expansion in powers of the potential V; and obviously requires for its convergence that the e¤ect of the perturbation V on the incident wave be small, hence ÁS << Á0 ; Á" : The solution obtained by truncating the series at order n is referredjj tojj as thejj njjthjj orderjj Born approximation to the scattered state. We defer till later an exploration of the approximate solutions obtained in this fashion, and instead introduce additional ways of looking at the problem.

9.4 Scattering Amplitudes and T-Matrices The form that we have developed for the scattering amplitude

m 3 ikr^ ~r0 f(µ; Á)=f(^r)= d r0 e¡ ¢ V (~r0)Á"(~r0) ¡2¼~2 Z describes the amplitude for measuring a de‡ected particle along the direction r^(µ; Á) with wavevector k: Thus, it measures a state of wavevector ~kf = kr;^ so we can write f(µ; Á)=f(^r)=f(~kf ;~k0) in the form m ~ ~ 3 ikf ~r0 f(k; k0)= d r0 e¡ ¢ V (~r0)Á"(~r0) ¡2¼~2 Z Scattering Amplitudes and T-Matrices 273

~ which has the form of a projection of V Á" onto the plane wave state kf associated with ~ j i j i ~ ikf ~r thewavefunction ~r kf = e ¢ (Note that the normalization of this plane wave state is a little di¤erent thanh j usual,i but is consistent with our choice of Á :)Thus,e.g., j 0i

3 ikf ~r0 ~kf V Á = d r0 e¡ ¢ V (~r0)Á (~r0) h j j "i " Z ~ ~ m ~ f(kf ; k0)= kf V Á" ¡2¼~2 h j j i which appears to be a matrix element of V; except that it is taken between states of di¤erent type. The state on the left is part of an ONB of free particle states, that on the right is an eigenstate of the same energy in the presence of the potential, which is “smoothly connected” to the plane wave state Á0 = k0 as V 0: It is useful to introduce an operator T; referred to as the T matrix,j i orj i transition! operator which is de…ned so that V Á = T Á = T ~k0 : j "i j 0i j i This allows us to express the scattering amplitude m f(µ; Á)=f(~k;~k0)= ~kf T ~k0 ¡2¼~2 h j j i as the matrix element of the transition operator between initial and …nal free particle states (which are the ones that we deal with in the laboratory, outside of the target region). Hence the T -matrix contains all information regarding the scattering transitions induced by the potential. How do we evaluate T ? We generate an integral equation for it from the Lipmann-Schwinger equation, which gives

Á = Á + G+V Á : j "i j 0i j "i = Á + G+T Á =(1 + G+T ) Á j 0i j 0i j 0i We can compare this with the Born series to obtain

Á = Á + G+V Á + G+VG+V Á + j "i j 0i j 0i j 0i ¢¢¢ to obtain the operator relation

G+T = Á = G+V + G+VG+V + j "i ¢¢¢ from which we deduce that

T = V + VG+V + VG+VG+V + ¢¢¢ which gives a Born expansion for T in powers of the potential V . Formally we can write

T = V [1 + G+V + G+VG+V + ] ¢¢¢ = V + VG+T which is the integral equation obeyed by the T -matrix that generates the Born series. The nth order Born approximation to the T -matrix is then obtained by truncating the series ath the nth order term. The …rst Born approximation to the T -matrix is just the scattering potential T = T (1) = V andsoweobtaintothisorder ~ ~ m ~ ~ fB(kf ; ki)= kf V ki : ¡2¼~2 h j j i 274 Scattering Theory

To evaluate this, we work in the position representation

~ ~ m 3 ~ ~ fB(kf ; ki)= d r kf ~r V (~r) ~r ki ¡2¼~2 h j i h j i Z m 3 i(~kf ~ki) ~r = d re¡ ¡ ¢ V (~r): ¡2¼~2 Z Clearly the vector ~q = ~kf ~ki ¡ is the momentum transferred in the collision, since ~kf = ~ki + ~q.Moreover,since~kf = ~ j j ki = k are the same, we can write j j ~q =2k sin µ=2 j j where µ is the direction between the incoming beam and the de‡ected particle. Thus, we can write in the …rst Born approximation m fB(~kf ;~ki)= V~ (~q) ¡2¼~2 where 3 i~q ~r V~ (~q)= d re¡ ¢ V (~r) Z is, up factors of 2¼; simply the Fourier transform of the scattering potential. Thus, in the Born approximation the di¤erential scattering cross section

d¾(µ; Á) m2 2 = f(µ; Á) 2 = V~ (~q) d j j 4¼2~4 ¯ ¯ ¯ ¯ is, up to constant factors, simply the squared modulus¯ of the¯ Fourier transform of the scattering potential evaluated at the wavevector ~q corresponding to the momentum transferred in the scattering event. As a special case of this formula, we can consider the case where the potential is sphericallysymmetric,sothatV (~r)=V (r); in which case

2¼ ¼ 1 2 iqr cos µ V~ (~q)=V (q)= dÁ dµ dr r sin µe¡ V (r): 0 ¼ 0 Z Z¡ Z The angular integrals are readily evaluated to give

2¼ ¼ iqr cos µ 4¼ dÁ dµ sin µe¡ = sin(qr) 0 ¼ qr Z Z¡ so the only remaining integral to perform is the radial one

4¼ V~ (q)= 1 dr r sin(qr) V (r) q Z0 which will depend upon the precise form of the scattering potential. For example, if we take the so-called Yukawa (or screened-Coulomb) potential

e2 V (r)= e ®r r ¡ Scattering Amplitudes and T-Matrices 275 then 4¼e2 V~ (q)= 1 dr sin(qr) e ®r q ¡ Z0 4¼e2 = ®2 + q2 Thus, in this case, f(µ;Á)=f(µ)=f(q); where q =2k sin µ=2; and

2me2 1 2me2 1 f(µ)= = ¡ ~2 ®2 + q2 ¡ ~2 ®2 +4k2 sin2 µ=2 and the cross section becomes d¾ 4m2e4 = : 2 2 d ~4 ®2 +4k2 sin µ=2 By taking the limit that ® 0 we obtain¡ the corresponding¢ cross section, in the Born approximation, for the Coulomb! potential

d¾ m2e4 e4 = 4 = 4 : d 4~4k4 sin µ=2 16"2 sin µ=2 As a second example, considering the elastic scattering of electrons by a neutral atom in its ground state with initial electron energies that are too small to excite the atom to any of its excited states. For an atom of atomic number Z; thechargedensity ½(~r) can be written ½(~r)=e [Z±(~r) n(~r)] ¡ where n(~r) is the number density of electrons in the atom at ~r and can be written

n(~r)= Ã ±(~r ~ri) Ã Ãn¤ i (~r)Ãni (~r) h j i ¡ j i' i X X where the second form holds in an independent electron approximation. The electric potential '(~r) at a point ~r due to the charge density of the bound electrons and the nucleus satis…es Poisson’s equation

2' = 4¼½(~r)= 4¼e[Z±(~r) n(~r)] r ¡ ¡ ¡ where charge neutrality implies that d3rn(~r)=Z. The corresponding potential energy seen by an incoming electron is given by R V (~r)= e'(~r) ¡ so that 2V (~r)=4¼e2 [Z n(~r)] : r ¡ We now note that if d3q V (~r)= ei~q ~rV~ (~q) V~ (~q)= d3qe i~q ~rV~ (~r) (2¼)3 ¢ ¡ ¢ Z Z then 3 2 d q 2 i~q ~r V (~r)= q e¡ ¢ V~ (~q) r (2¼)3 Z 276 Scattering Theory has as its Fourier transform the function q2V~ (q): We write, therefore, as the Fourier transform of Laplaces equation ¡

q2V~ (q)=4¼e2 [Z F (q)] ¡ ¡ where the atomic form factor

3 i~q ~r F (~q)= d re¡ ¢ n(~r) Z is the Fourier transform of the electronic charge density. Thus, we can solve for V~ (~q) to obtain 4¼e2 [F (q) Z] V~ (~q)= ¡ ; q2 which allows us to evaluate the scattering amplitude in the Born approximation 2e2m [F (q) Z] f(q)= ¡ ¡ ~2 q2 and the corresponding cross section

d¾ 4e4m2 F (q) Z 2 = j ¡ j : d ~4 q4 Thus, measurement of the cross section for all momentum transfer allows information to be inferred about the distribution of charge in the atom [as contained in the form factor F (q)]. For a spherically symmetric charge density it is possible, in principle, to invert this relation to determine the charge density ½(~r) directly.

9.5 Partial Wave Expansions In the last section we have not really used the fact that V (r) is a spherically symmetric potential. In this section we explore some of the simpli…cations that occur as a result of 2 this fact. Speci…cally, if V = V (r); then H commutes with L and Lz and we know that 2 there exists a basis of eigenstates common to H; L ;Lz : Let "; `; m = k; `; m denote such a basis for the positive energy subspace of the system ofj interest,i where,j asi usual, © ª k = 2m"=~2: We note, in particular, that the potential V (r)=0is spherically symmetric, so therep must exist a basis of this sort for free particles. Let us denote by "; `; m (0) = (0) 2 j i k;`; m the corresponding free particle eigenstates common to H0;L ;Lz : Both sets ofj statesi satisfy orthonormality relations © ª (0) (0) k; `; m k0;`0;m0 = ±(k k0)±`;` ±m;m = k; `; m k0;`0;m0 h j i ¡ 0 0 h j i and have functions of the following form Á (r) Ã (r; µ; Á)=F (r)Y m(µ; Á)= k;` Y m(µ; Á) k;`;m k;` ` r ` Á(0)(r) Ã(0) (r; µ; Á)=F (0)(r)Y m(µ;Á)= k;` Y m(µ; Á) k;`;m k;` ` r ` where the functions Ák;`(r)=rFk;`(r) obey the radial equation

`(` + 1) 2 Á00 + v(r) k Á =0 k;` ¡ r2 ¡ k;` µ ¶ Partial Wave Expansions 277 in which v(r)=2mV (r)=~2 and k2 =2m"=~2 0: Note that when V =0this reduces to ¸

(0) `(` + 1) 2 (0) Á 00 k Á =0: k;` ¡ r2 ¡ k;` µ ¶ The solutions to this latter equation that are regular at the origin are well-known and related to the spherical Bessel functions j`(z) of order ` Speci…cally, it is found that

2 Á(0)(r)= kr j (kr); k;` ¼ ` r 2 so the free particle eigenstates of H0;L ;Lz are f g 2 Ã(0)(~r)= kj(kr) Y m(µ; Á): k;` ¼ ` ` r On the other hand, provided that V (r) 0 as r faster than 1=r; then asymptotically both equations (V =0and V =0)obey! the equation!1 6 2 Á00 + k Á =0 r ; !1 ikr ikr which has the general solution Á(r) Ae + Be¡ ; so that the radial dependence of the F (r)=Á(r)=r has the form »

eikr e ikr F (r) A + B ¡ » r r of a superposition of incoming and outgoing spherical waves. To conserve probability, the ‡ux into the origin has to balance the ‡ux out of the origin, which imposes the requirement that A = B ; in which case we can asymptotically write j j j j Á (r) a` sin (kr ' ) : k;` » ¡ ` Indeed, it can be shown from the properties of the spherical Bessel functions that the free particle solutions have the asymptotic behavior

2 Á(0)(r)= kr j (kr) a sin (kr `¼=2) k;` ¼ ` ` r » ¡ so that `¼ '(0) = ` 2 for a free particle (V =0).WhenV =0it is convenient to write the phase of interest in the form 6 (0) `¼ ' = ' ±` = ±` ` ` ¡ 2 ¡ Á (r)=a` sin kr `¼=2+±`) k;` ¡ where ±`is the phase shift that arises due to the potential (±` 0 as V 0). and is ! ! uniquely determined by it (whereas a` scales with the normalization of the state). Our goal is to obtain an expansion for the stationary scattering state Á" in the complete set of states k; `; m and use it to obtain an expression for the scatteringj i amplitude f(µ; Á) expandedj in sphericali harmonics. In other words, we seek an expansion of the form m f(µ; Á)= f`;mY` (µ; Á): `;m X 278 Scattering Theory

As a preliminary simpli…cation, we note that, because of the spherical symmetry of the potenital, there is azimuthal symmetry along the z-axis associated with the incident beam, thus, only m =0components exist in the expansion:

0 f(µ; Á)=f(µ)= f`Y` (µ) (9.1) ` X To proceed we express the stationary scattering states of interest as an expansion

ikr ikz e Á (~r) e + f(µ) = A` Ã (~r) " » r k;`;0 ` X in states of well-de…ned angular momentum. The `th term in this expansion is referred to as the `th partial wave. In terms of the spherical harmonics, this expansion takes the form eikr Á eikz + f(µ) = A k;` Y 0(µ): r ` r ` ` X To make this useful, we now use the known expansion of the function eikz in free particle spherical waves: ikz (0) 0 e = c` Ãk;`;0(~r)= B` j`(kr) Y` (µ): ` ` X X ` The B` can be calculated exactly. The result is B` = i 4¼(2` + 1) so that

ikz ` p 0 e = i 4¼(2` + 1) j`(kr) Y` (µ): ` X p Thus we can write f eikr Á (r) B j (kr)+ ` Y 0(µ)= A k;` Y 0(µ): ` l r ` ` r ` ` X · ¸ X m Linear independence of the Y` ’s implies that

ikr B`rj`(kr)+f`e = A`Ák;`(r): Now asymptotically,

B` B` ikr i`¼=2 ikr i`¼=2 B`rj`(kr) sin(kr `¼=2) = e e¡ e¡ e » k ¡ 2ik ¡ h i and

A`Ák;`(r) a` sin(kr `¼=2+±`) » a ¡ ` ikr i`¼=2 i±` ikr i`¼=2 i±` = e e¡ e e¡ e e¡ : 2i ¡ h i Substituting these last two equations into our previous expansion and equating coe¢cients ikr ikr of e and e¡ we …nd that

a` B` e i`¼=2ei±` = e i`¼=2 + f 2i ¡ 2ik ¡ ` and a` B` ei`¼=2e i±` = ei`¼=2 2i ¡ 2ik Partial Wave Expansions 279

which gives two equations in the two unknown quantities a` and f`: Solving for f` we …nd that 1 f = 4¼(2` + 1)ei±` sin ± ` k ` so that p 1 f(µ)= f Y 0(µ)= 4¼(2` + 1)ei±` sin ± Y 0(µ) ` ` k ` ` ` ` X X p from which follows the expansion for the di¤erential scattering cross section

2 d¾ 1 1 i±` 0 (µ)= 2 4¼(2` + 1)e sin ±`Y` (µ) d k ¯ ¯ ¯`=0 ¯ ¯X p ¯ ¯ ¯ The total cross section can then be¯ written ¯ 2 1 1 ¾ = d 4¼(2` + 1)ei±` sin ± Y 0(µ) tot k2 ` ` ¯`=0 ¯ Z ¯X p ¯ 1 1¯ ¯ 1 ¯ i(±`¯ ± ) 0 0 = ¯ 4¼(2` + 1) 4¼(2` + 1)e ¡¯ `0 sin ±` sin ±` dY (µ)Y (µ) k2 0 0 ` `0 `=0 ` =0 X X0 p p Z which reduces to

4¼ 1 ¾ = (2` + 1)sin2 ± tot k2 ` `=0 X 1 = ¾` `=0 X where 4¼ 2 4¼ ¾` = (2` + 1)sin ±` (2` + 1) k2 · k2 is the scattering cross section to states with angular momentum `. Note that for free particles ±` 0 and ¾tot 0; as we would expect. ! ! Fkdswhu 9 JORVVDU\

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