THE THEORY OF QUANTUM MECHANICS: SFlVtiN LI.,lC'[URI.]S F OR S'l'UI)l.lN'l.SOl..- TN'fROI)UC'I'O RY UNI V I.]RSI'I'Y I' H YSICS

by

I)aniel'l'.Gillespie

ResearchDepartment; Naual WeaponsCenter China Loke, California gSSSs

Approvedfor publicrelease; distribution is unlimited"

Preparedfor the Final Reportof the

Carleton College Conference (August lg88)

of the

lntroductory University Physics Project I'REI.'ACE

I came to write these lectures in the course of participating in the [ntroductory University Physics Project (IUPP), which was organized in 1987 by the American Ph-vsicalSociet-v and the American Association of Physics Teachers under the sponsorship of the Nationarl Science Foundation. The ultimate goarlof IUPP is to develop some new models for the introductory, calculus-based, university physics course. As envisioned by IUPP chairman John lligden, the new course models should have a "leaner story line," and should if at all possible contain something meaningful about quantum mechanics. My own involvement in tUPP is as a membcr of the working group on quanturn mechanics,which is chilired by llugen Merzbacher and co-chaired by Thomas Moore. I musl emphusize tlwt lhe set of'lecture,spresente.d here. is rutt patterned after the syllabus 'l'he recommended hy the IUPP quctntum mechanics worhin14!:roup. approach taken in these lectures is frankly too abstract and too formal, too long on theory and too short on applications, for mttst of thc students rvho tuke the introductory carlculus-basedphysics course. But.sorne of those students, those with a more mathemartical and philosophical bent, might possibly prolit flrom thcse lccturcs. And pcrhaps also some physics tcachcrs might find onc or more fleaturesof thesc lectures worth incorporating into their own lecturesand rvritings. Wh-v should a univcrsity student bother to learn quantum mcchanics? For students intent on becoming prolessional physicists, the motive harsalways been clear and compelling: [f -vou don't learn quantum mcchanics, then you can't join the club. Obviously, a different motive must impel students on a broadcr spectrum, and pcrhaps should impcl physics majors as wcll. The motive that I have tried to implicitly establish in these lectures is roughly as [ollows. Nature has been found to behave, on the microscale, in ways thart seem to utterly dely comrnon sense. Mankind has thus been presented with an immense chtrllenge: N{akc scnse of Nature on thc microscarle!Quantum mcchanics, within carefully prescribed limits, has successfullymet that challenge. So the theory of quantum mechanicsrepresents an intellectuarlachievement of truly heroic proportions, arnd may rightly be regarded as "a principal jewel in the cultural crown of civilization." To appreciate the beauty and mystery of that cultural jewel is alone sufficient reason for any student of an.y university to undcrtake r.rscrious study of quantum mechanics. It seems that there are two arpproachcsto tcaching physics in general and quantum rnechanics in parrticular: In the ax,ioma,ticapproerch, one first sets forth a minimal number of assumptions or axioms, and one then rigorously deduces their various consequences. Ily contrast, in the organic approach one pays little attention to logical structure, and simply allows the various ftrctsof the theory to arrange themselves in whatever way seems convenient. While I have respect for the organic approach, and in fact suspect th.rt physicists of that camp are more likely than their opposites to make significant advances in physics, I havc delibcratcly writtcn these lectures in the axiomatic vein. I bclievc that students with an aptitude for mathematics and logic.rl reasoning strongly prefer t

I feel obliged to forcwarn teachers who are contemplating using these lectures that I have taken what some might consider to be an unnecessarily doctrinaire position, narmely, that obseruablesfor microscale systems do not ulways haue ualue,s. [n [act, I make thert premise to be a principal motive for devising quantum mechanics. A teacher who feels that lhis premise is unwarranted, or even false, will nol like these lectures. []ut bcfore such a teacher larysthese pages aside for that reason, I would respectfully entreat a reading of the entertaining article by N. David Mermin in the April 1985 issue of Physics Today. In that article, Mermin discusscs the implictrtions of Bell's theorem in terms of trn idealized cxperiment thut is equivalent to an electron-pair spin-correlation experiment, but which is happily dcvoid of the esotericjargon oI quantum physics. Mcrmin shows with rernarkablc clarit.v th.rt there is simply no way of expleriningthe results of his experiment without doing great violence to one's sense of "reasonableness." I think that a defensible interpretartion of the specific conclusions o[ the 'lhere Mermin experiment is this: is no way to assign fixed values (red or green) to each observable (1, 2 and 3) o[ both particles in such a way that the experimental data can be quantitatively acounted for, hence, we cannot generall-v ascribe simultaneous values, euen unh,nown ualues, to non-commuting observables. This is onc of several non-common-sensical features of quantum mccharnicsthat, have bccn exposed by.Iohn []cll's work in 1964, and more rccently confirmcd experimcntally by Alain Aspect and coworkers. I believe it is wrong not to bc up-front with studcnts about such issucs" I belicve it is wrong to pretend to students, on the prctcxt of "doing physics rather than philosophy," that although quantum mechanicsmay seem strangc and unusual there is nothing about it that should disturb our view of Reality. Indeed, I feeI th.rt the strong claim which quanlum mcchanics hars to prolound cultural significatnce stems largely lrom the fact that il hus raised serious and as yet unanswered questions about Rculity. We nccd not necessarily rnuke a philosophical study of such issucs in a physics course, arndwe do not do that in these lectures, but we should not deny the existence of these issues. As evidence in these lectures for the "fact" thut ohservablcs for microscale systems do not alwarys have values, I havc invoked thc doublc-slit experiment. Thc cvidencc provided by that experiment is strong, but not unassailnble. Evidence of a more compelling nature would have been providcd by an elcctron-pair spin-corrclation experimcnt, but unfortunatel-v etsattisfactory 'Ihe quantum analysis of such an experiment lics beyond the reach of thcse lectures" double-slit experiment on the o[her harndcunbe analyzed within the framework of thcse lcctures, arndin fact it provides us with a thcmatic bridge from Lecture I to l,ecture 7. As to the treatment givcn in thcsc lectures of the quantum theory itself, I havc indccd organized and presented the ingredients of the theory in a very different way than is usually done. Ilut the basic view of quantum mechanics taken in these lectures is quite orthodox, arndis fully in line with what one will find in such standard textbooks as l)irac, lllessiah and Merzbacher" I have tricd to prescnt in these lcctures a highly simplified and hcnce unconventional rendering of conventional quantum theory. I would like to th:rnk John lligdcn and Ougen Merzbacher for their encouragement in preparing these lectures. I am happy to acknowledge the benelits of conversations with other physicists who participated in the 1988 IUPP conferences at llarvey Mudd College and Carlcton College, cspecially those in the quantum mechanics working group. And I want to thank Ron [)crr, [lead of thc RcscarrchDcpartmcnt of thc Naval Wcapons Center, for allowing me to perrticip.rte in IUPP and develop these lectures within the frermework of the Cen[er's lndependent Rcsearch [)rogram"

China l,ilkc, Cal iflornra I)."1. Gillcspic May, 1989 Ilr

CONTENTS paSe Prelace Lecture l. Introduction and Motivation I l.l 'lhe fundamentalproblem of mechantcs. . . . I l.2 The failure of classicarlmechanics on the microscale I 1.3 The aim and planof theselcctures 3 1.4Complcx numbcrs 4 Lecture 2. The Mathematics of Generalized Vectors 6 2"I Vectors,scalars, components, basis expansions 6 2.2 Operators,lincarity, eigenvectorsand eigenvalues,eigenbarses . . . . l0 2.3Summarv ... t2 Lecture 3. Quantum Mechanics: Statics I l3 3.1 Statestrndobservables,vectorsandoperators ...... 13 3.2 ltesultsand effectsof measuremcnts.. 15 3.3AnswertotheF'POMforl=0...... 17 Lecture 4. Quantum Mechanics: Statics Il l8 4.1 Recapitulation r8 4.2 Compatible and incompatible observables 20 4.3 Intcrference. The "value" of an observarble 2T f LectureS. Quantum Mechanics: Statics III 23 5.1 The position and momentum opcrators. Wave lunctions 23 5.2 The wave-particleduality 25 5.3 The lleiscnbergUnccrtainty Principlc ,.7 Lectuie 6. Quantum Mechanics: l)ynamics I 29 'l'he 6.1 Hamiltonian operator and time-evolution 29 6.2 Conserved quantities. Stationary states 31 6.3 Answerto the FPOM lor any I > 0 32 6.4 '['ransitionamplitudes arnd virtual processes 33 t LectureT. Quantum Mechanics: Dynamics Il 34 7.1 llecapitulertion 34 7.2 (O ptio nal)'lhc Schr

t'Ihese two lccturesmay beomitted for a lessdemanding Iive-lecture scries LECTURF]I. INTROI)UCTION AND MOTIVATION

'fhe )l.l F'undamental Problem of Mechanics The subjectof "mechanics"deals with a systemand its observableproperties or obse.ruables.If A is an observableof a given system,then we can at any time "make a measurementof A on the system," and thereby obtain a result A, an ordinary number that we call the ualue of A. For example, the position (an observableA) of a ball (the system being considered)is measured,with the numerical result 4.'l (= !, the measuredualue of the observableA). In the "classical"mechanics of Isaac Newton, there is no need to distinguish between an observableA and its value A: The valuesof the observablesof a systemarc representedby ordinarry mathematical variables,and the centrarlprogram of mechanicsis to hnd (e.g.,by using F=md Lhe time dependenceof those"observarble variables." This program is very successfulon the macroscale, the scale of systems that we are familiar with from our everyday experience(such as thc moon; airplarnes,tennis balls and dust particles). But the clerssicalprogram runs into seriousdifficulties on the microscale,Lhescale of individual atomsand molecules.As a preludeto describingthe microscale difficulties of clarssicarlmechanics, and how "quantum" mechanicsultimately avoidsthose difficulties, let'sbegin by rephrasingthe goalof mechanicsin a more cautious"operational" way: ,THE F'UNDAMENTAL PROBT.EM OF MECTIANICS (FPOM): - At time zerowe measuresome observable A on the system,and obtain the result A. - Then we let the systemevolve on its own until sometime I (t>0). - n t ,, we measuresome observable B on the system. o Knowing A, A, t and B, what can wepredict qbout the result B of the secondmeasurement?

)1.2 The Failure of Classical Mechanics on the Microscale In the classicalapproarch to the FPOM, we know that in some circumstanceswe celnpredict B exactly,while in other circumstanccswe can predict R only probahilistically. b'orexample, consider a simple harmonic oscillator of mass rn and spring constant lc. Classically, it executessinusoidal motion with frequencyZn(hl m)l/2 and someamplitude a. Supposethe first-measuredobservable A is "total energy," and the secondmeasured observable B is "position." Sincethe energy value A will be related to the oscillation amplitude aby A= ha2/2,then after the A-measurementwe will know thart the amplitude of thc oscillator motion is a-I2A/hlll2. But that's all we will know. So we can't make a unique predictionfor the result Il of any subsequentposition measurement. We can, however,make someprobabilistic predictions: We know that B must lie somcwherebetween +I2A/hlttz'moreover, sincethe particle spendsmore time near the turning points LIZAIhlttz, where it movesslowly, than near 0, where it movesquickly, then we should expect B-values near *l2A/hlrtz to be more likely than lJ-valuesnear 0" [n fact, if we reason carefully from the classical equations of motion for a harmonic oscillator (we won't go through the details here), we can derive the "probability densiby" curve for the B-valuesshown in Fig. l-1. The shadedarea in that ligure is ecluulto the probahility that the result Il will lie betweenB1 and I}2" (The curve is the reciprocalof the oscillator'sspeed at positionB, multiplied by a constantthart makes the total area under the curve unity.t Well, what do we find experimentally?We get excellent quantitative agrcement with this classicalprediction for particles of "tangible" mass. But if we coulddo the experiment with a particle of uerysmall nlass,such as an atom,wc wouldsee markcd dcviartions from this prediction:I)cpcnding upon the energy result A, we would {ind that it is irnpossibleLo geL:;omeposition vtrluesB between t[2Athlttz, and po.s.si6leto get positionvalues I] with ltll>1ZAtny2. IWhen rn is very smull wc wor.rld alsonotice that the energyvalues A obtainedin l,hel'irst measurementare always integermultiplcs of a number proportionalLo &/ m)l/2,another fcarturethat is not predictedby clrrssicalmech:lnics. I We concludefrom this and similar experimentsthat, at the very lcast, the valuesof the observablesof a microscalesystem arc not interrelated by the farmiliarclarssical lormulas that work so well ltrr mucroscalesyste ms. tsut in fact, the fiailurcof classicalmcchunics on thc microscalcruns LECTURE1 INTRODUCTIONAND MOTIVATION

I ftIQA|h) - B2lr/2

Il-values

_(2A/h\r/2 81 82 0 (2A/hlr/2

I,'tG.1-1. The classicalprediction for the result B of a position measrlrement that is 'l'he performed on a harmonic oscillator with total energy A and spring consttrnt A. shaded area is equal to the probability that the result of such a measurement will fall between the position values B1 and 82.

much deeper than this: Other experiments indicate that it is pointless, even selfcttntrctdictory, to sa,v that microscalesystem observablesalways "have values." Of those experiments, perhaps the easiest to both describe and analyze is the so-callecldouhle-slit experiment. ln the double-slit experiment, schematizedin F ig. l -2, a very smirll particle - lct's sa-\'arn elcctron for definiteness - is fired with horizontal rnomentunr p1yat a vertical screen 51 (assume thut no gravityflorcesoperate).ScreenSlisoparquetotheelectronexceptfiortrvohorizontalslitsatJ--1'land y=y2. To actuarlly "seethe effects that we shall dcscribe, thesc trvo slits must be uerv nurrolo and uer,v close together, so this is indeed a "microscale" expcrimcnt. []e-vond51 is a second vcrtic:r.l screen 52, this one coatedlike a television screenwith a substancethi.rt cmits it spot of light whcrevcr it is struck

incident z2 electron slity1 -...-.---.t> (momentum pg) slity2

screenS1 screen 52

FIG. l-2. Schematic diagram of the doublc-slit cxpcriment, showing the hit probtrbility patterns with both slits open (curvc C12) and wiLh only onc slit open (curves C1 and C2). 'the length scale on screen 51 has been grcatly maenified in this drarving; compared to the "mtrcroscopic"sctrle of scrccn 52, the two slits in screen S1 are "microscopic" in both their sizeand their separation. LECTURE1. INTRODUCTIONAND MOTIVATION

by an electron. The coating allows us to record the point z where the electronhits 52. (We measure vertical positionalong Sr by y, and vertical positionalong 52 by z. Note that in Fig. l-2 the y-scaleis greatly magnified relative to the e-scale.) If we repeat this experiment many times, we can empirically determine Lheprobability that the impact point of the electronon screen52 will be at,any point z- Curve C12shows schematically the hit probability (or hit frequency)pattern that is actually observedwhen slits y1 and yz are both open. Now it so happensthat this curve looks exactly like thl interferenceintensity pattern we would get if we were transmitting through the two slits, insteadof particles,waues of soundor light with wavelength h = 2nh/po. (l-l) Here, pg is the measuredinitial momentum of the electrons,and i, called "h-bar", is a universal physicalconstant introduced by Max Planck with the numerical value h = 1.054...X l0-34 joule.sec. (r-2) So the shapeofcurve C12strongly suggeststhat an electronis a waue:yet the fact that the curve C12 is inferred from the statistics of one-at-a-time,point-like scintillations of equal intensity .rggurit equally strongly that an electron is a particle. This curious behavior is not peculiar to electrons alone,but is observedfor virtually all elementaryconstituents of matter and energy(such as protons, photons,etc.). Nature seemsto exhibit on the microscalea waue-particteduality that is simply noi understandablein purely classicalterms. Curve C1 in Fig. l-2 showsschematically the hit probability pattern thtrt is observedwhen only slit y1 is open. The curve C2 obtained when only slit y2 is open is virtually indistinguishablefrom curve C1, becausethe slit separationdistance on screen51 is minisculeon the length scaleof screen 52. Notice that there are somepoints on screen52, suchas 21,that are more likely to be hit when both slits are openthan when only one slit is open;that's not surprising. But what is surprising is the fact that there are other points on screen52, such as 22,that are les.slikely to be hit when both slits are open than when only one slit is open;i.e., by closingone slit we make it more lihely that point z2 will be hit on any one electron{-rring! Now, it is obviousthat any electron arriving at screenls2can only have comeby way of the two slits y1 and y2 in screen51. And it is equally obviousthat when only sl{t y1 is open,any electronarriving at 52 can only havecome through that slit, and hencemust have had at screen51 the y-positionvalue y1" When 6olh slits are open,common sense would seemto require that any electronarriving at 52 must have come either wholly through slit y1 or wholly through slit y2. But if that were true, then closing one slit could not possibly increasethe Iikelihood that an electronwill reach point 22,as we observethat it does. So we must concludethat "commonsense" is wrong: any electronthat reachesscreen 52 with both slits opendid notgo through either slit y1 or slit y2, but insteadsomehow made use of bothslits! It followsthat, for suchan electron,the observable'y- positionat 51" cannotmeaningfully be said to have"had a value." This linding is just one exampleof the astonishingconclusion that physicistshave been forcedto *haue by many carefully done experiments: Obseruablesfor microscale systemsdo notalways ualues"" Therefore, we cannot addressthe Fundamental Problem of Mechanicson the microscalewith the usual "common sense"classical approach, becausethat approach always starts from the assumption that system observablesdo always have values. (For example, we can't analyze the double-slit experimentby trying to figure out how the positionvalue o[ the electronchanges with time, becausea "position value of the electron" does not always erisl during that experiment.) Although classical mechanicsworks quite well for macroscalesystems, it seemsthat a radically different app.oachto microscalemechanics must bedevised.

)1.3 The Aim and Plan of These Lectures 'f he "replacementtheory" for classicalmechanics on bhemicroscale is calledquantum mechanics. Quantum mechanicswas developedduring the first quarter of the Twentieth Century by Werner lleisenberg,Erwin Schrridinger,Niels Bohr, Max Born, Paul I)irac, John von Neumann, and a number of others. We shall not delve into the history of quantum mechanicsin theselectures, but we shouldnote that the decisionof early Twentieth Century physiciststo abandonclassical mechanics on 'I 4 LECTURE. INTRODUCTIONAND MOTIVATION

the microscale was at the time both daring and traumatic. The theory that ultimately emerged from the heroic cfforts of those physicists, Lhestandard theory of quantum mechanics . . . - takes a radically different approach than classical mechanics; - is "non-intuitive" from t hc viewpoint of ordinary cxperience; - gives correct results on the microscale; - reduccs to classical mechanics on the macroscale; - is controversial, even among physicists; - is one of the most ingenious and significant intellcctual achievements in the history of mankind. There is no way that we can cover quantum mcchanicscompletely in seven lecturcs. Wc will have 'the to leave a lotout, and greatly simplify the rcst. limitcd goal of these lectures will rutlhe to show you how to solve lots of quarntum mechanicsproblerns, but rathcr to give you an accurate appreciation of the "essence" of quantum theory" Witl"r some effort on your part, you should acquire from these lectures an honest sense of the slrange kind of reasoning we have to use in order to successlully describe physical phcnornena on the microscale. Wc shall first have to learn somc new mathematics - calculus alonc is not enough lor quantum mechanics. We'll get a brief start on that task in.iust a moment by developing some facts ab

)1.4 Complex Numbers You have probably alreatdy encountered complcx numbers. tlere we are going to rcview a few basic facts about thcm that wc sharllnecd for our development of quantum mcchanics. A compler number is a numbe r of the form c = q,* ih, (l-3) where i=V-1, and o and 6 arc ordinary real numbcrs. We call a the "rcal part" of c, and 6 thc "imaginary part" o[c, and wc write o=ltc{c},6=lm{c}. If6=0wesaythatcispurereal,whileifa=0weseryth.rtcispureimaginary. Andifcanclbarcholh zcro, wc write c=0. It is important lo understand that, thc tcrms "rcal" and "imaginary" arc uscd LECTURE1. INTRODUCTIONAND MOTIVATION

here in a strictly technical sense;you should nol grant them their usual connotationsof "genuine" and "artilicial." The key feature of complex numbers is that they can be added and multiplied by using the ordinary rules of algebra,but supple.mentedby the rule that i2= - l. )ErerciseI-1. lf cr=at* i61 lnd c2=aZ* i62,show that c1* c2 = (ot * ail + i(h+ bz), (l-4a) crc2= (apz- bft2\ + i(a1h2*bp,il. tl-4b) The complerconjugate of c in (l-3) is definedto be the eomplexnumber c*=a-ib. (l-5) oExercise1-2. Showthat c is pure real if and only if c* = c. Also provethat (c*)* = c, ( l-6a) c*c* = 2Re{c}, ( l -6b) (c1*c2)* = ct**c2*, (1-6c) (c1c2)*= cl*c2*' ( l -6d) The squareof c, namely c2= cc = (a2-bz) + i(2abl, (1-7a) is obviouslya complexnumber. [Iowever, the squaremodulus of c, which we defineto be lcl2= 6c* = c*c= q} + b2, ( l -7b) is evidently a.non-negatiue real nu,mber, which vanishes if and only if c = 0.

oExercise I -3. Carry out the algebra leading to equations ( I -7). Also prove, using the results of Exercisel-2, that - l"r"zl l"rll"zl, (1-8a) lc1+c2l2= lc1l2+ lc2lz* 2Re{c1c2*}. (l-8b) Finally,for any realnumber u wedeline siu:cosu*isinu. (l-9) One rationale for denotingthe complexnumber on the right sideof (t-9) by the erponentialsymhol on the left is this: If a is any real constantand r any real variable, then (d/dr)siox= (d/dr)[cosax* isinorl = -esincr * iacosar = io[isinar * cosorl, (ddr)eior - iqsiax, (l-10) which is preciselywhat we would get if i were an ordinary real number. oExercise1-4. Using bhedefinition (l-9), provethat - eio l, (l-l 1a) - - (eiul* cosz-isinu e-iu, (l-l lb) - leialz t, (1-1lc) (siuxgiu;- gi(rr*u). (r-1 ld)

> Beforethe next lecture, you should do Exercisesl-l through l-4. You should becomelairly familiar with these facts about complex numbers before tackling the mathcmatics of generalized vectors,which is what we will do in the next,lecture. 6

LECTURE 2. THE MATHEMATICS OF GENER ALTZEDVECTORS

;2. 1 Vectors, Scalars, Components, Basis Bxpansions You have learned that there are some quantities in physics,such as mass, temperature ancl density,that have only "magnitude,"and are called scalars;other quantities, such as position, velocity and force,have both "magnitude" and "direction," and are called uectors.[,et's review some things you presumablyalready know aboutvectors" [SeeFig. 2-11We catnpicture a uectorv as an errow, whoselength is the rnagnitudeof the vector, and whosesense (from its ttril to its head)is the directionof the vector. If we translate the arrow, i.e., move it without changing its length or direction, we don't charngethe vector. We can multiply any vector v by any real number r, ctrlled a scalar, to get a new vector, written rv or vr, which has a magnitude that is lrl times the magnitude of v and a direction that is the same or opposite the direction of v accordinglyas r is positiveor negativc. Two vectorsv1 and v2 can be addedto form a new vector,vl +v2, by translating them so that the tail ofv2 lies on the headofvl and then drawing the sum vectorfrom the tail of v1 to the headof v2. And v2* v1 is the samevector as v1* v2.

also vector v vector v (0.5)v (- 1.25)v

vl -t\", ry"t*",ffi

FIG" 2-1. Vectors,scaltrr multiplication and vectoraddition.

[See Fig. 2-21 Any vector with length I is called a unit uector. The componentof any vector v relative to a unib vector e is the perpendicularlyprojected length of v onto e, a scalar that we denote by e'v. If the head-to-headangle betweenv and e is 0, then e.v is equal to the magnitudeof v times cosO;thus, e'v is positiveor negativeaccordingly as g is lessthan or greater Lhann/2. lf 0= rc12,Lhen e'v = 0, and we say that e and v are orthogonal. The e-componentof rv is r times thc e-componentof v; the e-componentof v1+ v2 is the sum of the e-componentsof v1 and v2. [SeeF'ig. 2-31 In two dimensions,any two orthogonalunit vectorse1 and e2 lorm a 6csl.s,and any vectorv can be "expanded"in that basisaccording to the rule v = el (er.v) f e2(e2.v), whereinthe scalarmultiplying the unit vectore, is just the er-componentof v. Similarly in three dimensions,any thrce mutually orthogonalunit vcclorsel, e2 and e3 constitutca basis,and any vectorv can be expandedin that basisaccording to v = el (e1.v)+ e2(e2.v)f e3(e3.v). LECTURE2. THEMATHEMATICS OF GENERALIZED VECTORS

--">

- *' unit vector,e "'o (length= 1) (a scalar)

e.(rv)€'(rv) ==7 r e.vs.y e.(v1+v2)=e.v1 *e.v2

F'lC.2-2. Unit vectors and components relative thereto.

(e3.v)

e2 (e2.v)

er (er"v) e1(e1"v)

v = el (er"v)* ez (ez"v) [email protected])

v = el (er.v) + ez(ez.v) * e3(e3.v)

FIG"2-3. Basesand expansionstherein.

The aforementionedfacts about vectorsshould not really be new to you. l,low,besides ueclor.s, you have no doubt also encounteredin your studies malhemuticians;they too are both atmusing and useful. One of the amusing and useful things about mathematiciansis their penchant for doing mathematicswithout relying on pictures,as we just did in our discussionof vectors. Let's sec how ar mathematician might try to "formalizc" the forcgoing vector notions, and lay out a mathematicat theory of vectorswithout ever refering to directed line segments.lNote to studentsol'linear algebra: The defrnitionsand postulatesthat follow will be u bit different from what you've learned. Wc're goingto take asdirect and elementaryan r.rpproachas our later ncedswill allow.I We dcfinea uectorspace to be a collectionof objectsv, callednectors, such that the followingthings [(a)through (d)l are true: (a) "Sc.rlarmultiplication" is defined: [f v is any vcctorin the spaceand r is any real number,also calledascalar, then rv ( =vr), calledthe productofrand v, is alsoa vectorin the space. LECTURE2. THEMATHEMATICS OF GENERALIZED VECTORS

(b) "Vector addition" is defined: If v1 and v2 areany two vectorsin the space,then v1* v2, called the sum of v1 and v2, is alsoa vectorin the space.And v2* v1 is the samevector as vl -r v2. Comment: From (a) and (b) we seethart, flor any two vectorsv1 and v2 in the spaceand any two scalars 11and 12,the linear combination rtvt * r2v2 is a well-definedvector in the space. (c) There exists in the spaceunit uectors,relative to which all vectors in the space have components.The componentof the vectorv relative bothe unit vectore, called the e-componentof v, is written e"v,and has the following properties: (c1) e-v is a scalar. (c2) e.(rrvr * r2v2)= rt(e.vt) + r2{re.v). (c3) e'e - l. (c4) e.e' = €'.e, for any two unit vectorse and e'. Cornment: (cl) stt that the e-componentol'any vector v is a real number. (c2) says that the e- componentof any linear combinationvector is the samelinear combinationof the e-components.(c3), saysthat the e-componentof itself is unity, and thus servesto definea unit vector. And (c4)says that for any two unit vectorse and e', the e-componentof e' is equal to the e'-componentof e. (d) If the vectorspace is N-dimensional,then there existsat leastone set of N unit vectors{er, ez, ...,eN) called a 6osissuch that (d1) e;'e2= 0 wheneverj*h. (d2) For any vectorv in the space, v = el (et.v) * ezbz.v| + ... + elrg(e1,'.v). (2-r) Comrnent: (dl) stipulatesthat the unit vectorscomposing a basisbe mutually orthogonal. (d2) says that any vector v can be written as a linear combinationof the basis vectors,and that the scerlar coefficientof e, in this e-6osisexpansion of v is just ey.v,the eJ-componentof v. Eq. (2-1)also suggests somethingthat is quite valid about how we actually specifyvectors in practice:.We generally specify a vectorv by giving its componentsrelative to some"known" basis,which in turn neednol be further specified.The point is that we can't really specifya length and direction lor v unlesswe say "length relative to what" and "direction relative to what," and a 6osis is the "what." So the actual specificationof a vectorin N dimensionsultimately comesdown to specifyingN ordinary numbers- the componentsof v relative to someparticular basis. Now at this point, if you're not a mathematician,you're probably thinking that all this formalism may be okay, but picturesare better: One picture is worth a thousandmathematicians! In fact, you probably suspect that even though the mathematician didn't d,raw picLures,he/she was thinhing pictures! But notice something: The mathematician'sformal vectorsare a little more general than our picture vectors: the dimensionality l/ of the mathematician's vector space can be any positiue integer,whereas we can draw picture vectorsonly for lV<3. Thus, the mathematician has come up with a significant generalizationof the vector concept,one that we will in fact make use of in our subsequentdiscussions.Howevern when we speakof "generalized"vectors in what follows, we shall mean something rnorethan arbitrary dimensionality. For generalizeduectors, the scalars are compler numbers instead.of real numbers. And then it's "goodbyepictures," even in two dimensions. [.'or generalizedvectors, we have no choicebut to usethe abstract,formal approachof the mathematician. Fortunately, though, we can lay out the theory of generalizeduectors by repeatingq,lmost uerbqtim the formal theoryof ordinary uectors.Here is how it goes: IFrom now on we shall usethe word "vector" to mcan "generalizedvcctor," and not:r dircctcd linc segrne nt. I We define a uectorspace Lo be a collectionof objectsrp, called rJectors,such that the following things [(A) through (D)l are true: (A) "Scalar multiplication" is defined: tf g is any vcctor in the sptrceand c is any complex number,also called a scalar,thcn crp1=r,trc), called the productof c and g, is also a vector in the space. LECTURE2. THEMATHEMATICS OF GENERALIZED VECTORS 9

(B) "Vector addition" is defined: If rp1and Vz are any two vectors in the space, then rpl * rp2, called the sum of rp1 and p2, is also a vector in the space. And rpz*g1 is the same vector as vr+ v2. Comment: F'rom(A) and (B) we see that, for any two vectorsrp1 and 92 in the spaceand any two scalarsc1 and c2,the linear combinq,tionc{pL* czpz is a well-definedvector in the spacc. (C) There exists in the spaceunit uectors,relative to which all vectors in the space have components.The componentof the vector rprelative to the unit vectorc, called the t-componentof g, is written (e,rp),and hasthe following properties: (Cl) (e,rp)isascalar. (C2)(e, cgpl*c29fl = c1(c,rp1)+ c2(t,p). (C3)(c,c) = l. (C4) (e,c') = (t',€)*, for any two unit vectorsa ands'. Comment'"(Cl) saysthat the s-componentof any vector rpis a complexnumber. (C2) says that the c- componentofanylinearcombinationvecboristhesamelinearcombinationofther-components.(C3) saysthat the e-componentof itself is unity, and thus servesto definea unit vector. And (C4) saysthat for any two unit vectorsc and e', the c-componentof e' is equal to the complexconjug,ate of the c'- componentof e. The conjugationoperation here marks an interestingand curious departure from the theory of ordinary vectors. IBut notice that we couldhave written (c4) in the ordinary vector theory as e.ef= (e'.e)* without altering anything, sincecomplex conjugation doesn't chernge real numbers.I (D) If the vectorspace is N-dimensional,then there exisbsat leastone set of N unit vectors{tr, tz, .. . , e,rr)cal led a basis such that (Dl) (e;,e6)= 0 wheneverj*h- (D2) For any vector12 in the space, V = €r(ct,?) * t2?2,q\ + ... + e1,,(c1s,rp)= Xj"j(c;,rp). Q-2) Cornment:(Dl) stipulatesthat the unit vectorscomposing a basisbe mutually orth

t2 ( e2,qt)

tl ct (c1,t/)

FIG" 24. Schematic representation of the exparnsion(2-2) for N =2. representation of the situartion. The reason, of course, is that the components (cj,g) arre genererlly complex numbers. But, if we can't draw an accuratc picture of rp,how can we specify thal vector? .Just as with an ordinary vector v in (2-l), wc can specify a 5;eneralizcd vector g by giving its componcnts, (rr,g), (t2,tp\,... , (cN,r/), rclative to some particular basis; thc only difference is that thesc N componcnts arc now complcx numbcrs instcad of rcal numbers. In our latcr work with generalizcd vectors we shall have occasionto use two theorems. Onc is the 10 LECTURE2. THEMATHEMATICS OF GENERALIZED VECTORS

Component Expansion Theorem: If rp is any vector, e' is any unit vector, and {c;} is any basis, then the e'-component of rpcan be "expanded in the basis {c;}" according to the formu"la (t' ,pt = ErG's1(tr,tttl, (2-3) wherethe sum runs over the dimensionalityof the space. IMnemonic: Sum over the "inside" e;'s. I Proof: (t' ,p) = G',2i eikr,p)) Iexpandingg in the {c;}-basisaccording to Q-Z)I = (d ,2i c,t.) t ki,W)= c.,is somescalar I = Elci(e',q) tinvokingthe fundamentarlcomponentproperty (C2)l = IiGr,u;) ({ ,q\ : X, (c',cy)(q,utl. lt's easier,in fact recommended,thatyouremember this important proof in the ftoo-slepform: - (t' ,rp)= (e',Ei erGl,vi ErG' ,er)Gi,ut\. The other theoremwe shall needlater on is the GeneralizedPythagoreanTheorem: If c'is any unit vector and {c;} is any basis,then the sum of the squaremoduli of all the {c;}-componentsof s' is one;i.e., X;l(c;,e')12= 1. (2-41 aUxercise2-1" Prove the generalizedPythagorean theorem. IHint: First apply the component expansiontheorem with rpreplaced by e'. Then make useof properties(C3) and (C4).I The nameof this secondtheorem comes from the fact that it gcneralizesthe following familiar result: For any two-dimensionalreal unit vectore', as shownin Fig. 2-5, (e1'e')2* (s2's')2- 12. But noticein (2-4)that we sum, not the squaresof the componentsof e', but their .squcremoduli.

{ I (ez.e') I lla *l I J- (e1"e'l ' FIC.2-5. A two-dimensionalunit vectore and a hasis {e1,e2}.

)2.2 Operators, Linearity, Eigenvectors and liigenvalues, Eigen bases In calculusyou learnedthat a function f transforms a number r into a new number, flr). Similarly, an operator O transforms a vector g into a new vector, written 0g. In the piciu.e terminologyof ordinary vectors,we can looselythink of 0 as generally "stretching" and "roltrting" the vectorg into a new vector0g. Ifthe actionofO on a particular vector@ is a pure "stretch" by a scalarfactor o, oQ = orb, (2.5) then we say that @is an e.ige.nuecrorof o, and o is the corresponding eige.nualue. o0xercise2'2. lf O is the rule "multiply the given vector by the scalar (3+i4)," find all the eigcnvectorsand eigenvalues of O. LECTURE2, THEMATHEMATICS OF GENERALIZED VECTORS 11

oExercise2-3. lf O is the rule "add to the given vector the vector(i2)P," wherep is some vector, showthatp is an eigenvectorof0, and find the correspondingeigenvalue. In quantum mechanics,we will only have to deal with operatorsO that have the following two specialproperties: (a) O is lineq,r.This meansthat, for any two vectors91 and q2andany two scalarsc1 and c2, 0(c1rp1I czrpil= crOqr I c2Op2. (2-6) (b) O hasan eigenbasis.This meansthut there is somebasis {@1, Qz, ...}, each vectorof which is an eigenvectorof O. We shall denotethe eigenvaluecorresponding to eigenvectortpi by or. We call the basis{@1, ,bz, ...1 the eigenbasisof 0, and the scalars{or, o2,... } the eigenuuluesei of O. Thus we have (Qi'Q) = t forallJ' (2-7a) @i"bi = 0 for allj*h' (2-7b) V = ELQt (Qj,rp)flor any vector rp, Q-7cl as with any basis,but additionally, Orbj= o;@;for allT' (2-7d) To deftnean operatorO, we must specifyhow it produces,for any given vector g, the vector Orp. Of the many ways in which we might do that, one especiallysimple way invokes the eigenbasisand eigenvalueset of 0. To show that a knowledgeof the eigenbasisand eigenvalueset of O sufhcesto determine Og for any given rp- and hencesuffices to define O - we reasonas follows: OW = OE.lQi(Q1,rp) [expandrp in the eigenbasisof 0] = Eroqi(qr,,p) linvoke the linearity of Ol = 2roiQlkbj,w) Iinvokethe eigenvectorconditionl Op = EiQ.1told7,dl. (2-8a) Equation(2-8a) tells us that the @7-componentof vector Orp is or(@;,rp);i.e., (Qi,Oql = or(Qi,tp). (2-8b) Thus, given the @7-componentof any vector ?, we can obtain the @;-componentof vectorOp by simply multiplying the former by the correspondingeigenvalue oi, as illustratcd in Fig. 2-6.

orp

oAQn,vl

FIG.2-6. illustrating schematicallyhow to constructvector Og for any given vectorg by usingthc cigenbasis{@1, Qz, ...1and eigenvalue set {ot, o2,... } of O.

oExercise2-4. Deriveequation (2-8b) directly by first substituting (2-7c)on the left (first change the summationindex to ft),and then applying (2-6),Q-7d), iCZ),(2-7b) and (2-7a),in that order. 12 LECTURE2. THEMATHEMATICS OF GENERALIZED VECTORS

)2.3 Summary

The table belowsummarizes generalized vector theory as we shall require it for our development of quantum mechanics. Beforethe next lecture, you should do three things: First, make ,r.u ynu understandeverything in the summary table. Second,work Exercises2-1 through 2-4; if you tu., master them, then you'll be able to handle all the mathematicsin the remaining lectures" i-inally, read again Sec. l"l to remind yourself of the FPOM; in the next lecture, we'll use the theory of generalizedvectors to showhow quantum mechanicsproposes to answerthe FPOM for l=0.

TABLE SUMMARIZINGGENERALIZED VECTORTHEORY

name item comment

A uector'. v A "very abstratct"arrow. A scalq.r: c Any compler number.

A Iinear combinartion: ctlpt + cZV)2 A vector in the space.

The component of q A scalar, relative to the (t,q) i.e.,a camplexnumber. unil uectore: The "c-componentof ?." Key (e,c1g1 * czrpil = c1(e,rp1) + czG,Vz) Usedagain and argain. properties (c,c) = I Definesar unit vector c. thereof: (e,e')= (e',c)* The curiousconjugation.

Gi,t) = I forallT Unit vectors, I A 6asis{c1, e2, ... }: Gr,til = 0 for all j- lc mutually orthogonal. = (er,g) v Ersj for all g The "{c;fexpansionof V." ',tp) 3 Component expansion thm.: (c = 2.,{t',t1 Gi,W) Sum over "inside" cy's.

3 Pythagorean theorem: Xyl(c;,c')12 = 1 Forany unit vectorc'.

An operator: o Transformsg into Og.

Linearity of O: O(c1q1* czyzl = crOqr * c2Og2 Used again and ar.gain.

2The eigenbasis{tpi} {Qr,Q2,...}isabasis Seeabove. and,eigenualue set {orl Ot, o2,...are SCalarS I.e.,complex numbers. of O: AQj = olQi for alli The eigenvectorcondition,

3 Effect of 0 on any V: Qti,Og) = oikpi,tpl SceF'ig.2-6.

Notes: I A.ty vector g in N dimensionalspace cun be specifiedby N complexnumbers, namely, the N componentsof y,k1,q),Gz,V),"..,(e61,rp), relartive to somebasis {c1, sz, ... , rN}. 2 n ny linear opcrator O defincdon an N dimensionalvcctor spacccan be specifiedby spccifying its cigenbasis{dt, Q2,...,rp,r,'} and eigenvalue set {o1, o2,... ,oru|. 3 A "theorcm,"derivablc from the preccdingdefinitions and properties. 13

LECTURE 3. QUANTUM MECHANICS: STATICS I

;3. I States and Observables, Vectors and Operators In our first lecture, we said that "mechanics" deals with systemsand their observables. If A is an obseruable (say, position) of a certain system (say, a ball), then we can at any time measure A on the system and get a result A (say, 4.7), which we call Lhe ualue of A. (So, the measured value of the ball's position is 4.?.) We also noted that the chief concern of mechanics is to.answer the following general kind of question (the FPOM): If we measure an observable A on a system with result A, and then a time I later measure observable B on the system, what can we predict about the result R of the second measurement? The answer to this question proffered by classical mechqnics is generally framed in terms of the system's "classical state." F'or example, supposeour system is a simple particle of mass nr that movcs on the r-axis in a potential cncrgy field V(r) [or equivalently, in a force field l'(r) = -V'(xll. Classic.rl mechanics defines the stq.teof this system at time J to be the pair of variables [x(t),p(t)I, the particle's position and momentum at time l. This definition of starteis motivated by two considerations: F irst, u knowledge of the particle's instantaneous state [rft),pft)l allows us to calculate the instantaneous values of all other meaningful observables,such as the particle's velocity p(t\/m, or its energy p2(tllzm+V(x(t)). And second,a knowledgeof the particle's initial state [r(0),p(0)l provides precisely the information needed(two integration constants) to uniquely solve Newton's equation F= ma, md2xldtz - _v,(r), (3-t) for the particle's state [r(f),p(t)=mdx(tl/dll at any time l)0. In general, then, the answer to the I.'POM from the classical vicwpoint hinges on how well we can know the systcm's state, which classical mechanics deftnes to be the valucs of some fixcd set of key observarblcs. We notcd in our first lccture that this classical approach to the I"POM works vcry wcll for macroscale systems, but not tor microscale systems. We explained that the [ailure of classical mechanics on the microscale is quite prof

> Rule l: Corresponding to arnyisolatcd physical system there is a gcneralizcd vcctor spacc.'['hc unit uectors in this space "represent" the possible physical stqtes of the system. The particul;rr unit vcctor rcprescnting the systcm state at time I is written rltl, and is c.rllcd Lhc stule uector ot thc system at time l; we say that the systern is "in t he state tPr" at time t.

) Rule 2: Flachsystcm obseruahleA is "rcprcscnted" by a linear operator A that has an eigenhasis {or, oz, ... } in the system's generalized vcctor space, and t real cigenvalue sct {4 1, A2, ... }; that is, thc linear opcrator A rcpresenl,ing thc observable A is such that 14 LECTURE3. QUANTUMMECHANICS: STATICS I

AoJ= Ap1 U--I,Z,... ) (3-2) where the eigenvectors{o1, o2,...} form a basisin the system'sgeneralized vector space,and whcrethe eigenvalues{A1, Az,...l;are all retrlnumbers. Now at this point, you must be uerypatient, because these rules don'tjust look etbstrerct,they look '"represents" pointless!What doesit meanto say that the vector tPr the system'ssttrte, or the operator A "represents"the observableA? How shouldwe go aboutfinding either that vectoror that operator? Why shouldwe even want Lo?In truth, Rules I and 2 are just getting things set up for Rules 3 and 4, which we'll write down in a minute, and which will answer most of thosevery reasonablequestions. lJut beforewe go on to thosenext rules,let's clarify someimplicationsof RulesI and 2 that will turn out later to be important. First, Rule I implies that every unit vectorin the system'sgeneralized vector'spacc corresponds to a possiblephysical state of the s-vstcm,and conversely,every possiblephysical statc of the systcm correspondsto some unit vector in thc system's generalizedvcctor space. ('lhc correspondence betweenphysical statesand unit vectorsis not quite one-toone, but we're not going to worry about that technicality here") ln writing the system'sstate vectorat time I as tltl, Rule I obviouslysuggests thaLthe stuteuector euolues with time, i.e., for two different times I and l', rll and tll' will usually be two different unit vectorsin the space. The time-dependenceof the state vector Ws is describedby Rule 5, but we won't get to Rule 5 until our Lecture6. In this and the next two lectures,we'll be concernedsolely with whertcan be said erboutthe systemat a single instant I - i.e., with the "static" featuresof quantum mcchanics. Rule2 similarly impliesthat there is an essentiallyone-to-one correspondence between physic.rl observableson thc one hand, and lincar operatorswith an eigenbasisand real eigenvalueset on the other. Of course,we miry not know, or even care, how Lomeasure all thoseobservables. I n fact, it is the high calling of the physicist arsa "diviner of Narture" to identify the physicall-vrelevant observablesof a system,and to mathematicallyconstruct their associatedoperators. Notice that, in the processof constructingopcra[or A, the physicistwill alsoconstruct, either explicitly or implicitly, the associatedbasis {crl, o2, .." }, and hencethe entire generalizedvector space of the system. Cle.rrly, this is not a trivial t.rsk! In Lecture5 we'll seehow to constructthe operatorsX and I) that correspond t

t{t=EnFp(pp,V1), wherethe componentsof W; relative to the B-eigenbasisare complexnumbers satisfying lpl{Pp,v,11z= 1- If the A-eigenbasis components of W1 happen to be known, then we can calculate from them the B- eigenbasis components of W1by using the component expansion theorem Isee Eq. (2-3t] $ p,V) = Ei (Bp,oi\(or,Vr). (3-5) Here,the coefficient (pp,o) is ofcourse the p2-component of the vector o; in theexpansion aj = Ep0t(pp,o). (3-6)

oExercise3-1.Eq.(3-6) istheexpansionoftheA-eigenvectororintheB-eigenbasis. Writedown the expansion of the B-eigenvector ppin the A-eigenbasis. oExercise3-2. Prove the following two relations between the p2-component of vector o, and the or- component ofvectorp2: (pp,o) = (oi,Fk\*, (3-7a) l(pe,o)1z= l(a,,fil12. (3-7b) oExercise3-3. What is the formula for calculatingthe A-eigenbasiscomponents of W; from the Il- eigenbasiscomponents of tF/ How are the coefficients in this formula related to lhe coefficients in formula (3-5)?

) 3.2 Results and Effects of Measurements Now that we have "set the stage"with Rules I and 2, let's "raise the curtain and start the play:" Let's addressthe crucial question of what happenswhen we measurear given obscrvableA on a system in a known state W;. The answer to that questioncomes in two parts. The first part, which we'll call Rule3, dealswith predictingthe resultof a measurement. > Rule 3: If observableA is measuredon a systemin state tP;,then the rno.sfthut can be predicted about the result of that measurementis this: The probability th'at the result will be the tlrl, eigenvalueA; is equal to the squaremodulus of the or-componentof namely l(or,W,)lz. Rule 3 implies that even though the sterteof the system is completelyspecified, in that the state vector W1is known, we can neverthelcssmake only probabillslic predictionsabout measurement results. This is in sharp contrast to the situartionin classicalmechanics, where the (classical)state uniquely determinesthe result of any observablemeasurement. Ilut in quantum mechanics,all we can say prospectivelyabout a measurementof A on the systemin starteW1 is that the probability that the resultwill beA1 is l(o1,Wr)12,the prob.rbilitythat the resultwill be42 is l(o2,tPr)12,etc. The addition law for probabilitiesimplies that we can calculate the probability that an A- measurementon stateW; will yieldany of the eigenvalues of A - i.e.,either A1 or zt2or 43 ...- by simplyadding up all the individuerlprobabilities. Thus, Prob{A1or 42 or... i = l(o1,\trr)|2+ l(az,Wr)|2+ ... = Ijl(oJ,Wr)12. But F)q.(3-4) tells us that the sum on the right is one,which in probabilitytheory means"absolute certainty."Thus,amcasuremcntof Aiscertqin toyicld someoneofthecigcnvalucsof A. [nolhcr words,the eigenvalues{A1 , Az,... } introducedin Rule 2 are the only valuesthat any measurementof A can cver yield. Now, dependingupon the observablc,eigenvarlue scts can be either discretely distributed(like the integer numbers)or continuou.slydistributed (like the real numbers). Ilistorically, the lact that many observableson the microscule(such as the energyof an electron insidean atom)turn out to havediscrele or "quantized"values, is what led to thc namc "quantum" mechanics.(Of course,you can seearlready that the differencebetween classical and quantum mcchanicsruns much deeperthat qu.rntizedmeasurcmcnt results.) [n our work here we are going to 16 LECTURE3 QUANTUMMECHANICS: STATICS I

pretend, insofar as possible,LhaL all observableeigenvalue sets are discretely distributed, because that's the easiestcase to treat mathematically. We have seen how Rule 3 illuminates the significanceof the eigenualuesof the operator A correspondingto the observableA: those eigenvaluesare just the allowed results o[ an A- measurement.Rule 3 alsosheds some light on the significanceof the eigenuector.sof A as well: It says that if we expand the system'sstate vector W1in the A-eigenbasis,as in Eq. (3-3), then the square modulusof the coefhcientof eigenvectorcry is the probability of measuring the corresponding eigenvalueAr. And this in turn implies another interesting fact: If the system'sstate vectorV1 coincideswith the eigenvectoroa, then an A-measurementon the systemis certq.into yield the eigenvalueA6. To provethis statement,we simply observcfrom llule 3 that an A-mcarsurementon the system in the starteW1=q6 will yield eigenvalueA, with prob.rbilityl(or,rP1)12=l(or,oa)|2. This probabilityis zeroif j * , owing to the orthogonalityof different eigenbasisvectors, and one if j = 17,owing to the fact that o6 is a unit vector. So we sec that Rule 3 implics th.rt the eigcnbasisvectors of A de{ine thc possible statesof the system for which an A-measurementwill harvea uniquelypredictable result (numely the correspondingeigenvalue of A). F'urtherinsight into thc significanceof the cigenbtrsisvcctors of A is providedby Rule 4. This rule deals wiLhthe effectof a meqsurementon the stqte. > Rule 4: [f a measurementof A does yield eigenvalue Ar, then immediately after Lhat measurementthc systcm'sstatc vector will coincidewith the correspondingeigenvector o1, regardlessof what the state vectorwas just beflorethe measurement. Rule 4 implics that whcn wc measureA on a system,the system'sstate vector immedi.rtely 'Jumps" to one of the eigenbasisvectors of A, namely to that eigenbasisvector correspondingto the measurcdcigcnvaluc. 'fhis is anothcr dramatic deparrturefrom classical mccharnics,where a measurement(of the ideal kind we're consideringhere) has no effect on the state. Notice that in quantum mechanicsa measurementresult tells us much more about the state of the system immediately after the measurementthan immediately befiore:If the measurementresult is Ar, then immcdiately after the measuremcntwe know that the state vectorof the systemis or; however,all we can inferabout the state immediately beforethe measurementis that its or-componentwas not zero (otherwise,we couldn'thave got the result Ar). aExercise3-4. Supposeour system'sgeneralized vector spaceis two dimensiontrl,so that the eigenbarsisof A consistsof the two orthogonalunit vectorso1 and o2. -T-q2 I t/stz

I lrr4

(a) (b)

(a) If Vt=oz, as indicatedschemertically in (ar)above, calcularl,c the probability th.rt an A- nlcasurementwill yield the varlucA1. I)itto for the valuc A2. What will the stertevector of thc systembe immediatelyafter an A-mctrsurcmcnt? (b) lf rV-fi/z)or *( t/3/21o2,as indicatedschcmatically in (b) above,calculate the probability that an A-measurementwill yield the varlueA1. I)itto for the value 42. Por each outcomc, describethe state vectorimmediately after the measurement. LECTURE3 QUANTUM MECHANICS:STATICS I 17

) 3.3 Answer to the F-I'OM for t= 0 With Rules1 through 4, you have now seenthe mostbizarre featuresof quantum mechanics;you havearrived at the "radical core"of the theory! Also with theselour rules, you are now in a position to seehow quantum mechanicsproposes to answerthe FPOM for d=0, i.e.,for no appreciabletime lapsebetween the A-measuremcntand the subsequentB-measurement. We reasonas follows: - When A is measuredon the system,the result must (by Rules2 and 3) be someA-eigenvalue; let'scall the measuredeigenvalue A1. - So immediatelyafter the A-measurement,the systemwill (by Rule 4) be in the state oj. And sinceI - 0, the systemwill stil/ be in stateo, for the B-measurement. - MeasuringB on the state c; will (by Rule 3) yield any B-eigenvalue86 with probability ltp6,o)12. ) Therefore,quantum theory's answcr to the F'POMin the /= 0 caseis: Prob{B=86,given thatA=A, and r=0} = l$n,o)12.(h=7,2, "..\ (3-8) Noticethat all we needto know to answerthe FPOM for I = 0 are: - the eigenvaluesofA and B, and - the componentsofthe A eigenvectorsrelative to the B eigenbasis. Noticealso, from Bq. (3-7b),lhat ppanday in (3-8)can be interchanged. As a simpleapplication of the result (3-8),suppose that A and B are the sameobseruable (c.rll it A). Then (3-8)implies that the probability that the secondA-mcasurement will give the result Ap when the FrrstA-measurement gave the result A; is r(op'o,)r2 r<'>rh*il =i ;trf=l l:il::::,::',?il;::;*"".' Thus, the sccondA-measurement is certainto give the same result 4, as thc first A-mcasuremcnt. More generally, the results obtainedin two rapid, successiuemeqsure.ments of any obseruable,will always agree with eoch other. So quantum mechanicsis not totally crazyt But noticc that this consistencyof immeditrteremeasurement results depends strongly on the "measurementjump" of the state vectorpostulated.by Rule 4: After the {irst A-measurement,with whatever result A;, the starte vectorrn a.sf coincide with the correspondingeigenvector o; in order for ltule 3 to guaranteethe result A; on the secondA-meersurement. oExercise.S-5.Suppose A and B are observablesof a system with a two-dimensionalstate space, and supposethe B-eigenbasisvectors arc given in tcrms of the A-eigenbasisvectors by - fr = i(l/3)U2qr+(213)U2o2and p2 (213)U2or+i(1/3)u2o2. (a) By inspectingthe aboveformulas, identify the components(or,fa) for allT and ft. Then usc Eq. (3-7a)to deduce(pp,o) for allT and lr. Using the latter numbersand t)q. (3-6),write down o1 and o2 in terms of p1 and p2. Checkyour answerby solving the abovetwo equationssimullaneously for o1and c2. (b) In the FPOM, supposethe A-mcasurementyields the result 42. What is the probability that the immediatelysubsequent B-measurement will yield the resultR1? The result82? (c) Supposethe B-meersurementgives the result Ii2, and then trn immediarteremeasurement of A is made. What is the probarbilitythat the result of this secondA-measurement will give the sarme value"A.ras obtainedin the first measuremcnt? 18

LECTURE 4. QUANTUM MECHANICS: STATICS II

14.1 Recapitulation Let's review our story so far. Classical mechanics founders on the microscale, essentially because it tries to identify the enduring "'state" of a physical system with the not-so-enduring values of certain observables of the "state" and system. Quantum mechanics attempts to salvage things by separating the notions of '"observable" using the mathematical theory of generalized vectors. Specifically, quarntum mechanics postulates, in its Rules I and 2, that to every physical systcm there corresponds an abstract generalized vector space, in which the system's stute.sare represented by certain uectors and the system's obseruablesare represented by certain operators. The vcctors representing system's states are all unil vectors. The operators representing system's observables trre all linear with eigenhases and real eigenvaluc sets. And just how are we supposed to find the operartorsthat represent specific system observables, and that in fact defrne through their eigenbases the system's abstract vector space? Th.rt is something that quantum theory leaves to the wit and rviles of the ph.vsicist, who thus ii given the challenging task of hooking up the rules of quantum mechanics to the real world. But we c4n't expect to form a meaningful physical theory by doing nothing more than erssociating states with vectors and observables with operators" Whal ties all these concepts together is the notion of measurement" Rules 3 and 4 describe what happens when we make a measurentetrt of itn observable A on a system whose state vector is W1. Those rules say, firstly, that the outcome of such a measurement must be one of the real eigenvalues iA1, A2, ... ) of the operator A arssociatedwith observableA. But which eigenvalue? Quantum theory says thab usually we can't be sure, and that the 6estwe can do is this: Expand the state vector Wsin the eigenbasistol, crz,... l of A, tUr = X"roy (oy,Vr), (-t-l ) and observe the complex component (or,tUr)of W1along the eigenbasis vector oJ;the square modulus of on thart component, namely l(ar,W1)12,is numerically cqual to the probahility thart an A-measurement the system in the startetPl will give the eigenvalue A, corresponding to eigenvector or- And while we cern'talways be sure of the result of an A-measurement, we can be sure of this: If the result i.sthe particular eigenvector Ar, then the state vector oI the sy'stcm immediately after the measurement will coincide with the corresponding eigenbasis vector crr. In other words, a measurement of A causes the 'jump" vector to to one of the eigenvectors of the observable's operittor A, namely to system's state 'l'his that eigenvector corresponding to the eigenvalue found in the measurement. measurement jump, like the measurement result itself, is inherently random and uncontrollable, and does not seem to be explainable in terms of any underlying deLerministic mechanism. That, in brief, is the gist of quantum mechanics as embodied by our Rules I through 4- At the end of our last lecture, we sh

(a)

FIG"4-1. Generalizedtwo-dimensional vector space for a hypotheticalsystem, showing in (a) the eigenbasesio1,o2) and, {fi,p21of two observableoperators A and B, and reminding us in (b)of the inherent limitation of this kind of pictorial representationof generalizedvectors.

()2,ct1)

(b)

FIG.4-2" Showingthe complexcomponents needed to answerthe I'I'OM for f =0, (a) when the result of the A-measurementis A1, and (b) when lhe result of the A-mearsurementis A2.

result of the B-measurement,we {irst calculate the complex components(fr,crt) and (p2,o1)of c1 relative to the B-eigenbasislsee Fig. 4-2a1" We then assert that the B-measurementwill yield the rcsult B1 with probabilityl(/lr,or)12, and thc resultIl2 with prob.rbilityl1Jz,o)|2. On the othcr hand,if the A-measurementhad given the result 42, then we would know that the system'sstate vectorjust belore the B-measurementwould be o2. [n that case,we would calculate the complex components (p1,o) and (p2,o) of o2 relartiveto the ll-eigenbasislsee Fig. 4-2b1,and then trssert that the B- meirsurementwill yield the result Il1 with probarbilityl(jt,oill2, and the result Il2 with probability lQz,oz\|z.That, in brief, is quantum theory'sanswer to the FPOM in the "static" case. Now let'spursue some other implicationsof the first four rulesof quantummechanics. 20 LECTURE4 QUANTUM MECHANICS:STATICS ll

14.2 Compatitrle and Incompatitrle Observables First we'regoing to discussa notion, or if you will a definition, that is very revealing of lhe non- classicalcharacter of quantum mechanics. Two obscrvablesA and B are said to be compatibleif and only if, in a rapid sequenceof three measurements- of A then of B then of A again - the results of the first and third measurementswill always agree with each other. But if, in those three measurements,it might happenthat the first and third measurementswill nol agree,then A and B are said tobe incompatible. In classicalmechanics this definition is rather useless,because there cll observablesare compatible: The intervening B-measurement,being "ideal," has no effecton the state of the system, and henceno effecton the valueof observableA. But in quantum mechanics,it is etrsyto seehow two observablesmight very well be incompatible.For example,consider a systemwith a two-dimensional state space,and supposethat the eigenbasesior,oz) and {p1,82}of the opera.torslor observtrblesA and B do nol coincidewith eachother, as is indicatedschematically in Fig. 4-3a. If the first

qz=Fz(orp1)

= plbr pz)

(a) (b)

t'IG"4-3. illustrating, for two observablesA and B with repectiveeigenbases {or,oz} and {l}1,[]21, the conditionsof (a) incompatibility,and (b) compatibility.

A-measurementyields eigenvalueA1, then the system'sstate vector will jump to state o1. 'fhe subsequentB-measurement will make the system'sstate vectorjump to either state p1 or state p2. But in neither of thosetwo statesis it guaranteedthat an A-measurementwill give the result A1 that was ohtainedin the first measurement. Of course, we might get the result A1 on the third measurement,but that is not a certainty; soby our definition, A and B areincompatible observables. On the other hand, supposeth.rt the eigenbarses{or,az} and {$1,p21ofthe A and B operators coincidewith eachother, as is indicated schematicallyin Fig. 4-3b (whether the eigenbasisindices match or not is immaterierlfor our arguments). Now what happensin our A-B-A measurement sequence?lf the first A-measurementgives the result A 1,then the systemwill jump to state o1. Since o1 is alsoan eigenbasisvector of B'soperator, then the B-measurementwill simply leauethe systemin that state;thus, the remeasurementof A will necessarilyyield the eigenvalueA1 again. Obviously,a similar argument will show that if the first meilsurementhad given the result 42, then the third measurementwould also have to give the result 42. So A and B in this case are compcttible observables" Our conclusionshere represent a general result in quantum mechanics: A necessaryand sufftcient condition for two obseruablesto be compatible is that their operators haue a common eigenbasis. If the operatorsfor A and B do nol have a common eigenbasis,then in a rapid A-B-A mcasurenlentseque nce the B-measurementalways has the potentialof "spoiling" the remcasuremenl of A. But it is very importan[ to understarndthat this spoilage,when it occurs,is nol the result o[ sloppymcasuring technique, but rather is intrinsic Lothe narturcof the measuredobservablcs. LECTURE4 QUANTUM MECHANICS:STATICS ll 21

) 4.3 Interference. The "Value" of an Observable A fascinatingproperty of incompartibleobservables is their apparrenttendency to "interfere" with each other. The most famousmar"rifestation of quantum interfcrence occurs in the double-slit experiment,which we discussedin our first leclure. Ilowever, t,hedouble-slit experiment is an exampleof "dynamic" interference,since it involves in a fundamental wa-vthe passageof time: wc shall considerthe double-slitexperiment in somedetail in Lecture 7. The general phenomenonof interferencecan be demonstratedin a static context rather easily by appealing once again to a hypotheticalsystem with a two-dimensionalstate space. SupposeA and B are two incompatible observablesfor sucha system,so that the eigenbases{o1,o2i and {51,p} of their associatedoperators A and B do not coincide,as schematizedin Fig. 4-4. Supposefurther that the system'sinstantaneous

FIG.4-4. A situation leadingto "interference"between the incompatibleobservables A and B. statevectorWsdoesnotcoincidewithanyofthosefoureigenbasisvectors.Asweknowquitewell by now,the measurementprobabilities for observablesA und B on state V1 are as follovi's: Prob{Meas(a)=A,ri= l(o;,W1)12(j--1,2)i Prob{Meas(B)=Ba}= l$p,rUSlz(h=1,2). Now, if we expandV1 in the A-eigenbasis, Wr = ql (o1,V1)* qz (oz,Wr), then we can write the p1-componentof W1in the form (pr,tl'r)= (0t,o1(o1,V1) * o2(o2,rlt;)) = (fr,or)(o1,tF1)* (p1,o2Xo2,t[1), which of courseis just the change-of-basisformula (3-5). Thereflore,the probability that a B- measurementon state tP1will yield the result fl1 can be computedas Prob{Meas(B)= Bri = l(p1,tP1)12 = l(/i1,o1Xo1,W1)* (\t,oil,oz,rV t)12. Observingthat the right side is just the squaremodulus of the sum of two complexnumbers, we need only invokethe identities( l -8) to get Prob{Meas(B)=Ilr} = l(p1,o1)12l(o1,tP,)12 + l(pyo)lzl{o2,V,;lz * 2Ret(fr,o1Xc1,W1Xp1,o2)*(o2,V1)*) (4-2)

oExercise4-1. Carry out the algcbra leadingto Eq. (4-2).

Notice in Eq. (4-2) that the first two terms on the right are strictly positive(none of the paired unit vectorsare orthogonal),while the third term may be either positiveor negativebuL not zero. 'lhe third term on thc right sideof Hrq"(4-2) is called the "interfercnce"term. 'l'hat name is partly tr holdoverfrom a waryof trying to interpret l',q.Q-2) thartwars popular in the early days of quantum rncchanics.In this (unrecommended)vicw, one rcgards A and B as'joint randomvariarbles" - i.e.,ls variablesthat can simullaneouslytakc valueswhich are predictableonly in a probabilisticscnse. I.'or 22 LECTURE4 QUANTUM MECHANICS:STATICS ll

suchvariables the eventB=Bl alwaysoccurs in conjunctionwith oneof the two mutually exclusive eventsA = At or A = 42, so the lawsof probability theory imply that - Prob{B Br} = Prob{B= Br and A: A r} + Prob{B= Bl andA= Azl. (4-3) One now tries to view Eq. @-2)as an instanceof F)q.(4-3). The difficulty is that there is no consistent way of doing that. ln particular, the most plausiblcassociation, of the {irst two terms on thc right side of Eq. (4-2)with the correspondingt,wo terms on the right side of Eq. (4-3),leaves one quite unable to accountfor the third term on the right sideof Flq.(4-2). What somepeople did was to still regard those associationsas meaningful,and then regard the "interference"term as a manifestati

Therefore, if tltl is a linear combination of two or more eigenbasis vectors of operator A, then okrservableAcq,nnot be said to have a value. [n that circumstance,it is the acl of measuringA that deuelopsan A-value;the measurementdoes this by causingthe system'sstate vecl,orto jump to one of the A-eigenbasisvectors, so that then A will have a value. But an A-value doesnot exist prior to the A-measurement. Now, in our interference problem, since it is obviously not possible for V1 to simultaneouslycoincide with an eigenvectorof both A and B, then it is not possiblefor A and B to simultaneoaslyhave values;hence, it is not possibleto regard A and B asjoint random variables,and Eq. G-3) doesnot upply. With Eq. (4-3)thus eliminated,therc is no discrepancy.C)r, if you prcfer, we have replacedthe "interference"mystery with the rnysterythat there are perfectlylegitimate system observablesthat sometimeshuue values and sometimesdo not havevalues. But then, the latter mystery was preciselythe conclusionlhat we seemeclL<>be forcecl to by the results of the double-slitexperiment. Referring to Fig. l-2, we concludedthat the results of that experiment implied that any electron reaching screen52 with 6ol[ slits yt and yz open could not plausiblybe saidto havecome through one slit erclusiueof the other; therefore,such an electroncould notbe said to have had ay-posibionvalue when it passedscrecn S1. In light of such experimental evidence,it doesnot seemterribly unreasonableto disalloruthe underlying premiseof Eq" (4-3) that incompatibleobservables will always simultaneouslyhave values. Before the next lecture you should review this and the preceding lecture, and work all the exercisestherein (the five exercisesin Lecture 3 and one in the present l,ecture 4). We will use the remaining time here to answerany questionsyou might have aboutour developmentso far. 23

LECTURE 5. QUANTUM MECHANICS: STATICS III

)5.1 The Position and Momentum Operators. Wave F-unctions In this lecture we're going to discusstwo speciftcphysical observables,the position X and the momentumP of a particle thab is constrainedto move in one physicaldimension - say along the r- axis. Our first task is to producethe operatorsX and P that physicists,in their role as "diviners of Nature," have decreedshall represent those two observables. There are several equivalent proceduresfor defining the operators X and P. Our procedurehere will be to simply specifytheir eigenbasesand eigenvaluesets; that this will completelydefine those operators lollows from the generalresultofExercise2-5[seealsoFig.2-61.Inourdefinition,youwillnoticethattheeigenvalue setsof X and P are taken to be continuouslydistributed. While that seemsvery reasonable,it will make for some mathematical complicationsfurther on - which is why we have thus flar been "pretending"that all observableeigenvalue sets were discretely distributed.

. DEFINITIONOF X AND P: - The eigenvalueset of X is the set {r} of all real numbersr, and the eigenvalueset of P is the set {p}of all real numbersp. - The X-eigenbasisvector 6. that correspondsto the eigenvalue r (X6'.=a[r), and the P- eigenbasisvector @othat correspondsto the eigenvaluep (PQp=pQr), are such thab the componentof @,relative to 6, is the complexnumber - (6r,0i 7 g-irp/k = r [cos(rplh)-i sin(xp/h)|, (5-la) wherei is Planck'sconstant in Eq. (1-2),and r is a real constantwhose value will not concern us here. [In a moreadvanced quantum mechanicscourse, you'd learn that r= (znh)-l/2.1

Noticethat we havecleverly specifiedthe two eigenbases{6'.} and {{lp}bV giving their components. relutiueto eq,chother; indeed,it followsfrom Bqs.(5- la) and ( l - 11b) that the componentof 6'.relative to @omust be given by (Qp,6r)= (6r,Qp\*= rerixp/h: rlcoskp/h)*isin(rpli)1. (5-I b) Eqs.(5-1) make ib rather plain that the eigenbases{6ri and {@r}do not coincidcwith eachother; thus, we can expectX and P to be incompatibleobservables. We will explore thartissue in more deteril.r little later. In allowing the eigenvaluesets of X and P to be continuouslydistributed, we have evidently createda generalizedvector spacethat has as many dimensionsas there are real numbers;because, for eachreal numberr there is a distinct basisvcctor dr, and similarly, for eachreal numberp there is a distinct basis vector ry'o.This "heavy" kind of infinite-dimensionalitygives rise to some thorny mathematicalcomplications, the most bizarreof which is this: Although the positionand momentum eigenbasisvectors are mulually orthogonalin the usualsense that (6;,6;,)= 0 if x*r' and (Qp,Qp')= 0 if p*p', (5-2a) it turns out that the "self-components"of thoseeigenbasis vectors, instead of being unity, areinfinite: (6',6') = - and (Qp,Qi = -. (5-2b) But we hastento add that all other unit vectorsin the spaceare assumedlo have self-componentsof one- i.e.,(e,e) = l. Somewhatmore reasonably,the expansionformulas for the state vector tP1in the position ancl momentumeigenbases, instead of being discretesums as in llq. (3-3),are taken L<>becontinuorr.s sums - i.g.,integrals: .r,=j-o(a.ru)-) ,u,= .u,= (b-3) L - J J. t j=L I- ,ou"rur,rr,ro*,i- ,,rorr,,,ru,ror. 24 LECTURE5 QUANTUMMECHANICS: STATICS lll

Similarly, the Pythagoreanformula (4-3)takes the "continuous-sum"forms

1 [, ) lto,,w,)12=t' [";,0-,,u,,12dx=r' t' [- 1,4.,*.,1'dp=t.' p' t'' 6-4) =,' J' J _-' J _.,'

And the change-of-basisformula (3-5)takes the continuous-sumforms

* rct ts) (ph,qt)= + (Qr,w,)= (dr,wr)= (6x,q)r)kbr),v,)dp.(5-5) L,Ou,o-xor,w,) J_*rf r,o,rrd,,w,)d.r, .l_,

Apart f.o- ti,ere mathematicaltechnicalities, a noteworthy modilication is made in our Rule 3, for predictingthe result of a measuremcnt:F'or position and momentummeasurements, Rule 3 is to be interp rete d as follo ws:

l(6'.,Vr;;z6r= probability that an X-measurement,made on the particle in the stateW;, will yield somevalue between r andr * dr, (5-6a)

lkbp,Vt\12 dp = probability that a P-measurement,made on the particle in the state W1,will yield somevalue betweenp and p * dp. (5-6b)

With these rules, Eqs. (5-4) tell us bhatthe result of an X- or P-measurementon the particle in any state W6must be somereal number between- - and a o, just as we shouldexpect. Of all the foregoingstatements, only parts (a) and (b) of our definition of the operatorsX and P contain really new information; all the other formulas are just restatementsin "continuous" eigenvaluelanguage of things we'vesaid before in "discrete"eigenvalue language. The dr-componentof the state vector rU1,namely (dr,W1),is evidently a complex number that dependson the two parametersr and t; hence,it is a complexfunction of r and l. As such, it is often written in the alternate functional form (6r,Wr) = Vx(x,t), (5-7a) and called Lheposition waue function of the system. If we know the positionwave function Fx(r,/) for all valuesof its argumentr, then we obviouslyknow all the componentsof the state vectorrelative to a basis, namely the X-eigenbasis{6r}; thus, knowing the position wave lunction is tantamount to knowing the "state" of the particle. Similarly, the rlrr-componento[ tltl, (@u,tPr),is a complexfunction of p and I that is often written Qtp,V) = Vp(p,t\, (5-7b) and called the momentum wauefunction of the system. If we know the momentum wave function Vte@,t)for all valuesof its argument p, then we obviouslyknow all the componcntsof the statc vector relative to the P-eigenbasis{dr}, so knowing the momentum wave function is c/.sotantamount to knowing the "state" of the parlicle. Wave functions are a convenientand frequently used way of representingthe stateofer particle. Since Eqs. (5-4) through (5-6) all involve the 6r- and @r-componentsof the state vector, we can rewrite all of thoseequations in terms of the wave fiunctioni. Le.t'sdo that now. l3eginningwith Eq" (5-6),we seethat the square moduli of the positionand momentum wave funcbionsharve the special significance that

ltttsak,t)12dr = probabilityth.rt an X-measurement,made on the particlein the state V1,will yield somevalue between r and -r* dr, (5-8a)

l'lte(p,t1lz4p = probability that a P-measurcment,made on the particle in the stateW1, will yield somevaluc bctween p andp*dp. (5-8b) LECTURE5. QUANTUM MECHANICS:STATICS lll 25

And Dqs.(5-4) tell us that thesesquare moduli also satisfy

= t and = 1. (5-9) [* *lr*o,rlzdr l* _lrr,ro,rl2dp Fig" 5-t showsschematic plots of the square moduli of the position and momentum wave functions. Eqs.(5-8) imply that the shadedareas in thosefigures are respectivclyequal to the probarbilitiesfor X- and P-measurementson the particle in the state W1to yield results in the indicatedintervals. And F)qs.(5-9) imply that the totarlarea under eachcurve is one.

lr4y(x,t)12 lVp(p,t)12

xl x,z pr p2

p'IG.5-1. Schematic plots of the square moduli of the position and momentum wave functions. The shaded areas equal the probabilities for X- and P-measurements on the particle in the state W1to give results in the indicated intervals. The total area under each curve is unity.

And frnally, ib follows frorn the change-of-basis formulas (5-5) and our lundamental Eqs. (5-l) th.rt either of the two wave functions can be calculated from the other through the formulas

v = . j, Plhv dx, (5-10a) ,,@,t) I "'' *{x,t\

v*k,t)= ,l:".-i'dh wr{p,t)dp. (5-l0b)

oExercise5-l. DeriveF)qs. (5-10).

Now let us deducesome of the interestingphysical consequences of this rather heavyformalism.

> 5.2 The Wave-Particle Duality In developingthe consequencesof our definition of thc position and momentum operators,we don't want to confusethe particle that is our system with the particle attribute of bcing "localizedat somepoint." So let'ssuppose that our systemparticle in an electronon the r-axis. Then the foregoing tonsiderationscnable us to make the followingdeductions: Deductionl.Iftheelectronhasapositionr',thenitspositionwavcfuncti<>n.Psq(r,t)iszeroforall r exceptx= x' , whercit hasan infinitespike. Proof'.If the electronhas:r positionr', then by definition its state vector tU1coincides with the X- 'l'he eigenbasisvector 6'r'. p

Vxk,t) = (6r,tltr)= (6.r,6;'), which LryEqs. (5-2) is zero if r=r'and inlinite if r=r', preciselyas claimed. Notice that this result is consistentwith the probability interpretationof the squaremodulus of Wy(x,t)in (5-8a), since it implies that a position measurementon an electron with position r' cannot yield any valueother than r'. Deduction2. If the electronhas a momentump', then its momentumwave function Vp(p,t)is zero for all p exceptp= p' , where it hasan infinite-spike. oExercise.5-2.Prove Deduction2. IHint: Repeatthe proofof Deduction1, exceptinterchange the rolesof positionand momentum.l Deduction3. If the electronhas a momentump, then in some sense it also has a position wauelength Lx = Zrchlp' (5-l l) Proof: If lhe electronhas a momentump, then its state vector tP1coincides with the P-eigenbasis vector@p. tts positionwave function is then VxG,t) = (6r,tUr)= (6r,0p)- ys-ixpth = r[cos(rplhl-isin(:"plh)|. Now since cos[(r* 2nh/p)plhl = cosfxp/h*2nl - cos[rplhl, and similarly for the sine function,it followsthat VxG*Znh/p,t) = wxG,t). Thus, the positionwave function of an electronwith momentump is periodicin r, with period or wavelength 2rchlp. Since this function completelydefines the state of the electron, then the electrontoo must, in someobscure sense, "have a wavelength"2rcNp. Notice that Deduction3 providessome glimmer of insight into the double-slit experiment discussedin l,ecture 1, wherein an electron with momentum p6 exhibited an interflerencepattern characteristicof a wave with wavelength2nh/ps. We shall give a fuller accountof the double slit experimentin Lecture7. aExercise5-3. Show th.rt if the electron has a position r, then in some sense it also htrs a momentumwauelength \p=2nh/x. Itlint: Repeatthe proof of Deduction3, except intercharnge the rolesof positionand momentum.I Deduction4. If the electronhas a momentump, then a positionmeasurement will yield,any ualue with equal probability; consequently,it makes absolutelyno senseto ascribea position value to an electronthat "has a momentum." Proof: If the electronhas a momentump, then its state vector V1 coincideswith the P-eigenbasis vector 0p, so its position wave function is Wx&,t) = (6r,{t) = (6x,0p\- 7s-ixp/ll. Then accordingto Eq. (5-8a), the probability that a position measurementwill yield a result betweenrandr*dris l{txk,t)|z dx = r2le-ixtthlz d,x= r2 d,x. where we have invoked Eq. (l-llc). Since this probability is independentof r, then we must concludethat all r-valuesare equallylikely. Deduction5. If the electronhas a positionr, then a momentummeasurement will yield any ualue with equal probabilily; consequently,it makesabsolutely no senseto ascribea momentum value to an electronthat "has a position." oCxercise54. ProveDeduction 5. Ilint: Modify the proofof Deduction4.1 We htrd already inferrcd from our fundamentalpremise (5-1) that positionand momentum were incompatibleobservables, and DeducLions4 and 5 evidently conlirm that inferencein the strongest LECTURE5 QUANTUMMECHANICS: STATICS tll 27

possibleterms. Apparently,when we measurethe positionof an electron,we forcethe electron'sstate vectorinto one of the X-eigenbasisvectors 6r; there the electroncan be said to have a positionvnlue -t but not a momentum value, and the electron'sspatial locq.lizationcauses us to think of it trs a "particle."tsutifwethenmeasurethemomentumofthatsameelectron.weforceitsstatevectorinto one of the P-eigenbasisvectors r1u; there the electroncannot be saiclto have a position value. tlut it canbesaidtohaveamomentumvaluep,andhencealsoaspatiolwauelengthirth/p,sowecarnthink of the electronas a "wave." That, in essence,is how quantum mechanicsexplains, or at least accommodates,the wave-particleduality of Nature.

> 5.3 The Heisenberg Uncertainty Principle We shall concludeour sojourn into quantum statics by saying somethingabout the celebratcd 'Ihe HeisenbergUncertainty Principle, which you have no doubt heard of" HeisenbergUnccrtainty Principleis a purely mathematicalconsequence of F)qs"(5-10); howcvcr, its actualdcrivation would take more time than we haveavailable here. We shall simply contentourselves with an explanation of what it says. [Jut beforewe begin, here is a suggestion: Forget anything you thinh you already know about the [[cisenbergUncertainty Principle - just pretendyou're hearing about it for the first time right now. Supposewe havea particleon the r-axis in somestate V1. As we have seen,the componentsof W1 relative to the X-eigenbasis{dr} are given by the (complex)values of the position wave function tP1 Vy(x,t), while the componentsof relative to the P-eigenbasis{r1ru} are given by the (complex) valuesofthemomcntumwavefunctionVp(p,t). Wcharvealsoseenthtrtthe $quaremoduliof those complex functions have special import for predicting the rcsults of position and momcntum measurements.Specifically, as illustrated in lrig. 5-1, the area under the l!zxtr,t)12-uersus-rcurve bctweenr1 and .rzis numerically equal to the probability that a positionmeasurcment on thc particlc in state W1will yield ervalue be[ween11 and 12;similarly,the area under thel.!p(p,tll2-versus-pcurve betweenp1 and p2 is numerically equal to the probability that a momentum measurementon the particlein startet[1will yielda valuebetween p1 and p2. Now, it is possibleto mathematicallyclefine for any lVtxk,t\|2curve a genericquantity dx that measuresthe "width" or "sprcard"or "fatness" of that curve. The magnitude of 41 will therefore characterizethe uncertainty we would have to contendwith in trying to predict the result of a position measurementon the particle in state W1.And the samemathematical delinition givesfor anylVp(p,t)12curve a quantity dp that measuresits width, and hencethe amount of uncertainty we would have to contendwith in trying to prcdict the result of a momentum measurementon the particle in state W;. We show in !'ig. 5-2 the particle's position uncertuinty4x and momentumuncertainty dp for a hypotheticalstate V1.

lvyG,t)lz lvprp,tllz

F'lG"5-2. Schematic plots of the squaremoduli of the positionand momentumwave lunctions of a patrticlein somestatte V1. The Ileisenberg UncertarintyPrinciple saysthal the productsof thc (suitablydefined) widths 4x and4p can neverbe smallerLh.an h/2. 28 LECTURE5 OUANTUMMECHANICS: STATICS lll

Now, since the two wave functions 9x(r,t) and V'p(p,l)are related accordingto F)qs.(5-10), it shouldcome as no surprisethat the quantitiesdx and Ap are relatedto eachother. In fact, if we starrt with Eqs.(5-10) and go through a lengthy but purely mathematicalargument, it is possibleto provc the following inequality: Ax'Ap > hlz. (5-12) This is the [IeisenbergUncertainty Principle, at leastas it appliesto the positionand momentum of a particle. [t implies that, for ony state V1,there is a fundamentallimit on the simultaneoussmallness ofthetwouricerttrintiesdxand4p:Foragivenpositionuncertaintydx,thcn4pcanbenosmarller Lhanh/2A7a,a lower boundthart erpproaches infinity as dx--+0. Or, for a given momentum uncertainty dp, then 4x can be no smaller than hl2Ap,a lower boundthat arpproachesinfinity as dp---t0. The most difficult thing to understundabout the HeisenbergUncertainty Principle is the precise physical meaningsof the "uncertainties"4x and dp. Perhapsthe best way to clarify those rneanings is to imagine that we have 100 idcntical electrons,all in the same state t[1. Supposcwe make identical position measurcmcntson 50 of thoseclcctrons. If the squaremodulus of the position wavc function correspondingto starteW1 looks anything like that shown in l'ig. 5-2, then those electrons obviouslydo not havear position valueprior to the positionmeasurement, arnd thc rcsults obtarinedin those50 measurementswill generallyall be different. The quantity d; ch.rracterizesbhe probable '"scatter" in those50 measurementresults - i.e.,the approximateamount by which any two of those measurementresults will be foundto differ from eachother. Similarly, if on the other 50 electronswe make idcntical momen &m mcelsurements,then dp will characterizethe scattterin the resulting 50 momentum values" Noticethat d; is not a measureof our ignoranceof the "true value" of X prior to an X-measurement;rather, it is a measureof the intrinsic uncertainty as to what X-value will be "dcvclopcd"in an X-measurement.The "uncertaintics"in the HeisenbergUnccrtainty Principle obviously have a very technical definition thert cannot be tully appreciartedoutside of the logic establishcdby Rules I through 4. That's why thc Hciscnbcrg Uncertainty Principlc, which is an important and often usefulfact of modernscience, is sooften misrepresentcdoutside of physics. We have alrcady secnthc implicationsof thc HeisenbergUncertainty Principle in t,wolimiting cerses:If W1=6'., so that the elcctron"has a positionr," thenour l)eductionI impliesthat 4x = 0 while Dcduction5impliesthat4p-c.,,justasrequiredby(5-12).Andattheotherextreme,ifWt-Qp,so that the electron"has a momentump," then our Deduction2 impliesthat 4p=0 while Deduction4 impliesthat dx - -, aguinin consonancewith (5-12). But if (5-12)is true, then how is it that we can eueruse classicalmechi.lnics, which assumesthat dx and 4p are both .rlwayszero? The trnswer is that i. is such a flantasticallysmall nutnber from u macroscopicpoint of view that dx and 4p can appearto be macroscopicallyzero and sdill satisfy (5- l2). 'fhe It is only on the microscalethat the HeisenbergUncertainty Principle makes itsclf felt. followingexercise is intendedto demonstratethis point. .Exercise5-5. Sincemomentum and velocityare relatedby p= rnu,then Ap= mAy, and (5-12)can be written in the form Ax.Av > h/2m. (a) Show that a l-gram particle can have its positioncertarin to within l0-tt cm arndits velocity certainto within l0- l0 cm/sec,arnd yet still bea very longway from violating(5-12). (b)Supposean elcctron( mass =lS-27 grams)is confinedto an atom (diameterof l0-u cm). What wouldbe the minimum valucfor the velocityunccrtainty 4y? Would it bc at all mcarningful to ascribca'"velocity valuc" to an clcctron that is insidc an atom? This concludesour discussionof rluanturnstatics. In thc ncxt lccl,urowc will take a lo

Lecture 6. QUANTUM MECHANICS: DYNAMICS I

) 6"1 The Hamiltonian Operator and'fime-evolution In Rules 1 and 2, we attached a time variable / to the state vector Wi but not to the general observableoperator A. The obviousimplication is thabthe state vectorchanges with time, while all observableoperators (along with their eigenbasisvectors and eigenvalues)are constantin time. In order to answerthe FPOM for any t>0, it is necessaryto specifyprecisely how tlrl evolveswith time. This is the subjectof Rule 5, and leadsus into the "dynamics"part of quantum mcchanics. Rule 5 comesin two parts. The first part introducesan operator called the "tlamiltonian operator,"and the secondpart introducesa closelyrelated operator called the "time-evolution operator." We'll first state Rule 5 in full, and then elaborateits two parts separately. > Rule 5: (a) Every physical system has an observableH called the "total energy." The operator H that representsthis observable is called the system's llomiltonian operalor, and we denote its eigenbasisby {rln}and ibsassociated eigenvalue set by {En}: Htln = E rrln. (n = O, 1,2, ...) (6-1) (b) If the system is in somestate Vs at time 0, then providedthe system is not disturbed(such as by being measured),its state vectorat time t will be V1 = UlVs. (t>o) (6-2) Here, U1, called lhe time euolutionoperator, is the linear operator with eigenbasis {r1,,}and - associatedeigenvalue set {e ifntli}, Uren = e-iEntlh q,, = [cos(E - isin(E,rtlh)l\n, (n = 0, 1,2,...\ (6-3) "t/h\ where{r1"} and {E,r}are as definedin part (a),and i is Planck'sconstant [see Eq. ( l-2)1.

Part (a) of Rule 5 sirnply assertsthat every physicalsystem hasa [Iamiltonian operator H, which is the operatorthat representsthe systemobservable "total energy." But Rule 5 doesnot tell us how to find that operator H. As with any observable,it's up to the physicistto "discover"the Hamiltonian operatorH for eachphysical system ofinterest. In discoveringH the physicistdefines, either directly or indirectly, its eigenbasisir1,,) and eigenvalueset {8,,}; the former characterizesthe generalized vector spaceof the system, while the latter characterizesthe allowed energy values of the system. The task of f-rndingH is obviouslynon-trivial, but not so impossibleas it might seem. This is because various rules-of-thumb have been discoveredover the years that give demonstrably correct Hamiltonians for many physical systems. Those rules-of-thumbhave in fact becomean important part of quantum mechanicsas an "applied science"" But we're not going to discussthose rules here; we'rejust going to describewhat can be doneonce the Hamiltonian operatorH is in hand. Part (b) of Rule 5 says that once we know the system'sHamiltonian operator H, then we can proceedto determine the time-evolutionof the state vector" We do that by first delining a new linear operator U1,called the time-euolutionoperator,as follows: U1is to havethe sameeigenbasis {r1,,} as H, but the eigenvaluecorresponding to r1n,instead of being 8,,, is to be e-fnli, where i is Planck's constant. That this specificationprocedure completely defines the operator U1 follows from our discussionof Fig. 2-6. Notice that U1cannot be regardedas an "observableoperator," becauseits eigenvaluespectrum is not pure real. Indeed,the role of U1in our theory is quite different from the role of H or any other observableopcralor: U; actson thc time-0 state vcctor tlg and transformsit into the time-fstate vector W1. SinceRule I requiresthe system'sstate vector to always be a unit vector, then we may expectthat the action of U1on Wg will be a "pure rotation" with no "stretching;" we'll verify shortly that that is indeedthe casc. Now we'regoing to useRule 5 to derive an cxplicit formula for the time-vetryingstate vector. Wc proceedas follows: 30 Lecture6 QUANTUMMECHANICS: DYNAMICS I

Wl = UrVo tby (6-2)| = Ur[Xn qn(4r,,W6)l Iby expandingtl,o in the H-eigenbasisl = X' Ure,,(qr,,Wg) IsinceU1 is a linearoperatorl = X' e - dnli rlu(rlu,Vo) lby (6-3)l rl'r = Xn rlnle-tEntlh(rln,Vo)1. (6-4) Eq. (6-4)shows how, in principle, we can calculnte tP1from a knowledgeof W6,{q"} and {8,,}. Notice that the coefficientof the vector4u in this formula isjust the productof two complexnumbers, namel-v s-"LErtlhand (qr,,V6),and henceis itself a complexnumber. [n fact, since(6-4) is just an expansionof V1 in the eigenbasis{q"}, it follows that the coefficientof r1,,in that formultr is none other than the rln-componentof V6;i.e., Eq. (6-4)tells us that (11n,Vs)- s -'LE,rtlh(r1r,,V6). (6-5) Soto get the r1n-componentof rI1,we merely multiply the r1o-componentof V9 b-vthe complexnumbcr s-"LErtth.F'ig. 6-l illustrates this important relationshippictorially. oExercise6-1.ShowthatEqs.(6-4)and(6-5).rreidentitiesfor/=0. lllint: Recall(l-1lu).1

= UrtPo

I l r, Qn --(q,,,,Pg) +l I t I e-tE,rt/h(q,r,V6) +i I FIG. 6-l" Showing,.h"*oti.olly how the statevector evolves with lime lrom the perspectiveof the eigenbasis{q,,i of the Hamiltonian operator H.

Notice that the state vector of thc system can change in two different ways: One way is the smooth,orderly evolutionaryprocess described by Rule 5 that occurs when the systenr is left undisturbed"The other way is the sudden,random changedescribed by Rule 4 th.rt ()ccurswhen the systemis metrsured.Much effort hasgone into trying to explain the measurementchange of Rule 4 in terrnsof the evolutionarychange o[ Rule 5, hut with no real success.Perhaps that's not surprising in view of the fact that the measurcmentchange mandated by Rule 4 is rutn-deterministic,while the evolutionarychange describcd by Rulc 5 is quite dcterministic. So we apparontly must accordR,ules 4 and 5 equal status in the theory" Thoughtful physicistsstill worry about whether these five rules are logicallyconsistent; however, the fact remainsthat theserules seemto work verv well together in the laboratory. Lecture6. QUANTUM MECHANICS:DYNAMICS I 31

) 6.2 Conserved Quantities. Stationary States We'renow in a positionto answerthe FPOM for l>0. But f-rrst,let's savor the result (6-4)a bit by drawing out someof its other interestingconsequences. If we take the squaremodulus of Eq. (6-5),we get l(q",Vrlz = le-rErtth(n,,,tlro)12= le-trntthl2l(rl",.lro)12 = l(q,r,tPs)12, (6-6) where the two final stepshave invokedthe identities(1-8a) and (1-1lc). Now this result has two interestingimpl ications. First, since Rule 3 tells us that l(r1,r,l{r)12is the probability that an energy measurementon the systemin the state Vl will yield the H-eigenvalueE,r, then Eq. (6-6)implies that Prob{Meas(H) = E n at time l} is independentof t. (6-7) So energy measurenTentprobability is always conseruedin time" provided of course that the system is not measuredor otherwise disturbed. Notice that this does no, say that the value of the system's energy is always conservedin time - indeed, such a value will not exist unless V1 happens to coincidewith one of the H-eigenbasisvectors. All that (6-7)says is that the probability for an energy measurementto yield any particular result is conservedin time. A secondimplication of (6-6)follows by summing both sidesover all n: - - X,, l(rl,r,trrl2 Enl(q,.,Vo)12 t, where the last equality is due to the fact that W6 is a unit vector. In other words, the fact that V6 is a unit vector guaranteesthat W1=UlWg will also be a unit vector, as is required by Rule l. This confirms our earlier conjecturethat the effect of the time-evolutionoperator U1 on Wg is a "pure rotation" in the generalizedvector space, with no "stretching." Another interesting consequenceof Eq. (6-4)arises when the system'sinitial state Ws coincides with some H-eigenvectorr1;, so that the systemat time 0 has energy E;. Accordingto Eq. (6-4),the state vectorof the systemat time I will then be given by Wr = Xr, \,rle-tEntlh(qr,,r1r)1. Since(q,r,4;) is zerofor ntj and one for n--j, then the sum herecollapses, and we obtain the theorem -tEJtJh WO= 4.r + Vt = eJe for all t > 0. (6-8) This theorem has two important implications: [f W0=ej, and the systemremains undisturbed,then: (a) the system will have energy Erfor aII t>0;:rnd (b) the measurementprobabilitiesfor all system observableswill be conserved.We'll leavethe proofsof thesetwo factsas an exercise. .Exercise 6-2. (a) Provethe lirst assertionabove by using theorem(6-8) to showthat Prob{Meas(H\=Ejatanytimel)0, givenVs=Q;} = 1. (6-9a) IHint: Start with Rule 3.1 (b) Provethe secondassertion above by using theorem(6-8) to showthat, for any observableA, Prob{Meas(A) = At at time l, given .Io = 4,} is independentof ,. (6-9b) Itlint: Start with Rule 3, and rememberthat (e,cA)= c(e,g).I The time-dependentvector 4i e-tEtun appearingon the right side of theorem (6-8) is called a stationary stateof the system. Obviously,there is one stationary state fioreach eigenbasis vector of the system'sHamiltonian operator" What we have provedhere is that if the system starts out in a stationary state, then the system will remain in that stationarrystate with a constantenergy value and constantmeasurement probabilities for all its observables. 32 LECIUTE6, QUANTUM MECHANICS:DYNAMICS I

) 6.3 Answer to the FPOM for Any t>0 Now let us see how quarntummechanics proposes to answer the Fundamental Problem of Mechanicsfor any l>0" We reasonas follows - When A is measuredon the system,the result must (by Rules2 and 3) be someA-eigenvalue; let's call the measuredeigenvalue A;. - So immediatelyafter the A-measurement,the systemwill (by Rule 4) be in the state c;. - Starting from the state W9=o, at time 0, the state vector will then evolve smoothly until time l, whenit will belby (6-a)lthevector - rlr = Xu Qu[e ,E,rtlh(r1,r,oi)|. - Then measuringB at time I will (by Rule 3) yield any B-eigenvalueBp with probability l(\uV)12 = l(Fa,X,,r1,,[e-rnuh(qu,o)Dlz = lXn (Frr,\n)[e-Ertlh(r1u,o)112. > Therefore,quantum lheory'sanswer to the FPOM in the generalf caseis: Prob{B= B h at time l, given A = A; at time 0} = lXn(\tr,rt) e-iEt/th(qo,o;)12. (h=1,2,...) (6-10)

Notieethat all we needto know to answerthe FPOM for any I > 0 are: - the eigenvaluesof A, B,and H, and - the componentsof the A and B eigenvectorsrelative to the H-eigenbasis. Let's verify that the generalresult (6-10)indeed reduces to the static result (3-8)when l=0. Since eiO- I by (l-l la), thenthe right sideof (6-10)becomes lEn$p,ttn) eio(r1n,o,,)12= llu (\n,rtu\(qu,or)1z = l(pp,o)1z, where in the last step we have invoked the componentexpansion theorem (2-3). This is preciselythe result(3-8) for the I = 0 case. oCxercise6-3. In the FPOM, supposethe second-meersuredobservable B is the salmeas the first- measuredobservable A. Provethat Prob{A=Arattimel,givenA=Ararttime0i= lX,,l(q,,,or)12e-|E}thlz. (6-tl) Showthat this probability is unity if l=0, as required by Rules 3 and 4, but need not be unity for anyl)0. oExercise6-4. Use (6-10)to show that an energy measurementat time 0 followedby an energy measurementat any time l>0 will alwaysgive equal results,in agreementwith theorem(6-9a).

*sta,ndard" Rules I through 5 form the logical foundations of the theory of quantum mechanics, and (6-10) is the answer that thoserules giue to the Fundamental Problem of Mechanics. All the rest of quantum mechanicsconsists of: generalizing those five rules somewhat to accommodate more complieatedsituations; finding specificoperators to representspecilic system observables, especially specific Hamiltonian operators for specific systems;and, of course, evaluating (6-10) for specific problems. We shall concludethis lecture by sketchinga philosophicallyinteresting shift of perspectiveon quantum theory,along lines originally suggestedby Richard Feynman. Lecture6. OUANTUMMECHANICS: DYNAMICS I 33

)6.4 Transition Amplitudes and Virtual Processes It is possibleto look upon our fundamental result (6-10)as giving the probability that the two consecutiveprocesses "f-passage" and "B-measurement"will jointly carry the systemfrom state o, at time zerolo statepa at time /. In this view, (6-10)gives the probability that the system will "make a transition" from state o; to statep6 in bimel, with the tacit understandingthat B is actually measured at time l. In the spirit of this view, here is analternate wuy of arriving at the result (6-10): ) Rule A. Imaginethat the or-to-pptransition occurs,in somevaguely defined way, uia the energy eigenstates{r1n}, and assign to the particular uirtuql processlo;--+r1,r--+pp in time tl the transition amplitude, II ,r(a;+\n+ppin t) = UJn,rl,r)s-iE,rt/h (r1,r,or). (6-12a) [The quantity on the right side is most suggestivelyread from right-to-left. Notice that this transition amplitude is simply the product of three complexnumbers, and hence is itself just a complexnumber; indeed,it isjust the summandin (6-10).1 > Rule B. Similarly, associatewith the nel process{o;+Fn in time /} a transition amplitude Il(oi+pu in /), and postulate that this net transition amplitude is equal to the sum of all the virtual processamplitudes: [I(or-->ppinf) = Xnfl,.{or-*r1n-+pzint). (6-12b) [This net amplitude,being a sum of complexnumbers, is then itself a complexnumber"l ) Rule C. Finally, postulatethat the probability of the net process{ot-0n in time f} is equal to the squaremodulus of its amplitude: Prob(or"+p6in l) = III(or+pu in t)lz. (6-l2c)

It is easyto seethat if(6-12a) is substitutedinto (6-12b),and then that is substitutedinto (6-12c), we get preciselythe result (6-10). The foregoingdescribes what might'be called a f'virtual process" '"conventional" formulation of quantum theory. It's relation to the formulation of quantum theory, as embodiedby Rules 1 through 5, can be briefly summarizedthis way: 'Ihe virtual processformulation retainsRules 1,2 and 5(a),but replace.sRules 3,4 and 5(b)with RulesA, B and C. We know that, when considering mutually exclusive real processes,we must sum their probabilitiestoobtain the total processprobability. But for the mutually exclusiveuirtual processes contemplatedabove, the rules are evidently different: We must sum their amplitudes, and then obtain the total processprobability arsthe squaremodulus of that amplitude sum. But, you might ask, how does Nature really work? Doesthe system go from state c; to state pp by way of one,or all, of the states{r1n}? Or doesthe system'sstate vectorsmoothly rotate from state o, to state W1and then suddenlyjump to stateBp when the B-measurementis made? [n truth, no one really knows. Moreover,no one has beenable to paint a detailed,consistent, plausible picture of either the mysterious virtual transition process o'frln+Fh, or the equally mysterious measurement jump processVt-0n- So what are we to make of this state of affairs? The answer seemsto be this: AII that present day quantum theory can do is prouide a computationalalgorithm - namely (6-10) - fo, quantitatiuely answeringthe FundamentqlProblem of Mechanics.That algorithm has beenamazingly successfulin laboratory applications,not only in describing phenomenathat were already known, but also in predicting phenomenathat were subsequentlydiscovered. Ilut no one has beenable to undergird that algorithm with a common-sense"picture of reality" analogousto the one suggestedby classical mechanicsfor macroscalephenomena. The suspicionamong some is thurtmicroscopic retrlity may not be "understundable"by minds whosecriteria for understarndinghave bcenconditioncd so thoroughly by macroscaleexperiences. Seriouscontemplalion of this prospectle:rves most physicists in some linear combinationstate of dismay,skepticism, awe and fascination. 34

Lecture7. QUANTUM MECHANICS: DYNAMICS II

) 7.1 Recapitulation In our last lecturewe introducedltule 5, orrr final rule of quantum mechanics.Rule 5 comesin two parts" Part (a) assertsthat every physicalsystem has a "total energy" - an observablethat we representin the system'sgeneralized vector space by an operator H called the system'sHamiltonian. We denotethe eigenbasisof H by {qr} and the correspondingeigenvalue set by lE,rl Hrln= Euq,,. (n=1,2,...1 Q-r) Rule 5 does nol tell us how to find Lhe Hamiltonian operator for any given physical system; the "discovery"of H is up to us. In discoveringH, we also discover,either directly or indirectly, its 'lhe eigenbasis{q,,} and eigenvalueset {f),,i. former effectively defines the system'sgeneralized vectorsperce, while thc lattcr determinesthe systcm'sallowed energy valucs. What nrakes the [Iamiltonian opcrator more important than any other observablcoperator of the systemiis the fact that H determinesthe time evolutionof the system'sstate vector. Specifically,patrt (b) of Rule 5 sarys that, given that the systemis in state V6 at time 0, then if we leavethe systemarlone until time I its state vectorwill be Wr= UrrUo. (r>0) ft-2) Here, U1,called Lhetime-euolution operator, is the linear operator with eigenbasis {r1n}and eigenvalue - set {e id"elnl' Utrlu= g-lE,rtth11,r.(n=I,2,...) (7-3) So specifying H determines U;, and hence the time-evolution of the system's state vector. If in Eq. (7-2) we expand rI0 in the H-eigenbasis,and then use the fact that U1 is linear and statisfiestlqs. (7-3),we obtarin rV r = lJ tX,, q,, (q,,,rPg)= X, U1r1,,(qu,rl'9) = X,, e-|E tJ/hqr, (4,r,!P6),

tPr = X,, \t.[e-iP:ntlh(q,,,Vg)1. (7-41 This is evidently an explicit formula for the time-evolvingstarte vector in terms of the l{amiltonian's eigenbasisand eigenvalue set. With the result ft-41 iL is straightforwardto use ltules I through 4 to lormulate a generalanswer to the FundamentalProblem of Mechanics.In the FPOM, we measuresome observable A at time 0, notethe result, and then let the systemevolve to sometime I when we measuresome ohservable B. To predicttheresultoftheB-measurement,wercasonasfollows:WhenwemeasureAattimezero,we will obtain someA-eigenvalue Ay, thereby leaving the system in the correspondingA-eigenstate oy. The state vectorthen evolvesto time I accordingto F)q.(7-4) with Wo= ol. At time l, a 8-measurement will yield any B-eigenvalue86 with probability Prob{B=Bkattimet,givenA=Arattime0l =lTh,Vtt)12, (k=1,2,...) (?-5) whereB6 is the B-eigenbasisvector correspondingto B7r. Substituting Eq. (7-4),with tl,6=o;, inlo Eq. (7-5),we obtain thc key result of our theory: Prob{B= B n at time t, givenA = A, at time 0} = fXn (ja,\n) e-i9 ntth(r1u,or)12. (k = 1,2, ...) (?-6)

) 7.2 (Optional) The Schriidinger Equations 'l'his ll m:ry happen that I,)q. (7-6) is computationally inconvenient. would be lhe carseif the system's Hamiltonian operator H is specifiedin somc way other than by calling out its eigenbasisand eigenvalues,and those then turn out to be analytically so complicated that the right side of Eq. (7-6) becomcs difficult to evaluate. In such casos, it is uscful to harvean altcrnate way of linding thc timc- '[o varying state vcctor from the system's I Iamiltonian opcrator. that cnd, consider Lhetime- LECturE7-QUANTUM MECHANICS: DYNAMICS II 35

deriuq.tiueof the state vector tP1,which we define by the usual rule w'*o].-w' = ,,- = rim (ror-,w,,.. (-Ar)-rrp, * t+ Lt + t ). dt [1+o al ar_o\ / (The last expressionshows that the quantity we are taking the limit of is simply a linear combintrtion of two vectorsin our generalizedvector space.) Let's see what we get by evaluating this derivative using the formula (7-4). We herve dqttldt = (d/dl)(X,, q,,[e-d,,,/n(qu,tfls)l) tbyF)q. (7-a)l a'Iinearopera'[ion' =i;l:, ll;'fi:lJ:il11"" = (llih)DnEr\ne-iEntlh (rl,.,,Wo) [since - i= l/il - Olih)Xu "yy:;:""::;::;',lll*,, Htlne-iEntlh (rl,,,Wo) tby Eq.(?-l)l = (l/ii)H X,,r1,,[e-r,,/n (q,r,V0)l IH is a linearoperatorl = (l/ii)Ht[r lby tiq. (I-a)l Thus we havederived the differentialequation ihdV/d,t = HtPr. ft-7) whichistobesolvedforVlsubjecttotheinitialconditionWr=0=W6" Thisprescriptionforfinding the time-evolvingstate vectoris of coursefully equivalentto Eqs.(?-2) and (7-4). Eq. (7-7) is called the time-dependentSchrddinge.r e.quation, and Eq. (7-l) is called the time- independentSchrddinger equation It is important that the similarity in the names of those two equationsnot obscurethe fundamentaldifference between their roles: The time-lgdependent Schrtidingerequation (7-1) is the eigenvalue-eigenvectorequation for the total energyoperator H. As such,it definesthe energyeigenbasis {q,,} and eigenvalueset iEn i in thosecircumstances when H has beenspecified in someother way. The time-dependentSchrodinger equation ft-71, on the other hand, is the fundamentaltime-cvolution equation for the system'sstate vector" It's purposeis to determine tl['1from V6 in those circumstancesin which the computationsrequired by Oq. (7-4) happen to be inconvenienttocarryout.NoticethatifwedofindtllbysolvingEq.(7-7)insteadofbyev.rluating formula (7-4),then to computean answer to the F.POMwe would want imposethe initial condition V1=s=o7 anduse Eq" (7-5) instead of Eq.(?-6). In practicalapplications, the Schrtidingerequations are nearly always expresscdin "component form" relative to the eigenbasisof someconvenient observable operator. To derive the A-component form ofthe time-independentSchr6dinger equation (7-l), we first observethat H.1,,= H E, o, (o;,nn)= X, Ho; (oy,n,,); thus, Eq. (7-1)can be written X, Hcrr(oy,n,r)= E,{ln. Now taking the op-componentof this equationand invoking the linearrityproperty, we get X; (c6,Ho;)(o;,nn) = E,r(o1r,r1o).&=1,2, ...;n=1,2, ...) (?-1') This is the "A-componentform" of the time-independentSchriidinger equation (7-l). In it, the set of complexnumbers {(ot,rln) for h=1,2,... } "represents"the vectorq' relativeto the A-eigenbasis,and the set of complexnumbers {(a;,Hor) forT- 1,2, ...and fr= 1,2, "..} "represents"the operatorH relative to the A-eigenbasis.If H is definedby specifyingthe set of numbers{(crp,Ho;)}, bhen we can ') solveEq. (7-1 for the set of numbers{(ou,nnrt and the number En- i"e., for q,, and its associated eigenvalue.Similar manipulationsapplied lo the time-dependentSchrtrdinger equation (?-?) will showthaf its A-componentform is (optionalerercise!) ih d(eh,rV)/dt = Lt (o2,Hcrr)(o;,tP1). &= 1,2,... ) (7-7',) This is a set of couplcd, first order differcntial equations for thc A-components {(op,tPs)}of the sttrtc tP1, vector and thesc cquationscan in principlc bc solvcd if wc know the set of numbers {(o6,Hc;)i. 36 LectUTe7 QUANTUM MECHANICS:DYNAMICS II

) ?.3 The Free Particle Perhapsthe simplestof all dynamical systemsis thefree particle, a particle of massrn on which no forcesact. You know the solution to this problemoffered by classicalmechanics: The particle moves with constant velocity, say along the r-axis. More precisely, if the particle's initial position and momentum are r0 arndpg, so that its initial velocity is pglm, then its position and momentum at any later time t will be r(l) = ro * p11t/m, p(t) = po" (7-B) To see how the free particle is analyzed in quantum mechanics, we first recarllfrom Lecture 5 that in quantum mechanics we represent the observables "position" and "momentum" for any particle, 'f free or otherwise, by two linear operators X and P, which are de{ined as follows: he eigenvalue sets of both X and P are all the real numbers, and their eigenbases{6'.} .rnd {@,r},which satisfy X6,.= -t6, (-oo(.r(o), ['Qp= pQp (-n1p<*1, (7-91 are defincdby the prescription (6b0p,- 7s-ItPth. (--{r,P<-) (7-10) llut what about the observable"total energy"? What shall we take to be the Hamiltonian operator H for a lree particle? You will recall that in clqssicel mechanics the total energy of a free particle of "llandbook momentum p is given by the formula im(plm)2=p2/2m. Well, the physicist's of Tried- and-True Hamiltonian Recipes"declares thab the [{amiltonian operator for the free particle is H = (ll2mlPz. (7-lI ) Fair enough, but what do we mean by the "square" of the operator P? We mean simplv this: [f q, i5 any vector, then to find the vector p2rp we first operate on g' with P to get the vector Pq;, atndrve then operateonthatvectorwithP. SofortheoperatorHinF)q.(?-ll),thevectorHqris H,p = 0l2m)P2W = 1l2m\(P( I)qr))" With this definition, we see that the free-particle Ilarniltonian is indeed a well-defined operator: furthermore, this definition implies the following importirnt result: - For the free particle, the eigenbasis of H is the mornentum eigenbasis {{rr,}, and the eigenvalueof H correspondingto the eig,-enbasisvector rhpisp2/2m. The proof of this assertion goesas follows: F or any momentunt eigenbasis vector 4p we hatve ( 2Q = ( = ( = ( = ( 1l2mlP p ll2m)(I'( I'@p)) ll2ml(P( pQfll p/2m)(PQ i p/Zm)(pQ ). Thus, for the free particle we indeed have the eigenvalue-eigenvector relation HQp = lp2/2m)Q,r.( -o(p(co) (7-l2l o Exercise 7-l . Of the three free-particle observables"position," "momentum" and "energy," which pairs are compatible and which are incompatible?

Since for the free particle the vector rfp is an eigenbasis vector of H with eigenvttlue p2l2m, then it follows from Eq. (6-8)that -- rV (7-13) WO Qp + t = Qps-itpll2ntttlh, (r>0) the latter being a "stationary state" of the system. Therefore,if the free particlc is in state 4p at time 0, then at any later time l)0 an energyrneasuremenbon the particlewould yield the certain value p2l2m, and a momentum measurementwould yield the ccrtain value p; however, a position measurementwould yield any real number with equal prohability. a Exercise7-2. Prove the three assertionsin the last sentenceby applying Rule 3 to the state vectortl,1in (7-13).

So we seethat if we initially measurethe momentumof a free parrticle,then the purticle'smomcntum value and energy value will be sharply lixed thereafter;however, the particle cannot lhereafter be said to havea positionvalue. LectuTe7 QUANTUM MECHANICS:DYNAMICS II 37

To calculateW; for initial statesW9 other than one of the vectors{rhol, *" must resort to formula (7-4). Making the arppropri:rtesubstitutions there, namely 4,r+ Qp, En-+ P212*, E,rt I dP, we seethat the generalformula for the time-varyingstate vectorof the free particle is " *, = {q (7-r4) i _ro*-'u'"'"'n r,v, ;rdp.

Sincewe'll be especiallyinterested in positionmeasurements at time l, let's take the d'.-componentof this vectorequation. We get / f- 2.,^, (dx,rrrr)= (d,, rdu,rpn)ldr) I _.r rtu-ip'tlznh

.2.^ |' ttzntn = (6 e-tp (Q . J _- ,,Qu) o,V]dn

UsingEq. (7-10), and recallingthat (6r,W, isjust the positionwarve function Vy(x,t), this is

v = , {.q dp . (7-15) *(x,t\ I:_"-;x/h "-ipztr2"'h r,v o)

This X-basisversion of Eq. (7-14)is usefulfor predictingthe results of positionmeasurements on the free particle because,as you will recall from Lecture 5 lsee F)q.(5-8a)1, lVxk1)l2rJx gives the probabilitythat a positionmeasurement at time I will give a result betweenr and r * dr. Let's evaluate Eq. (7-15)for the particular initial state tlto=6'n,which corresponds,tothe free. particlehaving position value a at time 0. Eq.(7-15) says that tPo=6n + Vy(x,t)= G(x,t,m,a), (Z-16) wherefor reasonsthat will becomeclear shortly we haveintroduced the [unction

G(x,t,m,a)= , {o dp . ft-r7) I:-"-rrpth "-rftr2"'h 0,6o)

Surprisingly,we can evaluateG(x,t,m,a) up to an unimportant multiplicative constant without actually performingany integration! [Iere'show: Rememberingthat - (Qp,6) = (6n,rfp)* 7e+iapli, andalso that eiugiu:si(a*u), we getfrom Oq.ft-17\, = G(x,t,m,a\,rl- u*o {-f #^ . YIo,

, I im(x-a)2 = r exPl %t 38 LectuTe7, QUANTUM MECHANICS:DYNAMICS II

Changing the integration variable from p to

/ t ,'''/ m(x-alt / t \r/2 u=\z^h/ dz=(r^t)dP' lo* , )' this becomes - Iimk-a)' | 2mh \t" f -iu2 G(x,t,m,a)- r"" cxp | zht \, )l_-"'*ctu'

The n-integral here is evidentlyjust somecomplex constant; thus we concludethat

= (7-18) G(x,t,m,a)"(7)to"*o ['#1, wherc c is a complcxconstant whosc precise value will not concernus here. If we now substitutethe result (7-18)into (7-16),and then take the squaremodulus, we conclude: .Io=6n + lrVxk,t)|z= lslz(a/[). (7-19) o Exercise7-J. Showin detail how Eq"(7-19) follows lrom Eqs.(7-16) and (7-18).

The fact that the right sideof Eq. (7-19)is independentof both r and o is the key result here. It means that if the free particle has the preciseposition o at time 0, then a position measurementat any {initely later time will yield any x-valuewith equal probability. So a free particle, if localizedto one point, will immediatelyand completely"de-localize" itselfl tlolding this surprising and peculiar result in abeyancefor the moment, let's now considerwhat would happenif the intial state were Wo=6t*d-,r. That initial state correspondsto a situation in which a time-zero measurementof position would produce,with equal probability, either x,=q,or x= -q. Substitutingthis initial stateinto Eq"(7-15) gives

'lt = , u-ixdh (o dr ,(x,t) I "-ioz""'h odo+d-n)

= G(x,t,m,al+ G(x,t,m,-a), where we first expandedthe componentin the usual way and then invokedour definition (7-l7) of G. Taking the squaremodulus of this functiongives, using the identity (l-8b), lrZs4k,t1l2= | G(x,t,m,a)* G(x,t,m,- d l2 = lG(x,t,m,a\12+ lcE,t,m,-dlz * 2ReiG(r,f,m,a) G*(x,t,m,-a)1. Substitutingour result (7-18)for G into this expressionand simplifying algebraically,we conclude:. --lcl2(mt$ Wo=6o+d_a + lVxk,t)|z 2[l + cos(2max/ht)|. (7-20) aErercise 7-4. (a) Show in detail how Eq. (7-20)follows from the preccdingequation and Eq. (7-18). [Hint: Besidesthe identitiesdeveloped at the end of Lecture t, you'll needto recarllthe trigonometric . identityfor cos(u- u)"I (b) Show that the r-distancebetween an adjacentmaximum and minimum of the function in Eq.(7-20) is A(l) = rthtl2ma (7-21)

I"ig. 7-l showsplots of lhe free-particleposition probability density function ltlxk,nlz for the tw

lVyk,t)12,in unitsof lc'lz1*1s'1. llto=6o+d-n

i-i iA$ | " tlto=6t FIG.7-1. Plotofl!vx(r,r)l2forthefreeparticle,giventhetwoinitial states and Wo=6L* 6 -o. The distanced(t) is as given in Eq. (7-2r). are now going to show that these seeminglyabsurd quantum predictionssuccessfully explain the resultsof the double-slitexperiment that we describedback in Lecture 1

) 7.4 Explanation of the Double-Slit Experiment 'fhe set-up of the double-slit experiment is sketched in Fig. 7-2. An electron of mass m and, horizontal momentum p6 is incident on a vertical scrcen 51, which is opaque exccpt for two narrow slits. We measure vertical distance on screenS1 by the variabley, and we'll take slit I to be dty=q and slit 2 to be aLy= -o- A distance L beyond scrcen51 is.r secondvertical screen52, this one coated with a phosphor that enables us to record the point of impact of any electron. We'll mcasure vertical distancc on screen 52 by thc variablc z, with z = 0 on lhe samc lcvel as y = 0.

1 v

electron'simpact incident electron y=a (slit l) + (mass rn, momentum p9)

screen S1

I"lG.7-2. Geometry of the double-slitexperiment. (Seel,'ig. l-2.)

'Ihe kcy to understanding the double-slit experiment in tcrms of thc dynamics o[ a onc- dinrensionarlfree particle is this: In quantum mechanics,just as in classical rnechanics,the horizontul 40 LectuTe7. QUANTUM MECHANICS:DYNAMICS II

and uertical componentsof the motion of a free particle are independentof eqch other. Thus, any electron that makes it through scrcen S1 retains its horizontatlmomentum componentp0 until it strikes screen52. But the vertical momentumcomponent is drastically effectedby screen51; in fact, any electronpassing through screen51 with only slit I open is forcedinto the vertical positionstartc do, while any electron passing through screen S1 with both slits open is forced into the vertical positionstate 6"*d-o. Sincethe horizontalvelocity in either caseis pslm,then the electronrvill in either casereach screen Sr at a time t=Ll(polm)=mLlpo (7-22) after it leavesscreen 51. Therefore,screen 52 is essentiallymeasuring, at time t= mLlpg,the vertical positionof an electronthat was preparedat time I = 0 in the vertical positionstate 6o if only slit I was open,or in the vertical position state 6o*d-., if both slits were open. So in terms of our querntum modelof a free particle confinedto the r-axis, the y-axis on screen51 in F'ig.?-2 correspondsto the .t- axis at time J=0, while the z-axison screen52 correspondsto the r-axis nt time t= mLlpo.

r-axis at I time/= 0 I

r=A* lVtyk,ml,/ps)lz

(relative units)

FIG. 7-3. Quantum free-particlepredictions for the double-slit experiment, assuming the slit openingsare infinitesimally narrow. Curve D1 is for only slit 1 open,and curve D12is for both slits open,both as deducedfrom Fig. 7-1.

In Fig. 7-3 we showthe consequentquantum predictionsof Fig. 7-1,as given by lVya{r,t=mLlpsll2, for the electronhit probability patternson screen52, curve D1 is the predictedpattern when only slit 1 is open,and curve D12is the predictedpattern when both slits are open. Noticefrom Flqs"(?-21) and (7-22)that the distance4* betweenan adjacentmaximum and minimum of curve D12is predictedby quantum mechanicsto be nh /mLt nhL A*=A(t=mLlpo) t-l=-. (7-23) 2ma\ p, / 2op'

Now let's comparethese quantum predictionswith the experimentalresults sketchedin Fig. l-2. We immedialely noticean apparentdiscrepancy: Whereas the experimentalcurves C1 and C12dic olf in amplitude with increasingdistance from the center line, our theoreticalcurves D1 and D12have constantamplitudes" But if we were to repeat our experimentwith smqller slit opening.s,we would actuallyfind that the curvesC1 and C12would then die off moreslowly. And we would infer that in thc idealizedlimitof zero-widthslits, which we assumedfor the sakeof mathematicalsinrplicity in our quantumcalculations, curves C1 and C12would not die off at all. So,for comparisonpurposes, we may ignorethe amplitude tail-offs in curvesC1 and C12.When we do [hat, the two setsof curvesseenr LEctuTE7. QUANTUM MECHANICS:DYNAMICS II 41

to match rather well. In eachcase, the two-slit curve oscillatessmoothly between about 0 and 4, on a scalewhere I is the constantamplitude of the one-slitcurve. A cruciarltest is providedby comparingthe wavelengthsof the oscillationsof the two-slit curves. Rememberin our first lecture,we said that the experimentalcurve lookedlike the two-slit intensity interferencepatternof awq,uewithwavelength .\-Znhlpg. InFig. T-4we showtheconditiondefining

]1 Central maximum lt (iftr

screenSr screen FIG" 74. Two-slit diffractionof an ordinary wave,showing the conditionfor the locationof the hrst minimum in the intensity interferencepattern. the locationof the l-rrstminimum in suchan interferencepattern: d is such that the differencein the distancesto the two slits is )t/2, resulting in destructive interference at that point. For the experimentalcondition tr( c, we haveby similar trianglesthat (L/2)l(2a)= dlL. Solvingfor d, and then invoking the aforementionedresult \=2nh/po, we get a=4=+(r!.\ Lq no r po / 2apo

This agreesexactly with our quantum predictionfor d* in Eq. (7-23\t Our conclusion: Quantum mechanicsprouides an accurate description of the double-slit erperiment- an erperimentthat defiesa plausible descriptionin terms of classicalmechq,nics. On that optimistic note, which is echoedagain and again for the other systemson Nature's amazing microscale,we shall concludeour brief lookat the theoryof quantum mechanics. .Exercise 7-5- (a) Supposethe electron in the double-slitexperiment acquires its horizontal momentum po by being acceleratedthrough a potentialdifference V, thus acquiring a kinetic energy eV, where e is the electron'scharge" Write the formula for the distanceA* in F ig. 7-3 in terms of m, e and V. (b) If 7=10 voltsand L=10 mcters,whart should a bc in ordcr to makc d*=l millimctcr? lAnswer: About 100angstroms, which is why this experimentis so hard to do.I (c) With the valuesof V, L and o as in (b),suppose the electronwere a "macroscopic"particle, say of massl0-5 grams. Calculaled* lor that case. Wouldthe oscillationsin the two-slitcurve [e detectablein this macroscopiccase?

***