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Rotations and Angular Momentum

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Rotations and Angular Momentum rotations.nb:11/3/04::13:48:13 1 RotationsandAngularmomentum Á Intro Thematerialheremaybefoundin SakuraiChap3:1-3,(5-6),7,(9-10) MerzbacherChap11,17. Chapter11ofMerzbacherconcentratesonorbitalangularmomentum.Sakurai,andCh17ofMerzbacherfocuson angularmomentuminrelationtothegroupofrotations.Justaslinearmomentumisrelatedtothetranslationgroup, angularmomentumoperatorsaregeneratorsofrotations.Thegoalistopresentthebasicsin5lecturesfocusingon 1.J asthegeneratorofrotations. 2.RepresentationsofSO 3 3.Additionofangularmomentum+ / 4.OrbitalangularmomentumandYlm ' s 5.Tensoroperators. Á Rotations&SO(3) ü Rotationsofvectors Beginwithadiscussionofrotationsappliedtoa3-dimensionalrealvectorspace.Thevectorsaredescribedbythreereal vx numbers,e.g.v = v .ThetransposeofavectorisvT = v , v , v .Thereisaninnerproductdefinedbetweentwo ML y ]\ x y z M ] M vz ] + / T M T ] vectorsbyu ÿ v =Nv ÿ^u = uv cos f,wherefistheanglebetweenthetwovectors.Underarotationtheinnerproduct betweenanytwovectorsispreserved,i.e.thelengthofanyvectorandtheanglebetweenanytwovectorsdoesn't change.Arotationcanbedescribedbya3ä3realorthogonalmatrixRwhichoperatesonavectorbytheusualrulesof matrixmultiplication v'x vx v' = R v and v' , v' , v' = v , v , v RT ML y ]\ ML y ]\ x y z x y z M ] M ] M v'z ] M vz ] + / + / M ]]] M ]]] N ^ N ^ Topreservetheinnerproduct,itisrequirdthatRT ÿ R = 1 u'ÿ v' = uRT ÿ Rv = u1v = u ÿ v Asanexample,arotationbyfaroundthez-axis(orinthexy-plane)isgivenby rotations.nb:11/3/04::13:48:13 2 cos f -sin f 0 R f = sin f cos f 0 z ML ]\ M ] + / M 0 0 1 ] M ]]] N ^ Thesignconventionsareappropriateforarighthandedcoordinatesystem:putthethumbofrighthandalongz-axis, extendfingersalongx-axis,andcurlfingersindirectionofy-axis. z y x Thedirectionofrotationforfiscounter-clockwisewhenlookingdownfromthe+zdirection,i.e.rotatethex-axisinto they-axis.Similarlytherotationsaroundthexandyaxesare cos f 0 sin f 1 0 0 R f = 0 1 0 andR f = 0 cos f -sin f y ML ]\ x ML ]\ M ] M ] + / M -sin f 0 cos f ] + / M 0 sin f cos f ] M ]]] M ]]] N ^ N ^ Thesignofsin finRy isrelatedtothehanded-nessofthecoordinatesystemandthesenseofrotation.For 3-dimensions,itisequivalenttotalkaboutrotationsaroundthez-axis,orrotationsinthexy-plane.Inanyothernumber ofdimensions,thecorrectlanguageistotalkaboutrotationsinthexi x j -plane,wherexi definesoneofthecoordinate directionsofthevectorspace.Thus,whilefor3-dimensionsthereare3independentrotations,inN -dimensions,there willbe ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅN N-1 independentrotations. + 2 / ü Directionkets Tomakethecorrespondencetoquantumstates,justasatranslationwasdefinedbyitsactiononpositioneigenkets,a rotationaroundtheoriginalsocanactonpositioneigenketsby è x ' = R x = R x ? ? ? è whereadistinctionhasbeenmadebetweentheoperatorR,whichactsonthestate,andtherotationmatrixRwhichacts onthecoordinates.Sincetherotationsdon'tchangethelengthofthevector,itispossibletodefinespherical coordinates,r, q, f,andsphericalpositionkets, x Ø r ≈ n` ,whererdeterminestheradialposition,andn` indicates ` thedirectionfromtheorigin.Therotationsactonlyonthe ? ? n ?degreesoffreedom. è ? R r ≈ n` = r ≈ R ÿ n` + ? ?/ ? ? Directionketswillbeusedmoreextensivelyinthediscussionoforbitalangularmomentumandsphericalharmonics, butfornowtheyareusefulforillustratingthesetofrotations.Thesetofalldirectionkets n` canbevisualizedbythe surfaceofasphere,andtherotationsarethesetofallpossiblewaystoreorientthatsphere. ? rotations.nb:11/3/04::13:48:13 3 ü OrthognalgroupSO(3) Thesetofallpossiblerotationsformagroup.Considerthefourproperties:closure,identity,inverseandassociativity. Usingthepictureofrotationsasreorientationsofasphere,onecanconstructvisualizationstoillustrateeachproperty. Withgreatermathematicalrigor,thesetofallpossiblerotationsformthegroupSO(3),whereOØorthogonal,3Ø3 dimensions,andSØspecial,whichinthiscasemeansthematrixhasadeterminantof1.Therotationsaredescribedby threecontinuous,butbounded,parameters.Fromthematrixpointofview,a3ä3matrixhasninedegreesoffreedom. T Theconstraintthatthematrixisorthogonal,Ri j R j k = di k yields6conditions,i.e.threefori = k andthreefori ∫ k . Thepropertiesofagroupareobeyed: closure: ForanytwoorthogonalmatricesR1 andR2 ,theproductR3 = R1 R2 ,isalsoorthogonal.Thecombination oftworotationsisalsoarotation. identity: The3 ä3unitmatrixactsasanidentityelementforthegroup. 1 R = R 1 = R inverse: EachelementhasaninverseR-1 = RT ,RT R = RRT = 1 associativity: R1 R2 R3 = R1 R2 R3 + / + / UnliketheTranslationgroup,SO(3)isnotabelian,i.e.ingeneralR1 R2 ∫ R2 R1 . ThesignificanceoftheS-condition,Det R = 1,isthatreflectionsarenotincludedinthegroup,i.e.forthreedimensions onecannotturnaright-handedobjectintoaleft-handedobjectbydoingarotation.Ifweallowedreflections,e.g. -1 0 0 0 -1 0 ML ]\ M ] M 0 0 -1 ] M ] N ^ Then,thegroupwouldbeO(3)insteadofSO(3).O(3)iscalled"disconnected"sincenotallelementsofthegroupcan bereachedbyasuccessionofinfinitesimaltransformations.SO(3)isconnected. TherotationmatricesRarejustone"representation"ofthegroupSO(3).Fortwodifferentrepresentations,therehasto bea1 ¨1mappingoftheelementsofonerepresentationtotheother.Themappinghastopreservethecombinationlaw. ConsidertworepresentationsRandS .Labelarotationbyasubscriptwhichrepresentsthethreeparameterstodefinea rotation,andidentifyRa ¨ Sa ,IfR3 = R1 R2 ,thenwemusthaveS3 = S1 S2 topreservethecombinatinlaw. ` ü Fullsetofrotations:+n, f/ Therearetwocommonmethodsforparameterizingrotations.Thefirstistochooseanaxisforrotationandthenperform arotationbyananglebetween0andp.Theaxisofrotationcanbechosenanywhereonthesphere.Whynot0 ¨ 2 p? Thenrotationswithpolesonoppositesidesofthespherewouldberedundant.AnexplicitformforR n`, f willbegiven afterdevelopingthelanguageofinfinitesimalrotations. + / Draw your own picture showing the rotation of a sphere around an off + axis pole. The sphere represents the set of states, n, i.e. the set of direction kets. The rotation reorients the sphere. rotations.nb:11/3/04::13:48:13 4 ü Eulerangles ThesecondparameterizationistogiveEulerangles.Inthismethodonedescribeswherethe"northpole"movesto underarotation,andtheorientationofthesphereafterthepolehasbeenmoved.Thelocationofthepoleisdetermined byfirstchoosingalongitudebyrotatingaroundthez-axis,thenalatitudebyrotatingaroundthenewy-axis.Finally, theorientationofthesphereisgivenbyafinalrotationaroundthenewz-axis.Pictorially, Draw more pictures, showing the sequence of rotations to move the pole, and then reorient the sphere around the new pole. IntheEulerparameterizationtherangeofanglesis a = 0, 2 p , b = 0, p , g = 0, 2 p + / + / + / andanarbitraryrotationisgivenby R a, b, g = Rz' g Ry' b Rz a + / + / + / + / Note thatthez'andy'rotationsarenotdefinedwithrespecttotheoriginalcoordinateaxes,butratherwithrespectto wherethoseaxeshavemovedwiththereorientationofthesphere.Lateritwillbeshownthat Rz' g Ry' b Rz a = Rz a Ry b Rz g + / + / + / + / + / + / wheretheorderhasbeenreversed,butnowallrotationsareconvenientlydefinedaroundtheaxesoftheoriginal coordinatesystem. ü Equivalencyofthetwoparameterizations Thetwoparameterizationsmaynotseemequivalent,buttheyare,ascanbeseenbyapictorialmappingof n`, f to a, b, g .ObservethattherearetwowaystoproducethesamesetofEuleranglesconsistentwiththerestrictionof+ / fto +0, p . / + / This picture didn't make it into the classroom presentation. it's a bit of work Thetwotechniqueshavedifferentuses.Euleranglestendtobemoreusefulforbuildingupactualrotationmatricesina calculation.ThisisbecauseRz andRy aregenerallyfairlyeasytoconstructforarepresentation,andthematrix multiplicationisstraightforward.The n`, f notationhasadvantagesinsomeanalyticmanipulations,aswewillsee below. + / rotations.nb:11/3/04::13:48:13 5 ü J asthegeneratorofinfinitesimalrotations. Inanalogytothediscussionoftranslationsandtimeevolution,itisusefultobuildupthefiniterotationsfrom generatorsofinfinitesimalrotations.Recognizingthatweareeventuallyinterestedinaquantummechanical formulation,itisusefultodevelopthisformalisminawaythatrealizestherotationsasunitaryoperations.Forexample, aninfinitesimalrotationaroundthez-axisisgivenby Rz d = 1 - i d Jz + / whereJz isthegeneratorofinfinitesimalrotationsaroundthez-axis.SinceRisunitary(note:orthogonalmatricesare unitary),J mustbeHermitian.Inthepresentcase,toleadingorderind 1 -d 0 0 -i 0 R d = d 1 0 orJ = i 0 0 z ML ]\ z ML ]\ M ] M ] + / M 0 0 1 ] M 0 0 0 ] M ]]] M ]]] N ^ N ^ Similarly,forthisrepresentation 0 0 0 0 0 i J = 0 0 -i ,J = 0 0 0 x ML ]\ y ML ]\ M ] M ] M 0 i 0 ] M -i 0 0 ] M ]]] M ]]] N ^ N ^ Note thatsincethisisstillaclassicaldiscussionIhaven'tputinanyfactorsof. ü Commutationrelations Thegeneratorsobeythecommutationrelations Ji, J j = i ei jk Jk # ' whereei jk = 1, if i j k is an even permutation of x y z = -1, if +i j k/is an odd permutation of x+ y z / = 0, if any+ two/ of i j k are equal + / + / asausefulasideei jk elmk = dil d jm - dim d jl . Thecommutationrelationsareapropertyofthegroup,notjustaparticularrepresentation.Thecollectionofallthe commutatorrelationsforthegeneratorsissometimescalledthealgebraofthegeneratorsofthegroup,orjustthe algebraofthegroup. ü Finiterotations Forrotationsaroundaparticularaxis,itshouldbeclearthatwecanbuildupanarbitraryrotationbyasequenceof infinitesimalrotations,similartotheprocedureforbuildingupafintetranslationasthelimitingproductofalarge numberofinfinitesimals. rotations.nb:11/3/04::13:48:13 6 R f = Lim P Rz f n nض i + / - + s /1 n = Lim 1 - i Jz f n nض = e-i Jz f+ + s // Itisnowpossibletogiveaformforanarbitraryrotationinthe n`, f parameterization.Theinfinitesimalrotation ` aroundthen-axisisgivenby + / R n`, d = 1 - i d n` ÿ J + / ThereisnoconcernaboutwhichcomponentofJ isoperatedonfirst,sincetheeffectsofcommutationamongstthe
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