Ch4 - SJP QM 3220 3D 1 AngularMomentum(warmupforHatom)

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Unformatted text preview: SJP QM 3220 3D 1 AngularMomentum(warmupforHatom) Classically,angularmomentumdefinedas(fora1particlesystem) y m x O Note: definedw.r.t.anoriginofcoords. (InQM,theoperatorcorrespondingtoLxis accordingtoprescriptionofPostulate2,part3.) Classically,torquedefinedas and (rotationalversionof ) Iftheforceisradial(centralforce),then Hatom: electron (Coulombforce) protonatorigin Inamultiparticlesystem,totalaveragemomentum: isconservedforsystemisolatedfromexternaltorques. sumoverparticles Internaltorquescancauseexchangeofaveragemomentumamongparticles,but remainsconstant. Inclassicalandquantummechanics,only4thingsareconserved: energy linearmomentum angularmomentum electriccharge Page H-1 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 BacktoQM.Definevectoroperator operatorunitvector Recall Claim:foracentralforcesuchasinHatom (willshowthislater) Thisimplies (justlikeinclassicalmechanics) AngularmomentumofelectronisHatomisconstant,solongasitdoesnotabsorb oremitphoton.Throughoutpresentdiscussion,weignoreinteractionofHatom w/photons. WillshowthatforHatomorforanyatom,molecule,solidanycollectionofatoms theangularmomentumisquantizedinunitsof. canonlychangebyinteger numberof's. Claim: and Page H-2 (i,j,kcyclic: xyzor yzxor zxy) M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Toprove,needtwoveryusefulidentities: Proof: allotherterms like[y,px]=0 (Haveused I'mdroppingthe^overoperatorswhennodangerofconfusion. Since[Lx,Ly]0,cannothavesimultaneouseigenstatesof doescommutewithLz. ,i=x,y,orz =0(Notecancellations) [L2,Lz]=0=>canhavesimultaneouseigenstatesof Page H-3 However, Claim: Proof: M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 LookingforwardtoHatom: Wewillshowthat =>simultaneouseigenstatesof energyqnbr Lzqnbr L2qnbr WhenwesolvetheTISE =EfortheHatom,thenaturalcoordinatestouse willbesphericalcoordinates:r,,(notx,y,z) z x=rsincos y=rsinsin z=rcos y x Justrewriting insphericalcoordinatesisgawdawful.But separationofvariableswillgivespecialsolutions,energyeigenstates,ofform TheangularpartofthesolutionY(,)willturnouttobeeigenstatesofL2,Lzand willhaveformcompletelyindependentofthepotentialV(r). * Page H-4 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Givenonly[L2,Lz]=0and hermiteanweknowtheremustexist simultaneouseigenstatesf(whichwillturnouttobetheY(,)mentionedabove) suchthat (willberelatedtol,andwillberelatedtom) Wewillshowthatfwilldependonquantumnumbersl,m,sowewriteitasflm,and that willbedeterminedlater. NoticemaxeigenvalueofLz(=l)issmallerthansquarerootofeigenvalueof So,inQM,Lz<|L|Odd! Alsonoticel=0,m=0statehaszeroangularmomentum(L2=0,Lz=0)so,unlike Bohrmodel,canhaveelectroninstatethatis"justsittingthere"ratherthan revolvingaboutprotoninHatom. Proofofboxedformulae:(Thisprooftakes2pages!) DefineL+=Lx+iLy="raisingoperator" L=LxiLy="loweringoperator" (NoteL+=L,L=L+,A=hermiteanadjointofA) NeitherL+orLarehermitean(selfadjoint). Note =>Considerf: Page H-5 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Claim:g=L+fisaneigenfunctionofLzwitheigenvalue=(+).SoL+operator raiseseigenvalueofLzby1. Proof: ToproveLzg=(+)g,needtoshowthat[Lz,L+]=L+ Now So,operatingonfwithraisingoperatorL+raiseseigenvaluesofLZby1butkeeps eigenvalueofL2unchanged. (Similarly,LlowerseigenvalueofLzby1.) OperatingrepeatedlywithL+raiseseigenvalueofLzbyeachtime:L+(L+f)has (+2)etc. ButeigenvalueofLzcannotincreasewithoutlimitsince cannotexceed Thereisonlyonewayout.Theremustbeforagivena"topstate"ftforwhich L+ft=0. Likewise,theremustbeforagivena"bottomstate"fbforwhichLfb=0. Lz ft L L L+ L+ allwithsame= eigenvalueofL2 fb Page H-6 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 WriteLzf=mf,mchangesbyintegersonly Lzft=ft,=maxvalueofm L2ft=?WanttowriteL2intermsofL+,Lz: => (Also, => So, Repeatfor ) where=maxm,sameforallm's. minvalueofm. (tryit!) Sommin=mmaxandmchangesonlyinunitsof1. =>m=,+1,...2,1, Nintegersteps =>2=N,=N/2=>=0,1/2,1,3/2,2,5/2,... Endofproofof Page H-7 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 m 2 1 0 0 1 1 0 1 2 0 1 2 3 We'llseelaterthatthereare2flavorsofangularmomentum: 1.Orbital Ang.Mom. (integeronly) 2.Spin Ang.Mom. (integerorintegerOK) Page H-8 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 TheHatom mp>>me=> proton(nearly) stationary Hamiltonianofelectron me mp1840me TISE: specialsolutions(stationarystates). GeneralSolutiontoTDSE: SphericalCoordinateSystem: z r =(r,,) volume Normalization: Need insphericalcoordinates HardWay: Page H-9 z=rcos x=rsincos y=rsinsin M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Alsoneed9derivatives: EasierWay:Curvilinearcoordinates(SeeBoas) pathelement: Sphericalcoordinates: * InClassicalMechanics(CM),KE=p2/2m=KE= (radialmotionKE)+(angular,axialmotionKE) O Page H-10 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 SamesplittinginQM: * (Notice dependsonlyon,andnotr.) SeparationofVariables!(asusual) Seekspecialsolutionofform: Normalization:dV||2= (Convention:normalizeradial,angularpartsindividually) Plug=RYintoTISE=> Multiplythruby : =>f(r)=g(,)=constantC=(+1) Page H-11 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 HaveseparatedTISEintoradialpartf(r)=(+1),involvingV(r),andangular partg(,)=(+1)whichisindependentofV(r). =>Allproblemswithsphericallysymmetricpotential(V=V(r))haveexactlysame angularpartofsolution:Y=Y(,)called"sphericalharmonics". We'lllookatangularpartlater.Now,let'sexamine RadialSE: Changeofvariable:u(r)=rR(r) Canshowthat Notice:identicalto1DTISE: except r:0>insteadofx:>+and V(x)replacedwith same! Veff="effectivepotential" Boundaryconditions: Page H-12 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 u(r=)=0fromnormalizationdr|u|2=1 u(r=0)=0,otherwise blowsupatr=0(subtle!) Notice that energy eigenvalues given by solution to radial equation alone. Seek bound state solutions E < 0 E > 0 solutions are unbound states, scattering solutions FullsolutionofradialSEisverymessy,eventhoughitiseffectivelya1Dproblem (differentproblemforeach) Powerseriessolution(seetextfordetails).Solutionsdependon2quantum numbers:nand(foreacheffectivepotential=0,1,2,...haveasetofsolutions labeledbyindexn.) Solutions:n=1,2,3,... forgivenn =0,1,...(n1) max=(n1) n="principalquantumnumber" energyeigenvaluesdependonnonly(itturnsout) (independentof) sameasBohrmodel,agreeswithexperiment! Page H-13 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Firstfewsolutions:Rn(r) normalization"Bohrradius" NOTE: for=0(sstates),R(r=0)0=>wavefunction"touches"nucleus. for0,R(r=0)=0=>doesnottouchnucleus. 0=>electronhasangularmomentum.Sameasclassicalbehavior,particlewith nonzeroLcannotpassthruorigin CanalsoseethisinQM:for0,Veffhasinfinitebarrieratorigin=>u(r)must decaytozeroatr=0exponentially. =>exponentialdecayin aswell. Page H-14 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Backtoangularequation: Wanttosolveforthe "sphericalharmonics".Before,startedwithcommutationrelations, and,usingoperatoralgebra,solvedfortheeigenvaluesofL2,Lz.Wefound where=0,,1,3/2,... m=,+1...+ Intheprocess,wedefinedraisingandloweringoperators: (cmissomeconstant) So,ifwecanfind(foragiven)asingleeigenstate others(otherm's)byrepeatedapplicationof ,thenwecangenerateallthe . It'seasytofindthedependence;don'tneedthe businessyet. ^ Lz = (showed in HW) i ^ Y = Y = mY (and you can cancel the ) Lz i Assume Ifweassume(postulate)thatissinglevaluedthan =>m=0,1,2,...Butm=,...+ Sofororbitalangularmomentum,mustbeintegeronly:=0,1,2,...(throwout integervalues) Page H-15 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 (algebra!) * Candeduce => Checks:Plugbackin. from Solution:(unnormalized) Now,cangetother Normalizationfrom Noticecase=0 Example: Conventiononsign: Page H-16 byrepeatedapplicationof : Somewhatmessy(HW!) Fall 2008 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock SJP QM 3220 3D 1 Thesphericalharmonicsformacomplete,orthonormalset(sinceeigenfunctionsof hermiteanoperators) Anyfunctionofanglesf=f(,)canbewrittenaslinearcomboof : Likewise: =>Hatomenergyeigenstatesare n=1,2,...;=0,1...(n1);m=...+ Arbitrary(bound)stateis (c'sareanycomplexconstants) energyofstate(n,,m)dependsonlyonn. En=constant/n2(states,mwithsamenaredegenerate) = 0 n= 4 3 2 1 4 s 3 s 2 s 1 s (1) 1 4p (3) 3p 2p 2 4d 3d (5) 3 4f (7) Degeneracyofnthlevelis n2 (2n2ifyouincludespin) Page H-17 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 RadialProbabilityDensity Prob(findparticleindVabout If=0,=(r)then Prob(findinr r+dr)= P(r)=radialprobabilitydensity Groundstate: NoticeP(r)verydifferentfrom(r): If0,=(r,,)=R(r)Y(,),then "solidangle" Prob(findinr r+dr)=r2|R|2dr P(r)=r2|R|2 evenif0 Note: if Hatomandemission/absorptionofradiation: IfHatomisinexcitedstate(n=2,=1,m=0)thenitisinenergyeigenstate= stationarystate.Ifatomisisolated,thenatomshouldremaininstate210forever, sincestationarystatehassimpletimedependence: Page H-18 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock )= Fall 2008 SJP QM 3220 3D 1 But,experimentally,wefindthatHatomemitsphotonanddeexcites:210>100 in107s>109s E 2p E=hf=E Thereasonthattheatomdoesnotremaininstationarystateisthatitisnottruly isolated.TheatomfeelsafluctuatingEMfielddueto"vacuumfluctuations". QuantumElectrodynamicsisarelativistictheoryoftheQMinteractionofmatter andlight.Itpredictsthatthe"vacuum"isnot"empty"or"nothing"aspreviously supposed,butisinsteadaseethingfoamofvirtualphotonsandotherparticles. ThesevacuumfluctuationsinteractwiththeelectronintheHatomandslightlyalter thepotentialV(r).Soeigenstatesofthecoulombpotentialarenoteigenstatesofthe actualpotential:Vcoulomb+Vvacuum Photonspossessanintrinsicangularmomentum(spin)of1,meaning Sowhenanatomabsorbsoremitsasinglephoton,itsangularmomentummust changeby1,byConservationofAngularMomentum,sotheorbitalangular momentumquantumnumbermustchangeby1. "SelectionRule":=1inanyprocessinvolvingemissionorabsorptionof1 photon=>allowedtransitionsare: s 5 4 3 2 p d 1s IfanHatomisinstate2s(n=2,=0)thenitcannotdeexcitetogroundstateby emissionofaphoton.(sincethiswouldviolatetheselectionrule).Itcanonlylose itsenergy(deexcite)bycollisionwithanotheratomorviaarare2photonprocess. Page H-19 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 n=1 SJP QM 3220 3D 1 MatrixFormulationofQM completeorthonormalset c1 c2 c = cn n , {c n } = 3 n c n 1 0 0 u = 1 u1 = 2 0 0 Ifket'sarerepresentedbycolumnvectors,thenbra'sarerepresentedbythe transposeconjugateofcolumn=row,complexconjugate. Operatorscanberepresentedbymatrices: nohatonmatrixelement where{|n>}issomecompleteorthonormalset. Page H-20 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Whyisthat?Wheredoesthatmatrixcomefrom? Considertheoperator and2statevectors Inbasis{|n>}, = cn n = n relatedby () n n n cn = dn n n = n n n dn Nowprojectequationonto|m>byactingwithbra: ^ ^ m = m A = cn m A n n dn = Amn c n n But,thisissimplytheruleformultiplicationofmatrixcolumn. d1 A11 A12 ... c1 d2 = A21 A22 c 2 d3 Sothereyouhaveit,that'swhytheoperatorisdefinedasthismatrix,inthisbasis! Now,suppose areenergyeigenstates,then Amatrixoperator isdiagonalwhenrepresentedinthebasisofitsown eigenstates,andthediagonalelementsaretheeigenvalues. Noticethatingeneraloperatorsdon'tcommute .SamegoesforMatrix Multiplication:A BB A Page H-21 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Claim:Thematrixofahermitianoperatorisequaltoitstransposeconjugate: Proof: ^ ^ ^ * m An = Am n = n Am * Amn = Anm Similarly,adjoint(or"Hermitianconjugate") Proof: * ^ ^ ^ Am n = m At n = n Am Ofcourse,it'sdifficulttodocalculationsifthematricesandcolumnsareinfinite dimensional.ButthereareHilbertsubspacesthatarefinitedimensional.For instance,intheHatom,thefullspaceofboundstatesisspannedbythefullset{n,, m}(=|nm>).Thesubset{n=2,=1,m=+1,0,1}formsavectorspacecalleda subspace. Subspace?InordinaryEuclideanspace,anyplaneisasubspaceofthefullvolume. Ifweconsiderjustthexycomponentsofavector ,thenwehavea perfectlyvalid2Dvectorspace,eventhoughthe"true"vectoris3D. Likewise,inHilbertspace,wecanrestrictourattentiontoasubspacespannedbya smallnumberofbasisstates. Example:Hatomsubspace{n=2,=1,m=+1,0,1} Basisstatesare (candropn=2,=1inlabelsincetheyare fixed.) ^ Lz m = m m ( = 1) ^ L m = ( + 1) m = 2 2 m (for all m) 2 2 ( Lz ) mn +1 0 0 ^ = m Lz n = 0 0 0 0 0 -1 Page H-22 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 1 0 0 ^ L mn = m L n = 2 0 1 0 0 0 1 (WhataboutLx?Ly?) Beforeseeingwhatallthismatrixstuffisgoodfor,let'sexaminespinbecauseit's veryimportantphysicallyandbecauseitwillleadto2DHilbertspacewithsimple2 x2matrices. ReviewofDiracBraKetNotation bracketorinnerproduct: 2 2 Whichintegralyoudodependsontheconfigurationspaceofproblem. Keydefiningpropertiesofbracket: <f|g>*=<g|f> c=constant <f|cg>=c<f|g>,<cf|g>=c*<f|g> <|(b|>+c|>)=b<|>+c<|> Diracproclaims:<g|f>=<g|nextto|f> bracket="bra"and"ket" Ket|f>representsvectorinHspace(HilbertSpace) "ket""wavefunction" Bothand(x)describesamestate,but|>ismoregeneral: Different "representations" ofsame H=spacevector |> Page H-23 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 positionrepresentation,momentumrep,energyrep. Page H-24 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Whatisa"bra"?<g|isanewkindofmathematicalobject,calleda"functional". insertstatefunctionhere input output function: number number operator: function function functional: function numbe <g|wantstobindwith|f>toproduceinnerproduct<g|f> Foreveryket|f>thereisacorrespondingbra<f|.Likethekets,thebra'sforma vectorspace. |cf> <cf|=c*<f| (?) |f+g> <f+g|=*<f|+*<g| <f+g|h>=*<f|h>+*<g|h> Complexnumberbra=anotherbra =>bra'sform anylinearcomboofbra's=anotherbra vectorspace Thevectorspaceofbrasiscalleda"dualspace".It'sthedualoftheketvectorspace. Def'nofA isaket.Whatisthecorrespondingbra? Definition:hermiteanconjugateoradjoint ^ Af g f At g forallf,g. (If ishermiteanorselfadjoint.) Someproperties: Proof : ^ g At f ^ t ^ f At g = At f g = * * ^ ^ = Ag f = f Af ( ) Page H-25 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Theadjointofanoperatorisanalogoustocomplexconjugateofacomplexnumber: The"ketbra"|f><g|isanoperator.Itturnsaket(function)intoanotherket (function): ( f g ) h = f gh ProjectionOperators (x) = c n un (x) = un un (x) n n = cn n = n n = n n n n n ="projectionoperator" picksoutportionofvector|>thatliesalong|n> ^ Pn = n n = c n n u2=|2> | > |2><2|> Page H-26 => "Completenessrelation" (discretespectrumcase) u1=|1> u1<u1|>=|1><1|> M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 ^ ^ ^ = n n like R = x ( x R) + y ( y R) n ^ ^ 1 = n n like 1 = x ( x _ ) ^ ^ + y( y _ ) Anywherethereisaverticalbarinthebracket,oraketorabra,wecanreplacethe barwith Example: => Ifeigenvaluespectrumiscontinuous(asfor thansum,overstates. CompletenessRelation (continuousspectrum) Example: ( p) = f p = )thenmustuseintegral,rather n dx fp x x = TheMeasurementPostulates3and4canberestatedintermsoftheprojection operator: Startingwithstate , wheresum{n}isoveranycompletesetofstates,ifwemeasureobservable associatedwithn,thenwewillfindvaluen0withprobability Probabilityoffindingeigenvaluen0=expectationvalueofprojectionoperator Andasresultofmeasurementstate|>collapsestostate . 1 2 dx e -ipx (x) . (apartfromnormalization) Page H-27 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Wecannowgeneralizetocaseofstatesdescribedbymorethanoneeigenvalue, suchasHatom. Ifwemeasureenergy(butnotalso ),findn0,thenweareprojectingonto subspacespannedby{,m}withsomen0. Statecollapsesto mustrenormalize Page H-28 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Spin RecallthatintheHatomsolution,weshowedthatthefactthatthewavefunction (r)issinglevaluedrequiresthattheangularmomentumquantum#beinteger:l= 0,1,2..However,operatoralgebraallowedsolutionsl=0,1/2,1,3/2,2... Experimentshowsthattheelectronpossessesanintrinsicangularmomentum calledspinwithl=.Byconvention,weusethelettersinsteadofl forthespin angularmomentumquantumnumber:s=. TheexistenceofspinisnotderivablefromnonrelativisticQM.Itisnotaformof orbitalangularmomentum;itcannotbederivedfrom (Theelectronisapointparticlewithradiusr=0.) Electrons,protons,neutrons,andquarksallpossessspins=.Electronsand quarksareelementarypointparticles(asfaraswecantell)andhavenointernal structure.However,protonsandneutronsaremadeof3quarkseach.The3half spinsofthequarksaddtoproduceatotalspinofforthecompositeparticle(ina sense,makesasingle).Photonshavespin1,mesonshavespin0,thedelta particlehasspin3/2.Thegravitonhasspin2.(Gravitonshavenotbeendetected experimentally,sothislaststatementisatheoreticalprediction.) . Page H-29 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 SpinandMagneticMoment Wecandetectandmeasurespinexperimentallybecausethespinofa chargedparticleisalwaysassociatedwithamagneticmoment. Classically,amagneticmomentisdefinedasavectorassociatedwith aloopofcurrent.Thedirectionofisperpendiculartotheplaneof thecurrentloop(righthandrule),andthemagnitudeis . Theconnectionbetweenorbitalangularmomentum(notspin)andmagnetic momentcanbeseeninthefollowingclassicalmodel:Consideraparticlewithmass m,chargeqincircularorbitofradiusr,speedv,periodT. i m,q r i r |angularmomentum|=L=pr=mvr,sovr=L/m,and Soforaclassicalsystem,themagneticmomentisproportionaltotheorbital angularmomentum: Thesamerelationholdsinaquantumsystem. InamagneticfieldB,theenergyofamagneticmomentisgivenby (assuming ).InQM, . . . Writingelectronmassasme(toavoidconfusionwiththemagneticquantumnumber m)andq=ewehave ,wherem=-l..+l.Thequantity Page H-30 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 iscalledtheBohrmagneton.Thepossibleenergiesofthemagnetic momentin Forspinangularmomentum,itisfoundexperimentallythattheassociatedmagnetic momentistwiceasbigasfortheorbitalcase: (WeuseSinsteadofLwhenreferringtospinangularmomentum.) Thiscanbewritten Theenergyofaspininafieldis verifiedexperimentally. Theexistenceofspin(s=)andthestrangefactorof2inthegyromagneticratio (ratioof )wasfirstdeducedfromspectrographicevidencebyGoudsmitand . (m=1/2)afactwhichhasbeen isgivenby . Uhlenbeckin1925. Another,evenmoredirectwaytoexperimentallydeterminespiniswithaStern Gerlachdevice,nextpage Page H-31 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 (ThispagefromQMnotesofProf.RogerTobin,PhysicsDept,TuftsU.) Stern-Gerlach Experiment (W. Gerlach & O. Stern, Z. Physik 9, 349-252 (1922). B ^ F = B = ( ) B (in current free regions), or here, F = z (z z ) (thisisalittle z ( ) crudeseeGriffithsExample4.4forabettertreatment,butthisgivesthemainidea) Deflectionofatomsinzdirectionisproportionaltozcomponentofmagnetic momentz,whichinturnisproportionaltoLz.Thefactthattherearetwobeamsis proofthatl=s=.Thetwobeamscorrespondtom=+1/2andm=1/2.Ifl=1, thentherewouldbethreebeams,correspondingtom=1,0,1.Theseparationof thebeamsisadirectmeasureofz,whichprovidesproofthat Theextrafactorof2intheexpressionforthemagneticmomentoftheelectronis oftencalledthe"gfactor"andthemagneticmomentisoftenwrittenas .Asmentionedbefore,thiscannotbededucedfromnonrelativistic QM;itisknownfromexperimentandisinserted"byhand"intothetheory. Page H-32 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 However,arelativisticversionofQMduetoDirac(1928,the"DiracEquation") predictstheexistenceofspin(s=)andfurthermorethetheorypredictsthevalue g=2.Alater,betterversionofrelativisticQM,calledQuantumElectrodynamics (QED)predictsthatgisalittlelargerthan2.Thegfactorhasbeencarefully measuredwithfantasticprecisionandthelatestexperimentsgiveg= 2.0023193043718(76inthelasttwoplaces).ComputingginQEDrequires computationofabinfiniteseriesoftermsthatinvolveprogressivelymoremessy integrals,thatcanonlybesolvedwithapproximatenumericalmethods.The computedvalueofgisnotknownquiteaspreciselyasexperiment,neverthelessthe agreementisgoodtoabout12places.QEDisoneofourmostwellverified theories. SpinMath Recallthattheangularmomentumcommutationrelations werederivedfromthedefinitionoftheorbitalangularmomentumoperator: . Thespinoperator doesnotexistinEuclideanspace(itdoesn'thaveapositionor momentumvectorassociatedwithit),sowecannotderiveitscommutation relationsinasimilarway.Insteadweboldlypostulatethatthesamecommutation relationsholdforspinangularmomentum: .Fromthese,wederive,justabefore,that Page H-33 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 (sinces=) (sincems=-s,+s=-1/2,+1/2) Notation:sinces=always,wecandropthisquantumnumber,andspecifythe eigenstatesofL2,Lzbygivingonlythemsquantumnumber.Therearevariousways towritethis: Thesestatesexistina2DsubsetofthefullHilbertSpacecalledspinspace.Since thesetwostatesareeigenstatesofahermitianoperator,theyformacomplete orthonormalset(withintheirpartofHilbertspace)andany,arbitrarystateinspin spacecanalwaysbewrittenas ) Matrixnotation: .Notethat (Griffiths'notationis IfwewereworkinginthefullHilbertSpaceof,say,theHatomproblem,thenour basisstateswouldbe .Spinisanotherdegreeoffreedom,sothatthe fullspecificationofabasisstaterequires4quantumnumbers.(Moreonthe connectionbetweenspinandspacepartsofthestatelater.) [Noteonlanguage:throughoutthissectionIwillusethesymbolSz(andSx,etc)to refertoboththeobservable("themeasuredvalueofSzis ")anditsassociated operator("theeigenvalueofSzis ").] Page H-34 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 ThematrixformofS2andSzinthe element.(Recallthatforanyoperator basiscanbeworkedoutelementby .) Operatorequationscanbewritteninmatrixform,forinstance, WearegoingaskwhathappenswhenwemakemeasurementsofSz,aswellasSx andSy,(usingaSternGerlachapparatus).Willneedtoknow:Whatarethe matricesfortheoperatorsSxandSy?Thesearederivedfromtheraisingand loweringoperators: TogetthematrixformsofS+,S-,weneedaresultfromthehomework: Forthecases=,thesquarerootfactorsarealways1or0.Forinstance,s=, m=-1/2gives .Consequently, ,leadingto and Page H-35 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 NoticethatS+,S-arenothermitian. Using Oftenwritten: ,where are yields Thesearehermitian,ofcourse. calledthePaulispinmatrices. Nowlet'smakesomemeasurementsonthestate . Normalization: . . SupposewemeasureSzonasysteminsomestate Postulate2saysthatthepossibleresultsofthismeasurementareoneoftheSz eigenvalues: .Postulate3saystheprobabilityoffinding,say , is . ,theinitial Postulate4saysthat,asaresultofthismeasurement,whichfound state collapsesto . ButsupposewemeasureSx?(WhichwecandobyrotatingtheSGapparatus.) Whatwillwefind?Answer:oneoftheeigenvaluesofSx,whichweshowbeloware thesameastheeigenvaluesofSz: .(Notsurprising,sincethereis Page H-36 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 nothingspecialaboutthezaxis.)Whatistheprobabilitythatwefind,say,Sx= ?ToanswerthisweneedtoknowtheeigenstatesoftheSxoperator.Let's callthese(sofarunknown)eigenstates (Griffithscallsthem ).Howdowefindthese?Wemustsolvetheeigenvalueequation: ,where aretheunknowneigenvalues.Inmatrixformthisis, whichcanberewritten linearalgebra,thislastequationiscalledthecharacteristicequation. Thissystemoflinearequationsonlyhasasolutionif .So Asexpected,theeigenvaluesofSxarethesameasthoseofSz(orSy). Nowwecanplugineacheigenvalueandsolvefortheeigenstates: ; . .In Sowehave Nowbacktoourquestion:Supposethesysteminthestate measureSx.Whatistheprobabilitythatwefind,say,Sx= ,andwe ?Postulate3gives Page H-37 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 therecipefortheanswer: Questionforthestudent:Supposetheinitialstateisanarbitrarystate andwemeasureSx.WhataretheprobabilitiesthatwefindSx= and ? Page H-38 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 SJP QM 3220 3D 1 Let'sreviewthestrangenessofQuantumMechanics. SupposeanelectronisintheSx= eigenstate .Ifweask:What .Butifweask:Whatisthe isthevalueofSx?Thenthereisadefiniteanswer: valueofSz,thenthisisnoanswer.ThesystemdoesnotpossessavalueofSz.Ifwe measureSz,thentheactofmeasurementwillproduceadefiniteresultandwillforce thestateofthesystemtocollapseintoaneigenstateofSz,butthatveryactof measurementwilldestroythedefinitenessofthevalueofSx.Thesystemcanbein aneigenstateofeitherSxorSz,butnotboth. Page H-39 M. Dubson, (typeset by J. Anderson) Mods by S. Pollock Fall 2008 ...
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