量子力学英文课件：Chapt7 Identical Particles

QuantumMechanicsChapter7.IdenticalParticles

WehaveseeninSection6.4thatthestatesavailab1etoidenticalparticlesofhalf-integerspin(fermions)arerestrictedbythePauliexclusionprinciple:

Notwoidenticalfermionscanexistinthesamequantumstate.Wenowexploreamultitudeofconsequencesofthisprinciple.§7.1IdenticalParticlesandSymmetryofwaveFunctionsIndistinguishability

Toanalyzesystemscontaininganumberofidenticalelectrons(e.g.,anyatomexcepthydrogen)wemustexpressourequationsinawaythatmakesnodistinctionbetweenoneelectronandanother.Itissometimesdifficulttograspthisfact,becauseinwritingequationswehavebecomeaccustomedtoidentifyingeachparticleinacollectionbyaseparatelabel.

Iftheparticlesaretrulyindistinguishable,wemustbeverycarefulinusinglabels.Forexample,supposethattwoelectronscometogether,sothattheirwavefunctionsoverlap.Whentheyflyapartagain,therecanbenoway,aftertheyhaveinteracted,todeterminewhichonecameinfromtheleftandwhichonecamefromtheright.Iftherewereaway,thiswouldmeanthatelectronsarenotidentical.SymmetryofWaveFunctionsHowcanwewriteawavefunctionfortwoidenticalparticles,ifwecannotuselabelstodescribethecoordinatesofeachelectron?Theanswerisclearwhenwedevelopthegeneralformofthewavefunctionofasystemofanynumberofparticles.(ThisisanextensionofthediscussioninSection5.1.)ThespatialpartofthewavefunctionforasystemofNparticlesisafunctionof3Ncoordinates,

uT(x1,y1,z1,┅,xN,yN,zN),whichweabbreviateasuT(1,2,┅N).•Whathappensifweinterchangethecoordinatesofparticles1and2?Afterthisinterchange,thefunctionuTdependsonthecoordinatesofparticle1inthesamewaythatitformerlydependedonthecoordinatesofparticle2,andviceversa.Thuswehaveformallyinterchangedtheparticles.Butaccordingtothedefinitionofindistinguishability,iftheparticlesareidentical,thestateresultingfromthisinterchangecannotbedistinguishedfromtheoriginalstate.Thismeansthat

uT(2,1,┅N)=AuT(1,2,┅N)(11.1)whereAisaconstant.Interchangingparticles1and2againmusthavethesameeffectonthewavefunction,yielding

uT(1,2,┅N)=AuT(2,1,┅N)(11.2)CombiningEqs.(11.1)and(11.2)thenyields

uT(2,1,┅N)=A2uT(2,1,┅N)(11.3)

andthusA2=1.IfA=+1,wesaythatthewavefunctionissymmetricwithrespecttointerchangeofthetwoparticles;ifA=-1,thefunctionisanti-symmetric.Figure11.1illustratesthesetwosituationsgraphicallyfortheone-dimensionalcase,wherethefunctionuisafunctionoftwovariablesonly:thecoordinatex1forparticle1,andthecoordinatex2forparticle2.Noticethatthelinex1=x2isalineofsymmetry.Interchangingx1andx2isequivalenttoreflectingthefigurealongthisline.

Figure11.1showsgraphicallytheindistinguishabilityofthetwoparticles.Althoughweputdifferentlabelsonthetwoaxes,anyphysicalresultmustbeindependentofthelabel.Forexample,wecanfindtheprobabilitythatatleastoneofthetwoparticleshasxcoordinatebetween+aand+b,byusingthefactthatu*uistheprobabilitydensityforbothparticles.Thatmeansthatu*uistheprobabilitydensityforaparticleateachoftwocoordinates.Tofindtheprobabilitythatoneorbothoftheparticlesislocatedbetweenx=aandx=b,weintegrateu*uovertheentireregionforwhicheithera<x1<bora<x2<b.ThisregionisenclosedbydashedlinesinFigure11.1.Figure11.1(b)illustratesanimportantgeneralfeatureofantisymmetricfunctions,namelythatu=0whenx1=x2.Toprovethatthismustbeso,letx1=x2=c.Then,fromEq.(11.1),wemaywriteu(x2,x1)=-u(x1,x2)(11.4)

oru(c,c)=-u(c,c)(11.5)whichcanbetrueonlyifu(c,c)=0.FIGLRE11.1Contourmapshowingvaluesofpossible(a)symmetricand(b)antisymmetricwavefunctionsuasfunctionsofthexcoordinatesx1andx2oftwoidenticalparticles.Contoursconnectpointsatwhichuhasaconstantvalue.Onthelinex1=x2,umustbezerointheantisymmetriccase.SeparationofVariables

Todeterminethewavefunctionforasystemoftwoormoreparticlesisaformidabletask.Thereforewestartwiththeapproximationthatthereisnointeractionbetweentheparticles.Thatis,weassumethateachparticlemovesinaknownexternalpotentialthatisindependentoftheposition(s)oftheotherparticle(s).ThusfortwoparticleswewritetheSchreodingerequationas[-(ћ2/2m)(▽12+▽22)+V(1)+V(2)]u(1,2)=ETu(1,2) (11.6)

where▽1operatesonthecoordinatesofparticle1,andV(1)isthepotentialenergyofparticle1andisafunctionofthecoordinatesofparticle1only.Wenextassumethattheparticlesaredistinguishable,andweseparatethevariablesbywritingu(1,2)=ua(1)ub(2),whereuaandubmaybedifferentfunctions.SubstitutionintoEq.(11.6)andregroupingtermsyields [-(ћ2/2m)(▽12+V(1)]ua(1)ub(2)+ [-(ћ2/2m)(▽22+V(2)]ua(1)ub(2)=ET

ua(1)ub(2) (11.6)Asbefore,wenowseparatethevariablesbydividingalltermsbythewavefunctionua(1)ub(2),Wenowconclude,asinSection9.1.thateachtermmustequalaconstant.LabelingtheseconstantsEaandEb,wehaveorwhereEa+Eb=ET.Exceptforthelabels,thetwoequations(11.9)arereallythesameequation,thesingle-particleSchreodingerequation.Therefore,ifbothparticlesaresubjecttothesamepotential,uaandubbelongtothesamesetofeigenfunctions.Forexample,asafirstapproximationwecanassumethateachofthetwoelectronsinaheliumatomseparatelyoccupiesoneofthestatesofaheliumion,whosewavefunctionsaregiveninSection5.2(Table9.1withZ=2).Theenergyofeachelectronistherefore-54.4eV[fromEq.(9.12)],andthetotalenergyis-108.8accordingtoEq.(11.8).Thisisfarfromthemeasuredenergyfortheheliumatom,becausewehaveneglectedthepotentialenergyofrepulsionbetweenthetwoelectrons.However,Chapter8showshowtoapproximatethisenergybyamethodthatgivesgreatagreementwithexperiment.§7.2SymmetryofStatesforTwoIdenticalParticlesThefunctionsua(1)ub(2)ofEq.(11.7)isingeneralneitherantisymmetricnorsymmetric;thusitisunacceptableasawavefunctionforastateoftwoidenticalparticles.Butwecanusethisfunctiontoconstructacceptablewavefunctions.Thesymmetricfunctionisthesum

Us(1,2)=[ua(1)ub(2)+ua(2)ub(1)]/(2)1/2(11.10)

TheantisymmetricfunctionisUA(1,2)=[ua(1)ub(2)-ua(2)ub(1)]/(2)1/2(11.11)Thedivisor(2)1/2isneededtopreservethenormalizationofthewavefunction,ontheassumptionthatuaandubareindividuallynormalized.Foreachofthesewavefunctionsthereisoneparticleinthesingle-particlestatewhosewavefunctionisua,andoneparticleinthesingle-particlestatewhosewavefunctionisub,butwehavenowaytosaywhichparticleisinwhichstate.Youmayverifythat

uS(1,2)=uS(2,1)(11.12)

uA(1,2)=-uA(2,1)(11.13)inagreementwithEq.(11.3).SpinStatesandSymmetryStudyofmanyphenomenahasshownthatallparticleswithhalf-integerspin(electrons,protons,neutrons,muons,neutrinos,andmanyothers)musthaveantisymmetricwavefunctions.Particleswithintegerspin(photons,pions,andothers)musthavesymmetricwavefunctions.ThesefactsaredirectlyconnectedwiththePauliexclusionprinciple,becausetheantisymmetricwavefunction(Eq.11.10)vanisheswhenbothparticlesareinthesamestate:

ua(1)ua(2)-ua(2)ua(1)≡0(11.14)

Thereforenotwoelectrons(orotherspin-1/2particles)cansimultaneouslyoccupythesamequantumstate.Theantisymmetryrequirementappliestoallcoordinates,includingspin.Itisconvenienttoseparatethewavefunctionintotwofactors,aspinfunctionandaspacefunction.Whenthespacefactorissymmetric,thespinfactormustbeantisymmetric,andviceversa.Forexample,therearefourindependentcompletelyantisymmetricstatefunctionsfortwoelectronsinstatesaandb.InDiracnotation,thenormalizedstatesare[|a>1|b>2+|a>2|b>1][|+>1|->2-|+>2|->1]/2(11.15a)[|a>1|b>2-|a>2|b>1]|+>1|+>2/(2)1/2(11.15b)•[|a>1|b>2-|a>2|b>1][|+>1|->2+|+>2|->1]/2(11.15c)[|a>1|b>2-|a>2|b>1]|->1|->2/(2)1/2(11.15d)Thefirstofthese,calledthesingletstate,issymmetricinthespacefunctions|a>and|b>butantisymmetricinthespinfunctions|+>and|->.Thusitisantisymmetricwithrespecttotheexchangeofallcoordinatesofthetwoparticles.Theotherthree,calledtripletstates,areantisymmetricinthespacefunctionsbutsymmetricinthespinfunctions.

Forthetripletstates,thezcomponentofthespinangularmomentumis+ћ,0,and-ћ,respectively.Fromthegeneralrulesforangularmomentum[Eqs.(10.18)-(10.20)],wethusdeducethatthissetofstateshasatotalspinquantumnumberofS=1.[exercise]Thesquareofthetotalspinangularmomentumis,accordingtothoserules,equaltoS(S+l)ћ2,or2ћ2.Youmayveritythesestatementsbyapplyingthespinoperators(seeExercise1).ThisdivisionofstatesintoatripletwithS=landasingletwithS=0ischaracteristicofstatesofanytwoparticlesofspin1/2,whetherornottheparticlesareidentical.Thisfacthasastatisticalconsequencethatiswellverifiedexperimentally:Iftwoparticlesofspin1/2cometogetheratrandom,theirspinsare“parallel”(S=1)threequartersofthetime,andtheyare“antiparallel”onequarterofthetime.ThismeansthateachofthefourstatesofEq.(11.15)isequallylikelytooccur.Forexample,inarandomcollectionofhydrogenmolecules,threequartersareortho-hydrogenwhoseprotonshavetotalspinnumberSp=1,andtheotheronequarterarepara-hydrogenwithzerototalprotonspin.

ExchangeEnergyTherequirementthatthetotalwavefunction,involvingallcoordinatevariables,beantisymmetricleadstoconsequencesthatresembletheeffectsofanewforcethatisunknowninclassicalphysics.This“force”isnotaforceintheclassicalsense.However,theeffectofthisrequirementisthattheelectrons'motionsarecorrelatedinawaythatsuggeststhepresenceofanotherforceinadditiontotheCoulombforce.(Althoughwecannotfollowthetrajectoriesoftheelectrons,wededucefromtheobservedenergylevelsthatthiscorrelationispresent.)Theeffectmaybemadeplausiblefromthefollowingconsiderations:Whenthespacepartofthewavefunctionisantisymmetric,thecombinedwavefunc