06 薛定谔方程

106TheSchrödingerEquationInclassicalmechanicsitispossibletocalculate,forexample,thevibrationalmodesofastring,membraneorresonatorbysolvingawaveequation,subjecttocertainbound-aryconditions.Attheverybeginningofthedevelopmentofquantummechanics,onewasfacedwiththeproblemoffindingadifferentialequationdescribingdiscretestatesofanatom.Itwasnotpossibletodeduceexactlysuchanequationfromoldandwell-knownphysicalprinciples;instead,onehadtosearchforparallelsinclassicalmecha-nicsandtrytodeducethedesiredequationonthebasisofplausiblearguments.Suchanequation,notderivedbutguessedatintuitively,wouldthenbeapostulateofthenewtheory,anditsvaliditywouldhavetobecheckedbyexperiment.Thisequationforthecalculationofquantum-mechanicalstatesiscalledtheSchrödingerequation;letusnow"derive"it.Inrelativisticclassicalmechanics,timecoordinatesandspatialcoordinatesaswellasenergyandmomentumaretreatedasthefourcomponentsofafour-vector,i.e.,(,i)xctr,,iEpcp,1,2,3,4.(6.1)Byenlargingtheoperatorrepresentationofthethree-dimensionalmomentumtoafour-dimensional,relativisticcovariantvector-operatorwegetiˆˆ,i,,,i(i)Ecxyzctxp.(6.2)Bothsidesofthisequationarefour-vectors.Bycomparison,theenergyisreplacedbythefollowingoperator:ˆiEt.(6.3)Werememberherethatwealreadyhadanoperatorfortheenergyin(4.83),namelytheHamiltonianĤofaparticle.Obviouslywehavetwooperatorsfortheenergy.BothÊandtheHamiltonianĤdescribethetotalenergyandcanthereforebesetequal.ThisgeneratestheSchrödingerequation:ˆˆ(,)(,)EtHtrrorˆi(,)(,)tHttrr,with22ˆˆ()()22HVVmmprr.(6.4)Usingthewavefunctionofafreeparticle(deBrogliewave),iii(,)exp()expexptAEtApxArprpr,(6.5)wefindthattheoperatorÊhasthetotalenergyEasaneigenvalue:ˆ(,)i(,)(,)EttEttrrr.TheSchrödingerequation(6.4)isnotarelativisticequation.Indeed,startingfrom222240Epcmc,(6.6)fortheenergyofafreerelativisticparticle,thefreeKlein-Gordonequationfollows,22222402(,)()(,)tcmcttrr.(6.7)TheSchrödingerequationandtheKlein-Gordonequationarelineardifferentialequa-tions;thismeansthatwith1and2,thefunctiondefinedby12abisasolution,too.Thisisthemathematicalformulationfortheprincipleofsuperposition,2whichwasdiscussedinChap.3.Schrödinger’sequationisoffirstorderintimeandsecondorderinspace;theKlein-Gordonequationisofsecondorderinbothspaceandtime.Wesupposethatthewavefunctionattimetocontainsalltheinformationabouthowthestatepropagatesiftherearenoexternalperturbations.OnlySchrödinger’sequationasafirstorderdifferentialequationintimesatisfiesthisrequirement.TheKlein-Gordonequation,beingimportantinrelativisticquantummechanics,needstobereinterpreted.TheSchrödingerequation(6.4)containstheimaginaryunitiasafactor,whichimpliesthatoscillatingsolutionsarepossible.Itisseparableintotimeandspace,iftheHamiltonianˆˆ(,)HHtrisnotexplicitlytimedependent:(,)()()tftrr,andthereforeˆi()()[()]()ftHfttrr.(6.8)(Since(,)trand()raretwodifferentfunctions,thisshouldnotleadtoanymisunderstanding.)Afterseparatingthevariables,onefindstheequationˆ()()iconst()()ftHEftrr.(6.9)Thismeansforthetime-dependentfunction()constexpiEtft.(6.10)Thefunctionwiththespatialargument()rsolvesthestationarySchrödingerequa-tionˆ()()HErr.(6.11)Thewavefunction(,)trisperiodicintime,withthephasefactorexp[i()]Et,andthatiswhythedensitiesandalso,asweshallsee,thecurrentsaretimeindependent.Equation(6.11)isaneigenvalueequationoftheHamiltonian,withEbeingtherealenergyeigenvalue.Thegeneralsolutionsof(6.4)areoscillatingfunc-tionsintime,(,)()expinnnEttrr,(6.12)withthenormalization(,)(,)d()()d1nnnnttVVrrrr.(6.13)Anystationarystatecorrespondstowell-definedenergyandtoaninfinitestabilityintime.Ithasthecharacterofastandingwavebecausethedensityofprobabilitygivenbyistimeindependent.Thisisnottrueforalinearsuperpositionofstationarystates.注.薛定谔方程是非相对论的,由它所得到的能量E不包含粒子的静能,这是因为哈密顿算符ˆˆˆHTV是动能势能之和.另一方面,在上述薛定谔方程的“推导”过程中E代的却是粒子的总能,因此这个“推导”当然是不严谨的.1Example6.1:AParticleinanInfinitelyHighPotentialWellAparticleofmassmiscapturedinaboxlimitedby0xa;0yb;0zc.Thecorrespondingpotentialisgivenby0if0;0;0,elsewhere.xaybzcV(Seefigure.)TheHamiltonianisgivenby2ˆˆˆ()2HTVVmr.(1)Insidethebox:Herewehavethepotential(,,)0Vxyzandthefollowingstation-arySchrödingerequation:2222222ˆ(,,)(,,)(,,)2HxyzxyzExyzmxyz.(2)Wearegoingtosolvethisproblemusingthewell-knownseparationofvariablesprocedure:123(,,)()()()xyzxyz.(3)Thisleadstothreeseparateequationsconnectedbytheconstantsofseparation2ik(theconstantssquaredarechosentobenegative,thisdoesnotconflictwiththegene-ralcase,sincetheconstantsthemselvesareallowedtobeimaginary):2312123()()()2()()()zxyEmxyz,or2111()()xkx,2222()()yky,2333()()xkx.(4)Thesolutionsofthesedifferentequationsaresimply111()constsin()xkx,222()constsin()yky,333()constsin()zkz.Thetotalsolutioninsidethepotentialwellisthereforegivenby112233(,,)sin()sin()sin()xyzAkxkykz,(5)whereAisanormalizationfactorandiarephaseswhichmustbedetermined.Outsidethepotentialwell:Thewavefunctionmustvanishhere,because(,,)Vxyzisinfinitelylarge;otherwisetherewouldbeaninfinitelylargepotentialenergy,since|()|Vrdiverges.Sincethewavefunctionshavetobesmooth,wegettwosetsofboundaryconditions:(0,,)(,0,)(,,0)0xyzxyzxyz,and(,,)(,,)(,,)0xayzxybzxyzc.(6)Thefirstsetrequires1230.Theothersetgivesaquantizationcondition:11akn,22bkn,33akn,andtherefore11kna,22knb,33knc.(7)2Here,123,,1,2,nnnareindependentquantumnumbers.Thepossibilityofchoosing0inmustbeexcluded,becausethecorrespondingwavefunction(seetheendofthisexample)wouldvanisheverywhere.Thetotalenergyis2222123()2Ekkkm;(8)itcanhaveonlydiscretevalues,namely12322221232nnnEnnnmabc.(9)Thisdiscreteenergyspectrumisconvertedintoaquasi-continuumwhenthemassmortheextensionoftheboxbecomesverylarge.Thelowestenergyvalue22221112Emabc(10)isnotzero,asonewouldexpectclassically.Thisisthefirstexampleofnonvanishingzero-pointenergy(seetheextendeddiscussioninconnectionwiththeharmonicoscillator,Chap.7).Thesolutioninsidethebox,123312123()sin()sin()sin()sinsinsinnnnnznxnyAkxkykzAabcr,(11)mustyieldaunittotalprobability,whichmeansthat1231231(,,)(,,)dnnnnnnxyzxyzV22222123000||sin()dsin()dsin()d||8abcabcAkxxkyykzzA,(12)andthereforethenormalizationfactorequals222||Aabc.(13)Theenergyspectrumisshowninthenextfigure.Wetookabc;thereforethelevel211Eisenergeticallyhigherthanthelevels121Eand112E,andtherelation211121112EEEholds.Inthecaseinwhich,ab,andcdonotdiffertoomuch,alltheselevelsareclosetogether.Thenwespeakofatriplet(ingeneralamultiplet),ofstates.Forabc,theparticlemovesinacube,andallstatesbelongingtoatripletaredegeneratedinenergy.Wehavethen211121112EEE.(14)3Thewavefunctionsofthethreestatesare211382()sinsinsinxyzaabcr,121382()sinsinsinxyzaabcr,112382()sinsinsinxyzaabcr.(15)Ifwebreakthisdegeneracyslightly,thevolumeisapproximatelythatofacubeandthethreelevelsareclosetogetherinenergy,aswejustpointedout.Forstatesofhigherenergyweobserveanequivalentphenomenon.Forinstance,therearetwotriplets(becauseofaslightbreakinthecube'ssymmetry)closetogether,namely211121112,,and311131113,,,followedbyonesinglestate(asinglet),namely222.Suchmultipletstructuresofstatesareidentifiedwith"shells".Shellmodelsexplainingshellstructuresareimportantinatomicandnuclearphysics.Forexample,innuclearphysicsallnucleonsinanucleusaresupposedtobeinapotentialwell.Ofcoursethispotentialwellissphericallysymmetric,butforsmallnucleiaboxlikepotentialisanacceptableapproximation.Becauseofthespinoftheprotonandtheneutron(comparelaterChap.12)andthePauliprinciple(comparelaterChap.14),onlytwoprotonsandtwoneutronscanbeputintoeachlevel.Westartbyfillingthelowestenergylevelsindividually,becausethesystemprefersthestateoflowestenergy.Here,the"last"particledeterminesthemost"visible"properties.Ifthislaststateisinsideamultiplet,thenasmallexcitationenergysufficestoliftthatparticleintoahigherstateofenergy;thenucleusistheneasilyexcitable.Ifweconsideranucleusthatcontainsjustthenumberofprotonsandneutronstofillashell,thenmuchmoreenergywillberequiredtoexciteanucleonintothefirstexcitedstate.Suchnucleiareparticularlystable,becausetheycanonlybedestroyed(i.e.,stronglyexcited)iflargeenergygapsareovercome.Insuchacasewespeakofmagicnuclei(comparablewithfilledelectronshellsintheatomsofinertgases)ordoublemagicnuclei.Foranextensivediscussionofthenuclearshellmodel,seeanothervolumeofthisseriesonnuclearmodels.1Exercise6.2:AParticleinaOne-DimensionalFinitePotentialWellSolvetheone-dimensionalSchrödingerequationforafinitepotentialwelldescribedbythefollowingpotential(seefigure):0if||,()0if||.VxaVxxaConsiderboundstates(0E)only.Solution(a)Thewavefunctionsfor||xaand||xa.ThecorrespondingSchrödingerequationisgivenby2ˆ()()()()()2HxxVxxExm.(1)Wedefine,forthesakeofbrevity,222mE,2022()mEVk,(2)andget:(1)ifxa:2110,111exp()exp()AxBx;(3a)(2)ifaxa:2220k,222cos()sin()AkxBkx;(3b)(3)ifxa:2330,333exp()exp()AxBx.(3c)(b)Formulationofboundaryconditions.Thenormalizationoftheboundstaterequiresthevanishingofthesolutionatinfinity.Thismeansthat130BA:11exp()Ax,33exp()Bx.Furthermore,()xshouldbecontinuouslydifferentiable.Allparticularsolutionsarefittedinsuchawaythat()xaswellasitsfirstderivative()xaresmoothatthatvalueofxcorrespondingtotheborderbetweentheinsideandoutside.Thesecondderivative()xcontainsthejumprequiredbytheparticularbox-typepo-tentialofthisSchrödingerequation.Allthistogetherleadsto12()()aa,23()()aa,12()()aa,23()()aa.(4)(c)Theeigenvalueequations.From(4)weobtainfourlinearandhomogeneousequa-tionsforthecoefficients1212,,,AABB;namely122exp()cos()sin()AaAkaBka,122exp()sin()cos()AaAkkaBkka,322exp()cos()sin()BaAkaBka,322exp()sin()cos()BaAkkaBkka.(5)Byadditionandsubtractionoftheseequations,wegetamorelucidformofthesystemofequations,whichiseasytosolve:132()exp()2cos()ABaAka,132()exp()2sin()ABaAkka,132()exp()2sin()ABaBka,132()exp()2cos()ABaBkka.(6)Assumingthat130ABand20A,thefirsttwoequationsyieldtan()kka.(7)Insertingthisinoneofthelasttwoequationsgives13AB;20B.(8)2Hence,asaresult,wehaveasymmetricsolutionwith()()xx:11exp()Ax,22cos()Akx,31exp()Ax.Wethenspeakofpositiveparity.Almostidenticalcalculationsleadfor130ABandfor20Btotan()kka,(9)andhence13AB;20A.Thethus-obtainedwavefunctionisanantisymmetricone,correspondingtonegativeparity()()xx:13exp()Bx,22sin()Bkx,33exp()Bx.(d)Qualitativesolutionoftheeigenvalueproblem.Theequationsconnectingandk,whichwehavealreadyobtained,areconditionsfortheenergyeigenvalue.Usingtheshortformska,a,(10)wegetfromthedefinition(2)2222022mVar.(11)Ontheotherhand,using(7)and(9)wegettheequationstanorcot.Thereforethedesiredenergyvaluescanbeobtainedbyconstructingtheintersectionofthosetwocurveswiththecircledefinedby(11),withinthe,-plane(seenextfigure).Atleastonesolutionexistsforarbitraryvaluesoftheparameter0V,inthecaseofpositiveparity,becausethetanfunctionintersectstheorigin.Fornegativeparity,theradiusofthecircleneedstobelargerthanaminimumvaluesothatthetwocurvescanintersect.Thepotentialmusthaveacertaindepthinconnectionwithagivensizeaandagivenmassm,topermitasolutionwithnegativeparity.Thenumberofenergylevelsincreaseswith0V,aandmassm.Forthecase202mVa,theintersectionsarefoundattan()kacorrespondingto212nka,1,2,3,n;cot()kacorrespondingtokan,1,2,3,n,(12)or,combined:2nka.(13)Fortheenergyspectrumthismeansthat22022nnEVma.(14)Onenlargingthepotentialwelland/ortheparticle'smassm,thedifferencebetweentwoneighbouringenergyeigenvalueswilldecrease.Thelowermoststate(1n)is3notlocatedat0V,butalittlehigher.Thisdifferenceiscalledthezero-pointenergy.Wewillcomebacktoitlaterwhendiscussingtheharmonicoscillator(seeChap.7).(e)Theshapeofthewavefunctionisshownforthediscussedsolutionsinthetwofigures.1Exercise6.3:TheDeltaPotentialSupposewehaveapotentialoftheform0()()VxVx;00V.Thecorrespondingwavefunction()xissupposedtobesmooth.(a)Searchfortheboundstates(0E)whicharelocalizedatthispotential.(b)Calculatethescatteringofanincomingplanewaveatthispotentialandfindthecoefficientofreflection2ref2in0||||xR,2trans2in0||||xT,whereref()x,trans()xandin()xarethereflected,transmittedandincomingwaves,respectively.Hint:Toevaluatethebehaviourof()xat0x,integratetheSchrödingerequa-tionovertheinterval(,)andconsiderthelimit0.Solution(a)TheSchrödingerequationisgivenby2202d()()()()2dxVxxExmx.(1)Awayfromtheoriginwehaveadifferentialequationoftheform2222d()2()()dxmExxx,22mE.(2)Thewavefunctionsarethereforeoftheform()xxxAeBeif0xor0x.(3)As2||mustbeintegrable,therecannotbeanexponentiallyincreasingpart.Fur-thermorethewavefunctionshouldbecontinuousattheorigin.Hence,()xxAe(0)x,()xxAe(0)x.(4)IntegratingtheSchrödingerequationfromto,weget20[()()](0)()d2VExxm.(5)Insertingnowresult(4)andtakingthelimit0,wehave20()02AAVAmor02mV,2022mVE.(6)Clearly(thoughsurprisingly)thereisonlyoneenergyeigenvalue.Thenormalizationconstantisfoundtobe02mVA.(b)Thewavefunctionofaplanewaveisdescribedby22i2d()()()dkxxkxxAex,22mEk.(7)Itmovesfromtheleft-handtotheright-handsideandisreflectedatthepotential.IfBandCaretheamplitudesofthereflectedandtransmittedwaves,respectively,wegetii()kxkxxAeBe(0)x,i()kxxCe(0)x.(8)2Conditionsofcontinuityandtherelation,022(0)(0)(0)(0)mVfwith022mVf,give,2ii()2i.2ifBAABCfkkCABfCkCAfk(9)Thedesiredcoefficientofreflectionistherefore2222ref0222242in00||||||||xmVBRAmVk.(10)Ifthepotentialisextremelystrong(0V)1R,i.e.,thewholewaveisreflected.Thecoefficientoftransmissionis,ontheotherhand,2242trans222242in00||||||||xCkTAmVk.(11)Ifthepotentialisverystrong,(0V)0T,i.e.,thetransmittedwavevanishes.Obviously,1RTasistobeexpected.1Exercise6.4:DistributionFunctionsinQuantumStatisticsLetaquantum-mechanicalsystemhavetheenergyeigenvaluesiwhicharedegene-rateigtimesandareeachoccupiedwithinparticlesinsuchawaythat23(10)inN(1)isthetotalnumberofindistinguishableparticlesandiinE(2)isthetotalenergy(forexample,anidealgasenclosedinabox).(a)Astatecanbeoccupiedbyonefermiononly,butwithanunlimitednumberofbosons.ThisistheconsequenceofthePauliprinciple.(WewillreturntoitlaterinChap.14.)Provethatitispossibletodistributeinparticlesoverigstateswith(I)FD!!()!iiiiiiiggWnngn(Fermi-Diracstatistics)possibilitiesinthecaseoffermions,with(II)BE1(1)!!(1)!iiiiiiiigngnWnng(Bose-Einsteinstatistics)possibilitiesinthecaseofbosons,andwith(III)BiniiWg(Boltzmannstatistics)possibilitiesforclassical,i.e.,distinguishable,particles.(b)Wedefine12{}{,,}innnasadistributionofparticleswitha"weight"{}iWn,whichissimplythenumberofpossibilitiesofdistributingexactlyinparticlesinenergylevelsi.Naturallytheonemostlikelytoberealized,i.e.,themostprobabledistribution,istheonewiththegreatestweight.Derivethevariationalprinciplefromtheseremarksandbyconsideringtheconstantnumberofparticlesandtheconstanttotalenergy:[ln{}]0iWnNE.(3)Here,theparameters,areLagrangemultipliersand{}inisthedistributionsearchedfor.Provethatinthecaseof1iigntheaveragenumberofoccupation,namelyin,forthelevelwithindexiisgivenby:exp[()]iiignkT,(4)with1forfermions,1forbosons,0forclassical,distinguishableparticles.(5)Hint:UseStirling'sformulaforcalculating!nforlargevaluesofn:12!2nnnnnnee,(6)andtheninsertixforin,i.e.,changefromadiscreteintoacontinuousvariableix.(c)DrawadiagramfortheFermi-DiracdistributionFDEnasafunctionoftheen-ergyEat0T.Howistheparametertobeinterpreted?2Solution(a)Inordertounderstandthedifferentstatistics,wefirstconsiderthepro-blemofinindistinguishableballsthataretobedistributedintoigboxes.(I)Fermi-Diracstatistics.Thereareindistinguishableballstobedistributedintoigboxes:(7)Eachboxcancontainonlyoneball(theFermi-Diraccase!).Thefirstonecouldbeplacedinoneoftheigboxes.Forthesecondonetherearethen1igpossibilitiesleft,becauseoneboxisalreadyoccupiedbythefirstball.Forthelastball,exactly1iignpossibilitiesarefoundtoexistifiign.Thetotalnumberofpossibilitiesisgivenbytheproduct!(1)(1)()!iiiiiiiggggngn.(8)Sofar,wehaveassumedthattheparticlesaredistinguishable;iftheyarenotdistin-guishable,however,thereareseveralidenticalcombinations.Forexample,thecom-binationisidenticalto(9)Thismeansthatwehaveoverestimatedthenumberofpossibilitiesofdistributionuptonowbythenumberofpermutationsamongtheinparticles.Thepermutationsofinelementsraiseafactor!in,sothatfinallyFD!!()!iiiiiiiggWnngn.(10)(II)Bose-Einsteinstatistics.Eachboxisnowabletocontainarbitrarilymany,indis-tinguishableballs(theBose-Einsteincase!).Herethefollowingmethodwillwork.Foreachparticleandbox,aslipofpaperismarkedwith1,,igKKfortheboxes,andwith1,,inBBfortheballs.Nowslip1Kissetasideandallotherslips(bothkinds,thoselabelledKandthoselabelledB)areplacedinanurn;thereshouldbe1iignleft.Nowtheslipsaretakenoutagain,onebyoneandinanarbitrarysequence,andplacedattherightof1K;forexample:18731324KBKBBKKB.(11)Thiscanbeinterpretedasfollows:ballslocatedbetweentwoboxesaresupposedtobeintheleftone.Inourexample,itwouldbeballnumber8thathastobeinboxnumber1,whereasballs3and1belongtoboxnumber7,noballtobox3,ball4intobox2,andsoon.Frompart(I)wealreadyknowthatthereare(1)!iignarrangementspossible(arrangementsasshowninexample(11)).Ontheotherhandthepositionsoftheballsandtheirboxesinthatrowareunimportant.Indeedtherearenonewarrangementsiftheinballsareexchangedamongthemselves.Thesameisvalidifthe1igslipsareexchanged.ThusthereareBE1(1)!!(1)!iiiiiiiigngnWnng(12)differentarrangements.3(III)Boltzmannstatistics:Supposewehavetwoballsandigboxes.Thereareex-actlyigpossibilitiesofdistributingball1.Ifmorethanoneballisallowedinasinglebox,therearealsoigpossibilitiesofdistributingball2.Altogetherthereare2igarrangements.Analogously,forinballsthereareBiniiWg(13)arrangements.(b)AswehaveindistinguishableparticlesinboththeFermi-DiracstatisticsandBose-Einsteinstatistics,thenumberofdistributionarrangementsforinparticlesintoenergylevelsi,ifwehavetheparticledistribution12{,,,}mnnn,isgivenbytheproductFDFD11!{}!()!mmiiiiiiiigWnWngn,(14)andBEBE11(1)!{}!(1)!mmiiiiiiiignWnWng.(15)LetustumnowtoBoltzmannstatisticswithdistinguishableparticles,wherethingsarefoundtobemorecomplicated.WeareobligedtosetiNnparticlesintothelevelsinsuchawaythatthereareinparticlesineachone.Thenumberofpossiblewaysofdoingthisiscalculatedinthefollowingway.Westartwith!Npossibilities,disregardingthegroupstructurecontainingmgroups.Wethencorrectthisnumberbyfactors!jncomingfromthearbitraryoccupationineachgroup.Thisyields12!!!!mNnnn(16)possibilities.Hence,BB111212!!{}!!!!!!immniiiiimmNNWnWgnnnnnn,(17)whichwecanalsowriteasB1{}!!inmiiiigWnNn.(18)Thenextstepistocalculatethemaximumofthesevariousdistributions{}iWn,inordertofindtheparticulardistributionwiththegreatestweight.ThemaximumofFD{}iWnagreeswiththemaximumofFDln{}iWn,whichiseasiertohandlemathe-matically.ThereforeusingStirling'sformula(6),weobtainFD1!ln{}ln!()!miiiiiigWnngn1ln[lnln()ln()]miiiiiiiiiggnngngn.(19)Inthelasttwostepswehavemadeuseof1ig,1in.Tofindthemaximaofadistributionweadmitcontinuousvaluesforin,sothatiinx,andintroducetwoLagrangemultipliersandtoincorporatetheconditionsiinEandinN,respectively.4ThevariationalprincipleyieldsFDln{}jjjjWxxxFDln{}ijjjjiixWxxxxlniiiiiigxxx.(20)Thelaststepisfoundasaresultof(19),thus:FDln{}ln[ln()ln()]lniijiiiiiiiiigxWxxxgxgxxxx.(21)Anecessaryconditionfortheexistenceofthemaximumisthatthetermsinparen