量子力學(xué)英文課件格里菲斯Chapter5

OutlineForasingleparticle,thewavefunction

(r,t)isafunctionofthespatialcoordinatesrandthetimet(we’llignorespinforthemoment).Thewavefunctionforatwo

particlesystemisafunctionofthecoordinatesofparticleone(r1),thecoordinatesofparticletwo(r2),andthetime:Itstimeevolutionisdetermined(asalways)bytheSchr?dingerequation:whereHistheHamiltonianforthewholesystem:thesubscriptonindicatesdifferentiationwithrespecttothecoordinatesofparticle1orparticle2,asthecasemaybe.

Thestatisticalinterpretationcarriesoverintheobviousway:istheprobabilityoffindingparticle1inthevolumed3r1andparticle2inthevolumed3r2.Evidentlymustbenormalizedinsuchawaythat

Fortime-independent

potentials,weobtainacompletesetofsolutionsbyseparationofvariables:wherethespatialwavefunction

(r1,r2)satisfiesthetime-independentSchr?dingerequation:andEisthetotalenergyofthesystem.

Supposeparticle1isinthe(one-particle)statea(r),andparticle2isinthestateb(r).Inthatcase,

(r1,r2)isasimpleproduct:

Ofcourse,thisassumesthatwecantelltheparticlesapart.

Otherwise,itwouldnotmakeanysensetoclaimthatnumber1isstatea(r)andnumber2isinstateb(r).Ifweweretalkingaboutclassicalmechanicsthiswouldbeasillyobjection.

Youcanalwaystelltheparticlesapart,inprinciple,justpaintoneofthemredandtheotheroneblue,orstampidentificationnumbersonthem,orhireprivatedetectivestofollowthemaround.

Allwecouldsayisthatoneofthemisinthestatea(r)andtheotherisinstateb(r),butwewouldn’tknowwhichiswhich.Butinquantummechanicsthesituationisfundamentallydifferent:

Youcan’tpaintanelectronred,orpinalabelonit,andadetective’sobservationswillinevitablyandunpredictablyalterthestate,raisingdoubtsastowhetherthetwohadperhapsswitchedplaces.

Thefactis,allelectronsareutterly(全然;完全地)identical,inawaythatnotwoclassicalobjectscaneverbe.

Itisnotmerelythatwedon’thappentoknowwhichelectroniswhich;Goddoesn’tknowwhichiswhich,becausethereisnosuchthingas“this”electron,or“that”electron;allwecanlegitimatelyspeakaboutis“an”electron.Quantummechanicsneatlyaccommodatestheexistenceofparticlesthatare

indistinguishableinprinciple:

Wesimplyconstructawavefunctionthatisnoncommittalastowhichparticleisinwhichstate.Thereareactuallytwowaystodoit:Thusthetheoryadmitstwokindsofidenticalparticles:

bosons,forwhichweusetheplussign,and

fermions,forwhichweusetheminussign.TheindistinguishableisonefundamentalprincipleofQuantumMechanics!ItsohappensthatThisconnectionbetweenspinand“statistics”canbeprovedinrelativisticquantummechanics;inthenonrelativistictheoryitmustbetakenasanaxiom.Itfollows,inparticular,thattwoidentical

fermions(forexample,twoelectrons)cannotoccupythesamestate.

Forifa(r)=b(r),then

andweareleftwithnowavefunctionatall.ThisisthefamousPauliexclusionprinciple.

Itisnotabizarre(怪誕的)adhoc(特別的)

assumptionapplyingonlytoelectrons,butratheraconsequenceoftherulesforconstructingtwo-particlewavefunctions,applyingtoall

identicalfermions.

Thereisamoregeneralwaytoformulatetheproblem.

Letusdefinetheexchangeoperator

Pwhichinterchangesthetwoparticles:Clearly,P2=1,anditfollowsthattheeigenvaluesofPare1.

Ifthetwoparticlesareidentical,theHamiltonianmusttreatthemthesame:ItfollowsthatPandHarecompatibleobservables,andhencewecanfindacompletesetoffunctionsthataresimultaneousofboth.Thatistosay,wecanfindsolutionstotheSchr?dingerequationthatareeithersymmetric(eigenvalue+1)orantisymmetric(eigenvalue

1)underexchange:Moreover,ifasystemstartsoutinsuchastate,itwillremaininsuchastate!Thenewlaw(symmetrizationrequirement)isthat:foridenticalparticlesthewavefunctionisnotmerelyallowed,

butrequired

tosatisfy

Eq.[5.14]

,withthe

plus

signforbosonsandthe

minus

signforfermions.

Thisisthegeneralstatement.

Eq.[5.10]isaspecialcase.Example.

Supposewehavetwonon-interactingparticles,bothofmassm,intheinfinitesquarewell(Section2.2).Theone-particlestatesare

Iftheparticlesaredistinguishable,thecompositewavefunctionsaresimpleproducts:Forexample,thegroundstateisthefirstexcitedstateisdoublydegenerate:Ifthetwoparticlesareidenticalbosons,thegroundstateisunchanged,butthefirstexcitedstateisnon-degenerate:stillwithenergy5K.

Iftheparticlesareidenticalfermions,thereisnostatewithenergy2K;thegroundstateisanditsenergyis5K.Whatthesymmetrizationrequirementactuallydoes?

Supposeoneparticleisinstatea(x),andtheotherisinstateb(x),andthesetwostatesareorthogonalandnormalized.

Ifthetwoparticlesaredistinguishable,number1istheoneinstatea(x),andnumber2istheoneinstateb(x),

thenthecombinedwavefunctionis

We’regoingtoworkoutasimpleone-dimensionalexample.

Iftheyareidenticalbosons,thecompositewavefunctionisandiftheyareidenticalfermions,itis

Let’scalculatetheexpectationvalueofthesquareoftheseparationdistance

betweenthetwoparticles,Case1:Distinguishableparticles.

ForthewavefunctioninEq.[5.15].wehavetheexpectationvalueofx2intheone-particlestatea(x),

andInthiscase,then,Theanswerwould,ofcourse,bethesameifparticle1hadbeeninstateb(x),andparticle2instatea(x).Case2:Identicalparticles.

ForthewavefunctionsinEqs.[5.16]and[5.17],

Similarly,ButwhereEvidentlyidenticalbosons(theuppersigns)tendtobesomewhatclosertogether,andidenticalfermions(thelowersigns)somewhatfartherapart,thandistinguishableparticlesinthesametwostates.

ComparingEqs.[5.19]and[5.21],weseethatthedifferenceresidesinthefinalterm:Noticethatxab

vanishesunlessthetwowavefunctionsactuallyoverlap(部分重疊)

Soifa(x)representsanelectroninanatominChicagoandb(x)representsanelectroninanatominSeattle,it’snotgoingtomakeanydifferencewhetheryouantisymmetrizethewavefunctionornot.Asapracticalmatter,therefore,it’sokaytopretendthatelectronswithnonoverlappingwavefunctionsaredistinguishable.

Theinterestingcaseiswhenthereissomeoverlapofthewavefunctions.Thesystembehavesasthoughtherewerea“forceofattraction(吸引力)”betweenidenticalbosons,pullingthemclosertogether,anda“forceofrepulsion(排斥力)”betweenidenticalfermions,pushingthemapart.Wecallitanexchangeforce,althoughit’snotreallyaforceatall—nophysicalagencyispushingontheparticles;rather,itisapurelygeometrical

(幾何學(xué)的)consequenceofthesymmetrizationrequirement.Itisalsoastrictlyquantummechanicalphenomenon,withno

classicalcounterpart.Butwait.Wehavebeenignoringspin.

Thecompletestateoftheelectronincludesnotonlyitspositionwavefunction,butalsoaspinor,describingtheorientationofitsspin:Whenweputtogetherthetwo-electronstate,itisthewholeworks,notjustthespatialpart,thathastobeantisymmetricwithrespecttoexchange.Now,aglancebackatthecompositespinstates(Eqs.[4.177]and[4.178])revealsthat:

thesingletcombinationisantisymmetric(andhencewouldhavetobejoinedwithasymmetric

spatialfunction),whereasthethreetripletstatesareallsymmetric(andwouldrequireanantisymmetric

spatialfunction).

Evidently,then,thesingletstateshouldleadtobonding(鍵合),andthetriplettoantibonding.Sureenough,thechemiststellusthatcovalent(共價(jià)的)bondingrequiresthetwoelectronstooccupythesingletstate,withtotalspinzero.Homework:

Problem5.4,Problem5.7.

Aneutral(中性的)atom,ofatomicnumber

Z,consistsofaheavynucleus,withelectricchargeZe,surroundedbyZ

electrons(massmandchargee).TheHamiltonianforthissystemis

Thetermincurlybracketsrepresentsthekineticpluspotentialenergyofthejthelectronintheelectricfieldofthenucleus;

Thesecondsum(withrunsoverallvaluesofjandkexceptj=k)isthepotentialenergyassociatedwiththemutualrepulsion(排斥)oftheelectronsTheproblemistosolveSchr?dingerequation.forthewavefunction

(r1,r2,,rZ).

Becauseelectronsareidenticalfermions,however,notallsolutionsareacceptable:onlythoseforwhichthecompletestate(positionandspin),

isantisymmetricwithrespecttointerchangeofanytwoelectrons.Inparticular,notwoelectronscanoccupythesamestate.

Unfortunately,theSchr?dingerequationwiththeHamiltonianinEq.[5.24]cannotbesolvedexactly(atanyrate,ithasn’tbeen)exceptfortheverysimplestcaseZ=1(hydrogen).Followingisonlytosketchsomeofthequalitativefeaturesofthesolutions,obtainedbyneglectingtheelectronrepulsiontermaltogether.

Afterhydrogen,thesimplestatomishelium(Z=2).TheHamiltonian,consistsoftwohydrogenic

Hamiltonians(withnuclearcharge2e),oneforelectron1andoneforelectron2,togetherwithafinaltermdescribingtherepulsionofthetwoelectrons.Itisthislasttermthatcausesalltheproblems.

Ifwesimplyignoreit,theSchr?dingerequationseparates,andthesolutionscanbewrittenasproductsofhydrogenwavefunctions:onlywithhalftheBohrradius,andfourtimestheBohrenergies(seeProblem4.16).

ThetotalenergywouldbewhereEn=13.6/n2

eV.

Inparticular,thegroundstateoftheHeliumwouldbeanditsenergywouldbeBecause0isasymmetricfunction,thespinstatehastobeantisymmetric,sothegroundstateofHeliumisasingletconfiguration,withthespins“oppositelyaligned(排列)”.

Theactual

groundstateofheliumisindeedasinglet,buttheexperimentallydeterminedenergyis78.975eV,sotheagreementisnotverygood.

Butthisishardlysurprising:Weignoredelectronrepulsion,whichiscertainlynotasmallcontribution.Itisclearlypositive,

whichiscomforting---evidentlyitbringsthetotalenergyupfrom(108.8)to(78.975)eV.Inthesolidstate,afewofthelooselyboundoutermostvalence(化合價(jià))electronsineachatombecomedetachedandroamaroundthroughoutthematerial,nolongersubjectonlytotheCoulombfieldofaspecific“parent”nucleus,butrathertothecombinedpotentialoftheentirecrystallattice.

Inthissectionwewillexaminetwoextremelyprimitivemodels:

(i)theelectrongastheoryofSommerfeld,whichignores

all

forces(excepttheconfiningboundaries),treatingthewanderingelectronsasfreeparticlesinabox(thethree-dimensionalanalogtoaninfinitesquarewell);and(ii)theBloch’stheory,whichintroducesaperiodicpotentialrepresentingtheelectricalattractionoftheregularlyspaced,positivelycharged,nuclei(butstillignoreselectron-electronrepulsion).Supposetheobjectinquestionisarectangularsolid,withdimensionslx,ly,lz,andimaginethatanelectroninsideexperiencesnoforcesatall,exceptattheimpenetrablewalls:TheSchr?dingerequation:separatesinCartesiancoordinates:(x,y,z)=X(x)Y(y)Z(z),withandE=Ex+Ey+Ez

.Lettingweobtainthegeneralsolutions

TheboundaryconditionsrequirethatX(0)=Y(0)=Z(0)=0,soBx=By=Bz=0,andX(lx)=Y(ly)=Z(lz)=0,sowhereeachnisapositiveinteger:The(normalized)wavefunctionsare

andtheallowedenergiesarewherekisthemagnitudeofthewavevector

k=(kx,ky,kz).

Ifyouimagineathree-dimensionalspace,withaxes

kx,ky,kz,andplanesdrawninatkx=(/lx),(2/lx),(3/lx),,atky=(/ly),(2/ly),(3/ly),,andatkz=(/lz),(2/lz),(3/lz),Eachintersectionpointrepresentsadistinct(one-particle)stationarystate(Figure5.3).

Eachblockinthisgrid,andhencealsoeachstate,occupiesavolumeof“k-space,”whereV

lxlylzisthespatialvolumeoftheobjectitself.

NowsupposeoursamplecontainsNatoms,andeachatomcontributesqfreeelectrons.

Butelectronsareinfactidentical

fermionssubjecttothePauliexclusionprinciple,soonlytwoofthemcanoccupyanygivenstate.

Theywillfilluponeoctant(八分圓)ofasphereink-space,whoseradiuskFisdeterminedbythefactthateachpairofelectronsrequiresavolume

3/V(seeEq.[5.40]):Thuswhereisthefreeelectrondensity(thenumberoffreeelectronsperunitvolume).

Theboundaryseparatingoccupiedandunoccupiedstates,ink-space,iscalledtheFermisurface(hencethesubscriptF).

ThemaximumoccupiedenergyiscalledtheFermienergy

EF

;evidently,forafreeelectrongas,Thetotalenergyoftheelectrongascanbecalculatedasfollows:ashellofthicknessdk(Figure5.4)containsavolumesothenumberofelectronstatesintheshellis

Eachofthesestatescarriesanenergy

?2k2/2m(Eq.[5.39]),sotheenergyoftheshellis

andhencethetotalenergyisThisquantummechanicalenergyplaysaroleratheranalogoustotheinternalthermalenergy(U)ofanordinarygas.

Inparticular,itexertsapressureonthewalls,foriftheboxexpandsbyanamountdV,thetotalenergydecreases:andthisshowsupasworkdoneontheoutside(dW=PdV)bythequantumpressure

P.EvidentlyHere,then,isaparticleanswertothequestionofwhyacoldsolidobjectdoesn’tsimplycollapse:Thereisastabilizinginternalpressurethathasnothingtodowithelectron-electronrepulsion(whichwehaveignored)orthermalmotion(whichwehaveexcluded)butisstrictlyquantummechanical,andderivesultimatelyfromtheantisymmetrizationrequirementforthewavefunctionsofidenticalfermions.

Itissometimescalleddegeneracy(簡(jiǎn)并)pressure,although“exclusion(排斥)pressure”mightbeabetterterm.We’renowgoingtoimproveonthefreeelectronmodelbyincludingtheforcesexertedontheelectronsbytheregularlyspaced,positivelycharged,essentiallystationarynuclei.

Thequalitativebehaviorofsolidsisdictatedtoaremarkabledegreebythemerefactthatthispotentialisperiodic—itsactualshapeisrelevantonlytothefinerdetails.Wearegoingtodevelopthesimplestpossibleexample:aone-dimensionalDiraccomb(梳),consistingofevenlyspaced-functionwells(Figure5.5).Butbeforewegettothat,weneedtoknowabitaboutthegeneraltheoryofperiodicpotentials.Consider,then,asingleparticlesubjecttoaperiodicpotentialinonedimension:Bloch’stheorem:ThesolutionstotheSchr?dingerequation,

forsuchapotential

Eq.[5.47],canbetakentosatisfytheconditionforsomeconstantK.Proof:

LetDbethe“displacement”operator:

ByvirtueofEq.[5.47],DcommuteswiththeHamiltonian:andhence(seeSection3.4.1)wearefreetochooseeigenfunctionsofHthataresimultaneouslyeigenfunctionsofD:D=,orforsomeconstantK.QED.Nowiscertainlynotzero,so,likeanynonzerocomplexnumber,itcanbeexpressedasanexponential:

AtthisstageEq.[5.53]ismerelyastrangewaytowritetheeigenvalue

,butinamomentwewilldiscoverthatKisinfactreal,sothatalthough(x)itself

isnot

periodic,|(x)|2is:asonewouldcertainlyexpect.

Ofcourse,norealsolidgoesonforever,andtheedgesaregoingtospoiltheperiodicityofV(x)andrenderBloch’stheoreminapplicable.However,foranymacroscopiccrystal,containingsomethingontheorderofAvogadro’s(阿伏伽德羅)numberofatoms,itishardlyimaginablethatedgeeffectscansignificantlyinfluencethebehaviorofelectronsdeepinside.ThissuggeststhefollowingdevicetosalvageBloch’stheorem:Wewrapthex-axisaroundinacircleandconnectitontoitstail,afteralargenumberN1023ofperiods;formally,weimposetheboundaryconditionItfollows(fromEq.[5.49])thatsoeiNKa

=1orNka

=2n,orInparticular,forthisarrangementKisnecessarilyreal.ThevirtueofBloch’stheoremisthatweneedonlysolvetheSchr?dingerequationwithina

singlecell(say,ontheinterval0

x

a);recursiveapplicationofEq.[5.49]generatesthesolutioneverywhereelse.

Nowsupposethepotentialconsistsofalongstringof-functionwells(theDiraccomb):Thewellsaresupposedtorepresent,verycrudely,theelectricalattractionofthenucleiinthelattice.Noonewouldpretendthatthisisarealisticmodel,butremember,itisonlytheeffectofperiodicitythatconcernsushere;theclassicstudyusedarepeatingrectangularpattern,andmanyauthorsstillpreferthatone.

Inther