量子力学英文课件：Chapt5 The Hydrogen Atom

QuantumMechanicsChapter5.TheHydrogenAtomThesolutionoftheproblemofatomicspectrawasagreattriumphfortheSchreodingerequation.Althoughthisequationdoesnotyieldexactsolutionsforatomscontainingmorethanoneelectron,itpermitsapproximationswhichcanbeapplied,inprinciple,toanyproblemandtoanydesireddegreeofaccuracy.

Theultimateresultisenormousaccuracyintheoreticalcalculationsofhydrogenenergylevels,takingintoaccountsuchfactorsasthespinoftheelectron,apreviouslyunknownquantity.

Subsequently,P.A.M.Diracshowedthatelectronspinemergesasanaturalconsequenceofarelativisticwaveequation..Experimentershaverespondedtothesetheorieswithequallyaccurateexperimentstotestthesecalculations.Thisaccuracyhasnotbeensoughtsimplytodemonstratetheprowessofphysicsandphysicists;proofoftheexistenceofeachsmallcontributiontotheenergyintheamountpredictedbythetheoryisanindicationofthecorrectnessofourfundamentalideasconcerningthenatureofmatter.§5.1WaveFunctionforMoreThanOneParticleIftheprotonwereofinfinitemass,itwouldbeafixedcenterofforcefortheelectroninthehydrogenatom,andwecouldsolvetheproblembythemethodsofChapter2.Thewavefunctionwouldbeafunctionofthecoordinatesofasingleparticle,theelectron.However,becausetheprotonalsomoves,wemustincorporatethisfactintoourwavefunction.

AccordingtoPostulates1and2(Section3.3),eachdynamicalvariableforeachparticlemustberepresentedbyanoperatorwhoseeigenvaluesaretheallowedvaluesofthevariable.Asalogicalextensionofthesepostulates,wenowassertthattheremastbeawavefunctionforasystemwhichiscapableofgeneratingallthedynamicalvariablesofthesystem.Therefore,forahydrogenatomthewavefunctionmaybewrittenasΦ(xp,yp,zp,xe,ye,ze,t),wherexpisthexcoordinateoftheproton,xeisthexcoordinateoftheelectron,etc.

Itisnowclearthatthewavefunctionissimplyamathematicalconstruction;thereisnophysical“wave”inthesenseofasimpledisplacementthatexistsateachpointofspaceandtime,forthewavefunctionisafunctionofsixspacecoordinatesinthiscase,ratherthanthree.Oursingle-particleproblemsofthepreviouschaptersenabledustomakeusefulanalogieswithconventionalwaves.butwemustnowgointoamoreabstractrealmoftheory.Thisdoesnotmeanthattheproblemsarenecessarilymoredifficulttosolve.Itsimplymeansthatwemustbewareofvisualizationsofthesolutions;

thelimitedexperiencethatwehavegainedthroughoursensesisnotsufficienttopermitthis.Asusual.thewavefunctionisaneigenfunctionoftheSchreodingerequation.TheSchreodingerequation,inturn,containstheoperatorforthetotalenergyofthesystem,asbefore.Thetotalenergyoftheproton-electronsystemisET=pp2/2mp+p2/2me+V(r)(9.1)or,inoperatorformwhereristheproton-electrondistance,mpistheproton'smass,meistheelectron'smass.V(r)=-e2/4πε0ristheCoulombpotential,andthesymbol▽2representsortheequivalentexpressioninsphericalcoordinates;▽p2operatesontheproton'scoordinates,and▽e2operatesontheelectron'scoordinatesinthewavefunction.Thetime-independentSchreodingerequationforthehydrogenatomisthereforebutwhentheequationiswritteninthisform,theenergyeigenvalueETincludesthekineticenergyoftranslationofthecenterofmassofthewholeatom.aquantityinwhichwearenotinterestedatthemoment.Thestatesthattellusaboutthehydrogenspectrumaretheinternalenergystates—statesoftherelativemotionofprotonandelectron.Fortunately,thepotentialenergyisafunctiononlyoftherelativecoordinatesofprotonandelectron,andwecanrewriteEq.(9.3)intermsofthecoordinatesX,Y,andZofthecenterofmassandthecoordinatesx,y,andzoftheelectronrelativetotheproton.Thesecoordinatesarerelatedtothecoordinatesoftheindividualparticlesasfollows:ItisnotdifficulttouseEqs.(9.4)towritethekineticenergyintermsoftherelativevelocityandthevelocityofthecenterofmass;theresultisTotalkineticenergl

=(me+mp)(X2+Y2+Z2)/2+mr(x2+y2+z2)/2 (9.5)wherex,y,andzarethex,y,andzcomponentsoftherelativevelocity,andmr=memp/(me+mp)isthereducedmass,whichwepreviouslyencounteredindiscussingtheBohrmodelofhydrogen.IntermsofmomentumvariablesdefinedasPx=(me+mp)X,px=mrx,etc.,thekineticenergymaybewrittenassothat.accordingtotherulesforwritingtheenergyandmomentumoperators,theSchreodingerequationbecomes

wheretheoperators▽c2and▽2operateoncenterofmassandrelativecoordinates,respectively.Equation(9.7)couldalsohavebeenobtainedbydirecttransformationofthepartialderivativesinEq.(9.3),makinguseofEqs.(9.4)tofindthederivativeswithrespecttothenewvariables.SeparationofVariablesThewavefunctionψisnowafunctionofX,Y,Z,x,y,z,andt.Letusassumethatitispossibletowritethespacedependenceofψasafunctionofx,y,andzmultipliedbyanotherfunctionofX,Y,andZ.Wewriteψ(x,y,z,X,Y,Z,t)=u(x,y,z)U(X,Y,Z)e-i(E+E')t/ћ(9.8)whereEistheenergyoftherelativemotionandE'theenergyoftranslationofthecenterofmass.FromEq.(9.7)wemaynowextractthetwotime-independentequations

•Equation(9.9)issimplytheequationofmotionofthecenterofmassofthewholehydrogenatom;ittellsusnothingabouttheatom'sinternalenergylevels.Equation(9.10)istheSchreodingerequationforthemotionoftheelectronrelativetotheproton.ItisidenticaltotheequationforasingleparticleofmassmrmovingundertheinfluenceofafixedpotentialenergyV(r).AsintheanalystsoftheBohratom,thefactthattheprotonandelectronbothmoveiscompletelyaccountedforbyusingthereducedmassmrinsteadoftheactualmassofthemovingelectron(orproton).TheeigenvaluesEaretheenergylevelsofthehydrogenatomintheframeofreferenceinwhichthecenterofmassoftheatomisatrest.§5.2EnergyLevelofTheHydrogenAtomCoulombPotentialandEffectivePotentialBecauseEq.(9.10)isidenticaltotheone-particleequationtreatedinChapter2,wealreadyknowthatthisequationcanbeseparatedintoanangularequationandaradialequationandthatthefunctionucanbewritteninsphericalcoordinatesasu(r,θ,ф)=R(r)Yl,m(θ,ф),whereYl,m(θ,φ)isasphericalharmonic,asolutionoftheangularequation.

Tofindtheenergylevels,weneedtosolveonlytheradialequation,or,inthiscasewherel(l+1)h2/2mrr2isthe“centrifugal”potential.whoseintroduction,aswesaw(Chapter4),resultsfromeliminatingtheangulardependenceintheequation.

Equation(9.11)isidenticaltotheequationforone-dimensionalmotionofapanicleinthepotentialfieldVeff=V(r)+l(l+1)h2/2mrr2.Thiseffectivepotential,thesumoftheCoulombpotentialandthecentrifugalpotentialVcent,issketchedinFigure9.1.Inhydrogen,anelectronofnegativetotalenergyistrappedinthe“potentialwell”formedbytheeffectivepotential.Classically,theparticlewoulddescribeanellipticalorbitundertheseconditions;itwouldoscillatebetweenthetwovaluesofratwhichitstotalenergywouldbeequaltothepotentialenergy.

FIGURE9.1TheeffectivepotentialVeff(solidline),thesumofthecentrifugalpotentialVcent,andtheCoulombpotentialV(r)=-e2/4πε0r2forthehydrogenatom.Inquantumtheory,wemayfindawavefunctionforthiswelljustaswedidfortheone-dimensionalwellsconsideredbefore.ThegeneralmethodofsolutionofEq.(9.11)beginswiththeassumptionthatthesolutionistheproductofapolynomialandanexponential.DetailsofthissolutionaregiveninAppendixE,whichtreatsthegeneralcaseofaone-electronionwithnuclearchargeZe.Thepolynomialcontainsn–lterms,wherelistheangularmomentumquantumnumberandnisanewquantumnumber,theradialquantumnumber..ThestatementthatR(r)containsn-l,termsmaybeconsideredtobethedefinitionofn.DegeneracyofSolutionsItisacuriousfeatureofthesolutionsthattheenergydependsonlyonn,notonl.Forexample.theenergyisthesameforthel=1solutionwithatwo-termradialsolutionandforthel=2solutionwithaone-termradialsolution;inbothcasesn=3.TheenergylevelswhichresultareidenticaltothelevelspredictedbyBohrforthehydrogenatomoranyone-electronion:althoughnnowhasacompletelydifferentinterpretationfromthatofBohr.Figure9.2ashowstheeffectivepotentialsforl=0,1,and2andtheenergylevelsforn=l,2,3,and4.Becausetheenergydependsonlyonnandnotonl,thesamelevelswhichareallowedforanygivenlarealsoallowedforalllowervaluesofl.Theonlyeffectoflonthelevelsappearsthroughtheconditionthatn≥l+1,sothatlowerenergylevelsarepossibleforsmallerl,becausesmallernvaluesarethenpossible.

FIGURE9.2(a)Effectivepotentialandenergylevelsofthehydrogenatomforl=0,l=1,andl=2.Thelowestfourlevelsareshown;thenumberoflevelsisinfinite.(b)RadialprobabilityamplitudesrRnl(r).NoticethepointsofinflectionwhereE=Veff,attheclassicalturningpointsofthemotion.

Forexample.E3isanenergyeigenvalueforallthreeeffectivewellsshowninFigure9.2a.butE2isaneigenvalueonlyforl=1andl=0.ThefactthatallofthesedifferentwellshavethesamesetofenergylevelsisaremarkablepropertypeculiartotheCoulombpotential.

BecauseEq.(9.11)isidenticaltotheone-dimensionalSchreodingerequation,wecanuseagraphicalanalysis,togainmoreunderstandingoftheformoftheeigentunctions.AswesawinChapter3,theproductrR(r)istheprobabilityamplitudefortheradialcoordinate.Figure9.2bshowsgraphsoftheprobabilityamplitudesrRnl(r)fortheradialprobabilityamplitudesrR20,rR21,andrR31,whoseenergyeigenvaluseareshowninFigure9.2aasE2,E2,andE3,respectively.NoticethattheprobabilityamplitudecurvesawayfromtheaxisintheregionwhereE<Veff

(theclassicallyforbiddenregion)anditcurvestowardtheaxis,tendingtooscillate,whereE>Veff.theclassicalturningpoint,whereE=Veff,isapointofinflectionfortheeigenfunction.YoumayalsonoticethatthefunctionrR31curvesmorerapidlythanrR21intheallowedregion,becausethecurvatureisproportionaltothekineticenergy.Asusual,eacheigenfunctioncontainsonemorenodethantheeigenfunctionimmediatelylowerinenergy.ThispointisreflectedinthefactthatthepolynomialfactorinR(r)hasn-lterms,andthustherearen-lrootstotheequationR(r)=0.BecauseeachofthefunctionsrR(r)isaprobabilityamplitude,itsabsolutesquareistheprobabilitydensityP(r)forfindingtheelectronataradiusbetweenrandr+dr.(SeeSection4.1fordetails.)ThusFigure2showswheretheelectronislikelytobe,asfarasthercoordinateisconcerned.Itindicateshowtheaverageradiusofthe“orbit”increasesasnincreases;itisevidentfromthefigurethatthisaverageradiusmustbeclosetothevaluegivenbytheoriginalBohrtheory.TableofWaveFunctions

Thecompletenormalizedwavefunctionsunlm(r,θ,φ)forthelowestenergystatesofthehydrogenatomaregiveninTable9.1.Thesewavefunctionsalsoapplytoanyone-electronion.ifoneusetheappropriatevaluesfortheatomicnumberZandthenuclearmassM.

TheprobabilitydensitiesassociatedwithsomeofthesewavefunctionsareshowninFigure9.3.TABLE9.1NormalizedwaveFunctionsforHydrogenAtomsandHydrogenlikelonsComparisonwiththeBohrModel:CorrespondencePrincipleandOrbitsinHydrogen

Figure1.7showsellipticalorbitsintheBohrmodelofthehydrogenatom,forn=4.Themajoraxisofeachellipsehasalengthof16a0.Ifyoumeasuretheorbits,youseethatthemosteccentricofthesehasaminimumvalueofrthatislessthana0(0.053nm),andthemaximumvalueofrinthatorbitisgreaterthan31a0,orabout1.64nm.Inthatorbittheangularmomentumisequaltoh,sothiswouldcorrespondton=4andl=1intheSchreodingerequation.ThegraphoftheprobabilitydensityforthispairofquantumnumbersisshowninFigure9.4a.Youcanseethattheclassicallyallowedregiondoesindeedextendfromlessthan0.5nmtogreaterthan1.6nm.

FIGURE9.4RadialprobabilitydensitiesForthreedifferentstatesofthehydrogenatom.(a)Forn=4,l=1,theallowedregionstretchesfromr<0.5nmtor>l.6nm.(b)Forn=4,l=3.thecenteroftheallowedregionisunchanged,buttheregionisnarrower,extendingfromr≌0.6tor≌1.1(c)Forn=100,l=99,theallowedregioniscenteredonr=530nm.theradiusoftheBohrorbitforn=1OO.Itextendsfromr≌490tor≌570.Thisisthenarrowestpossibleallowedregionforthisvalueofn.anditismuchnarrowerthancurve(b),relativetothevalueof<r>.

InFigure9.4b,withthesamenbutl=3,theallowedregionismuchnarrower.becausethelargervalueoflmeansalesseccentricorbit.Foranotherexample,youseeinFigure9.4cthattheallowedregionforn=100,l=99iscenteredonapproximately530nm,justtheBohrradiusforn=100.Yourecallthatthisstatehasthemaximumangularmomentumofallthestateswithn=100,andassuchthefunctionhasnonodes.Noticehowmuchnarrowertheclassicallyallowedregionappearswithn=100.Itisactuallybroader,butasafractionoftheradiusithasbecomesmaller.(SeeExercise9.)SpectroscopicSymbols

Forhistoricalreasonsassociatedwithobservationofthevariousseriesoflinesinatomicspectra,thelvalueofeachstateisdesignatedbyaletter,asfollows:Letter:sPdfghil:0l23456Thelettersgo,inalphabeticalorderforl>3.Eachstateisthenidentifiedbythenunberfornfollowedbytheletterforl:forexample.3dforn=3.l=2§5.3SolutionoftheRadialEquationfortheHydrogenAtomSimplificationoftheradialequationTheradialSchroedingerequationisgivenby

(E1)Tosimplifythesubsequentequations,weintroducethesymbolsWiththesesubstitutionsEq.(E1)becomes(E2)Tosolvethisequation,webeginwiththelimitInthatcase(E3).OnesolutionofthisequationisanditcanbeshownthatanyfunctionoftheformisalsoasolutionofEq.(E3)•Thereforewewrite,whereg(r)isapowerserieswhoseformwenowseek.•SubstitutionintoEq.(E2)yields(E6)SubstitutionofthepowerseriesintoEq.(E6)yieldsthefollowingfourseries:TosatisfyEq.(E6),thesumofthesefourseriesmustequalzeroforallvaluesofr.Thiscanbetrueonlyifthesumofthecoefficientsofeachpowerofriszero.Thelowestpower,withexponents-2,appearsonlytwice.ThesumofthecoefficientsofisthusThereasonablesolutionforsiss=l+1