Mathematical Analysis of the Photoelectric Effect

ESITheErwinShrodingerInternationalBoltzmanngasse9InstituteforMathematialPhysisA-1090Wien,AustriaMathematialAnalysisofthePhotoeletriEetVolkerBahFrederiKloppHeribertZenkVienna,PreprintESI1121(2002)January30,2002SupportedbyFederalMinistryofSieneandTransport,AustriaAvailableviaanonymousftpfromFTP.ESI.AC.ATorvia:¨at;D-55099Mainz;Germany;email:[e-mail]´ed´ericKloppLAGA,InstitutGalil´ee;Universit´eParis-Nord;F-93430Villetaneuse;Franceemail:[e-mail]ik;JohannesGutenberg-Universit¨at;D-55099Mainz;Germany;email:[e-mail]1,2001AbstractWestudythephotoelectriceffectontheexampleofasimplifiedmodelofanatomwithasingleboundstate,coupledtothequantizedelectromagneticfield.Forthismodel,weshowthatEinstein’spredictionforthephotoelectriceffectisqualitativelyandquantitativelycorrecttoleadingorderinthecou-plingparameter.Morespecifically,consideringtheionizationoftheatombyanincidentphotoncloudconsistingofNphotons,weprovethatthetotalionizedchargeisadditiveintheNinvolvedphotons.Furthermore,ifthephotoncloudisapproachingtheatomfromalargedistanceorismonochro-matic,thekineticenergyoftheejectedelectronisshowntobegivenbythedifferenceofthephotonenergyofeachsinglephotoninthephotoncloudandtheionizationenergy.MSC:81Q10,81V10,47N50.Keywords:PhotoelectricEffect,ScatteringTheory,QED.BKZ-Sep1,20011IIntroductionThephotoelectriceffectwasdiscoveredinincreasinglypreciseexperimentsbyHertz[15,1887],Hallwachs[14,1888],Lenard[16,1902]1,andMillikan[19,18,1916].Itwasobservedthat,whenlightisincidentonametalsurface,electronsareejectedfromthesurface.ThestrikingfactaboutthisphenomenonistheseeminglyodddependenceofthemaximalkineticenergyTmaxoftheejectedelectronsonthefrequencyofthelightanditsindependenceofthelightintensity.Thelattercontradictstheprinciplesofclassicalphysics,andin1905,Einsteinsuggestedanexplanationofthisphenomenon[8,1905]whichexplicitlyinvolvesthequantumnatureofelectromagneticradiation.HefoundthatTmax=hE;(I.1)providedthatthefrequencyofthelighttimesPlanck’sconstanth,i.e.,thepho-tonenergyhislargerthanthe(materialdependent)workfunctionE.Con-versely,ifhE;(I.2)thennoelectronsleavethemetalsurface.OurultimategoalisthederivationofEinstein’spredictions,Eqs.(I.1)–(I.2),fromfirstprinciplesofquantummechanicsandquantumfieldtheory.Inthepresentpaper,weanalyzeasimplifiedmodelwhichisfarfromtheap-propriatemodelforametalinteractingwithelectromagneticradiationandcan,atbest,beregardedasacaricatureofahydrogenatominteractingwiththera-diationfield.Yet,itcontainsmanyofthemathematicaldifficultiesweexpecttoencounterintheanalysisofamorerealisticmodel,andweproveEqs.(I.1)–(I.2)forthissimplifiedmodel.Ouremphasisliesinthefollowingaspects:GiventhemodelasdescribedinSect.I.1,below,ourderivationismath-ematicallyrigorous,andnounjustifiedapproximationsareused.Toourknowledge,thepresentpaperisthefirsttotreatthephotoelectriceffectwithmathematicalrigor.Wedrawfrommanyfactsaboutnonrelativisticquantumelectrodynamicswhichhavebeenpreviouslyestablishedin[2,3,4,5].Ourresultsarerelatedtoscatteringtheoryandtheasymptoticsoftheuni-tarytimeevolutionoperatorformodelsofnonrelativisticquantumelectro-dynamics.Resultsinthiscontext,butonotheraspectscanbefoundin[1,6,9,10,11,20].1Insomephysicstextbooks,e.g.,[13],Lenardisnotmentioned,anditseemsthatin1905,EinsteinderivedhisfamoustheoryfromnothingbutaGedankenexperiment.BKZ-Sep1,20012Whileourmodelfortheparticlesystem,i.e.,themetaloratom,isacrudemodelinvolvingonlyasingle,spinlesselectron,theparticlesystemiscou-pledtoaquantizedscalarfield.(Thedifferencebetweenthequantized[vec-tor]electromagneticandaquantizedscalarfieldisirrelevant,forthescopeofthiswork.Forcertainotherfactsinnonrelativisticquantumelectrody-namics,however,thisdifferenceiscrucial,see,e.g.,[4].)Ourmodeldescribesasingleatomratherthanametalorevenagasin-teractingwiththeelectromagneticfield.ThusourderivationofEqs.(I.1)–(I.2)showsthatthephotoelectriceffectisnotacollective,statisticalphe-nomenon,visibleonlyifmanyparticlesormanyphotonsareinvolved.Mosttextsonlasertheoryorquantumoptics,e.g.,[7,12],immediatelyproceedtoastatisticaldescriptionofboththemetalorgasofatoms,say,andthephotonfield,andthequestion,whetherthisisreallynecessaryormerelyamatterofmathematicalconvenience,isleftopen.Withinourframework,weproveEqs.(I.1)–(I.2)tobecorrectinleadingnonvanishingorderinthecouplingparametergwhich,inappropriateunits,equalsp23=2,where1=137isthefinestructureconstant.Moreprecisely,giventheinteractingatomatrestplusanincidentphotoncloudconsistingofNphotons,weshowthatinleadingorderthecontributiontothechargeejectedfromtheatomisadditiveineachphotonandindependentofallotherphotonsoftheincidentphotoncloud.Moreover,weprovethatthiscontributionisinaccordancewith(I.1)–(I.2)incasethatthephotonsareinanincomingscatteringstateorinthelimitofmonochromaticlight,i.e.,whenthesupportofeachphotonissharplylocalizedonenergyshellsinphotonmomentumsp