Coherent states for the hydrogen atom

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1、CoherentStatesfortheHydrogenAtombyJohnR.KlauderDepartmentsofPhysicsandMathematicsUniversityofFlorida,Gainesville,FL32611AbstractThelong-standingproblemofndingcoherentstatesforthe(boundstateportionofthe)hydrogenatomispositivelyresolved.Thestatesinquestion:(i)arenormalizedandparameterizedcon

2、tinuously,(ii)admitaresolutionofunitywithapositivemeasure,and(iii)enjoythepropertythatthetemporalevolutionofanycoherentstatebythehydrogenatomHamiltonianremainsacoherentstateforalltime.Harmonic-oscillatorcoherentstatesInmodernterms,thedistinguishedsetofstatesSchrodingerintroducedforthe12harm

3、onicoscillatorarenowcommonlyknownascoherentstatesandaregiven,forallz2C,Iby1nX1y2z(za?za)?jzj2pjziej0i=ejni;(1)n!n=0ywhere[a;a]=1asusual.Herejni;n=0;1;2;:::,denotenormalizedeigenstatesofthenumberoperatorN,Njni=njni,whichmaybeidentiedwiththeeigenstatesoftheyharmonicoscillatorHamiltonianH=!

4、N=!aa(h=1).Assuchitfollowsthat01n?in!tX12ze?iHt?jzj?i!t02pejzi=ejni=jezi(2)n!n=0illustratingthefactthatthetimeevolutionofanysuchcoherentstateremainswithinthefamilyofcoherentstates;weshallrefertothepropertyembodiedin(2)astemporalstabilityofthecoherentstatesunderH,or,morebrie y,justastempora

5、lstability.0Furthermore,thesestatesareevidentlycontinuousintheirlabelz=x+iyandadmitaresolutionofunitygivenbyZ11=jzihzjdxdy=(3)integratedoverC.IContinuityinthelabelsplusaresolutionofunityestablishthatthe3setfjzigisasetofcoherentstatesinthemodernsenseoftheterm.GivingupthegroupOfcourse,theree

6、xistmanyothersetsofstates,whichwealsorefertoascoherentstates,thatarecontinuousintheirlabelsandadmitaresolutionofunity.Suchstatesmay,forexample,taketheform,forall2C,Igivenby1nX2ji=M(jj)jni(4)pnn=0whereMischosensothathji=1,solongasapositiveweightkexistssuchthatwhenintegratedovertheplan

7、e(=+i)Z211=jihjk(jj)dd=:(5)?iHt?i!t0Evidentlyeji=jeiholdsforall2CIforthisalternativesetofstatesjustaswell.Toensure(5)itsucestohaveZ1n=u(u)du;n(6)02(u)M(u)k(u);4andtogeneratesuchexamplesitiseasiesttochooserst.Asoneexampleofsuchasetp?uof