量子力学-斯坦福大学-Chapter09-08.ppt

Non-degeneratePerturbationTheory,Problem:,cantsolveexactly.,CopyrightMichaelD.Fayer,2007,Solutionsof,complete,orthonormalsetofstates,witheigenvaluesand,Kroneckerdelta,CopyrightMichaelD.Fayer,2007,Expandwavefunction,and,CopyrightMichaelD.Fayer,2007,alsohave,Sumofinfinitenumberoftermsforallpowersoflequals0.,Coefficientsoftheindividualpowersoflmustequal0.,CopyrightMichaelD.Fayer,2007,Firstordercorrection,Wanttofindand.,Expand,Then,Aftersubstitution,CopyrightMichaelD.Fayer,2007,Aftersubstitution,Leftmultiplyby,CopyrightMichaelD.Fayer,2007,Wehave,Then,CopyrightMichaelD.Fayer,2007,Firstordercorrectiontothewavefunction,Againusingtheequationobtainedaftersubstitutingseriesexpansions,Leftmultiplyby,Equalszerounlessi=j.,Coefficientsinexpansionofketintermsofthezerothorderkets.,CopyrightMichaelD.Fayer,2007,isthebracketofwithand.,CopyrightMichaelD.Fayer,2007,Firstordercorrections,CopyrightMichaelD.Fayer,2007,SecondOrderCorrections,Usingl2coefficient,Expanding,Substitutingandfollowingsametypeofproceduresyields,l2coefficientshavebeenabsorbed.,SecondordercorrectionduetofirstorderpieceofH.,SecondordercorrectionduetoanadditionalsecondorderpieceofH.,CopyrightMichaelD.Fayer,2007,EnergyandKetCorrectedtoFirstandSecondOrder,CopyrightMichaelD.Fayer,2007,Example:x3andx4perturbationoftheHarmonicOscillator,Vibrationalpotentialofmoleculesnotharmonic.Approximatelyharmonicnearpotentialminimum.Expandpotentialinpowerseries.,Firstadditionaltermsinpotentialafterx2termarex3andx4.,CopyrightMichaelD.Fayer,2007,perturbationcandqareexpansioncoefficientslikel.,CopyrightMichaelD.Fayer,2007,InDiracrepresentation,Firstconsidercubicterm.,CopyrightMichaelD.Fayer,2007,hastermswithsamenumberofraisingandloweringoperators.,Therefore,CopyrightMichaelD.Fayer,2007,Sumofthesixterms,Therefore,With,CopyrightMichaelD.Fayer,2007,PerturbationTheoryforDegenerateStates,and,normalizeandorthogonal,and,Degenerate,sameeigenvalue,E.,Anysuperpositionofdegenerateeigenstatesisalsoaneigenstatewiththesameeigenvalue.,CopyrightMichaelD.Fayer,2007,nlinearlyindependentstateswithsameeigenvaluesystemn-folddegenerate,Canformaninfinitenumberofsetsof.Nothinguniqueaboutanyonesetofndegenerateeigenkets.,Canformnorthonormal,CopyrightMichaelD.Fayer,2007,Wantapproximatesolutionto,zerothorderHamiltonian,perturbation,zerothordereigenket,zerothorderenergy,CopyrightMichaelD.Fayer,2007,Hereisthedifficulty,perturbedket,zerothorderkethavingeigenvalue,CopyrightMichaelD.Fayer,2007,Tosolveproblem,ExpandEand,Somesuperposition,butwedontknowthecj.Dontknowcorrectzerothorderfunction.,CopyrightMichaelD.Fayer,2007,Tosolve,substitute,CopyrightMichaelD.Fayer,2007,thispiecebecomes,Leftmultiplyingby,CopyrightMichaelD.Fayer,2007,CorrectiontotheEnergies,Twocases:im(thedegeneratestates)andim.,CopyrightMichaelD.Fayer,2007,isasystemofmofequationsforthecjs.,CopyrightMichaelD.Fayer,2007,Solvemthdegreeequationgetthe.Nowhavethecorrectionstoenergies.,Tofindthecorrectzerothordereigenvectors,oneforeach,substitute(oneatatime)intosystemofequations.,Getsystemofequationsforthecoefficients,cjs.,Thereareonlym1conditionsbecausecanmultiplyeverythingbyconstant.Usenormalizationformthcondition.,Nowwehavethecorrectzerothorderfunctions.,Knowthe.,CopyrightMichaelD.Fayer,2007,Thesolutionstothemthdegreeequation(expandingdeterminant)are,Therefore,tofirstorder,theenergiesoftheperturbedinitiallydegeneratestatesare,Havemdifferent(unlesssomestilldegenerate).,CopyrightMichaelD.Fayer,2007,Correctiontowavefunctions,Againusingequationfoundsubstitutingtheexpansionsintothefirstorderequation,CopyrightMichaelD.Fayer,2