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1、Lecture29:QuantumMechanics(Continued)ReviewoTunnelingoSTMoImagingWavefunctionsTodayoHarmonicOscillatoroHydrogenAtomoWavefunctionsoNewQuantumNumbers(l,m)oEigen-values(energies)HarmonicOscillatorConsidersimpleharmonicoscillatoragain.Hereweconsidertwomassesm1andm2attached
2、toanelasticspring.Thepotentialenergyofthesystemdependsonthedegreeofstretchingorcompression:WecanthenimmediatelywritedownthecorrespondingSchrodinger’sequation.wheremisthereducedmassofthesystemgivenby:ThecorrespondingwavefunctionsaresketchedbelowTheenergylevelsaregivenby
3、HarmonicOscillatorAlthoughtheabovewavefunctionsmayappearsinusoidaltheyarebitmorecomplexTheyhavemoreofabell-shapedorgaussianappearance.Alsonotethatforquantumnumberu=0wehaveafiniteenergyknownaszeropointenergy.Thefundamentalfrequencyoftheoscillator,n0isstillgivenbyclassic
4、alvalue:Howeverasbeforetheenergyvaluesarequantized.ThemostinterestingaspectoftheHarmonicoscillatorproblemisthezeropointenergy.Whichmeansatruequantummechanicalharmonicoscillatorwillcontinuetooscillateevenatzerotemperature!HydrogenAtomAswehaveseenpreviously,developmentof
5、quantummechanicswasspurredbytheobservationofthespectrumofhydrogen.Now,weapplytheSchrodingerequationtocalculatethewavefunctionsandenergiesofforthisproblem.Inthisproblem,thepotentialenergyissimplycoulombicinteraction.Thefinalenergiesare:Thecorrespondingwavefunctionisbitm
6、oreinteresting.UnlikeBohrpicturewefindthereare2additionalquantumnumbersassociatedwiththeazimuthalandpolaranglesfandq.(RecallthatBohrhadinitiallyproposedquantizationofangularmomentum).Todescribetheelectronbehaviorinhydrogenatomproblem,weneedtospecifythreequantities,posi
7、tion(r)andtwoanglesi.e.qandf.HydrogenatomWavefunctionsForl=0wehavesphericalsymmetricwavefunctions,thesorbitals.Forl=1,wehaveporbitals;notethex,y,zdependenceoftheporbitalsinthetableabove.Physically,lquantumnumbercanhavevaluesfrom0ton-1,andmquantumnumbercanassumevaluelik
8、e–l,-l+1..0..l-1,l,altogether2l+1values.Thereasonmiscalled,asthemagneticquantumnumber,isthatonlyduringthepresenceofma
