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1、Contents3FineStructureandLambShift23.1FineStructure...................................23.1.1Kineticcontribution............................33.1.2Spin-OrbitInteraction...........................33.1.3TheDarwinTerm.............................43.1.4Evaluationo
2、fthefinestructureinteraction...............43.2TheLambShift...................................41Chapter3FineStructureandLambShift3.1FineStructureImmediatelyadjacenttoMichelsonandMorley’sannouncementoftheirfailuretofindtheetherinan1887issueofthePhilosophicalJou
3、rnalisapaperbythesameauthorsreportingthattheHlineofhydrogenisactuallyadoublet,withaseparationof0.33cm−1.In1915αBohrsuggestedthatthis“finestructure”ofhydrogenisarelativisticeffectarisingfromthevariationofmasswithvelocity.Sommerfeld,in1916,solvedtherelativisticK
4、eplerproblemandusingtheoldquantumtheory,asitwaslaterchristened,accountedpreciselyforthesplitting.Sommerfeld’stheorygavethelietoEinstein’sdictum“TheGoodLordissubtlebutnotmalicious”,foritgavetherightresultsforthewrongreason:histheorymadenoprovisionforelectrons
5、pin,anessentialfeatureoffinestructure.Today,allthatisleftfromSommerfeld’stheoryisthefinestructureconstantα=e2/¯hc.ThetheoryforthefinestructureinhydrogenwasprovidedbyDiracwhoserelativisticelectrontheory(1926)wasappliedtohydrogenbyDarwinandGordonin1928.Theyfoundt
6、hefollowingexpressionfortheenergyofanelectronboundtoaprotonofinfinitemass:⎡⎤2E1=⎣⎦(3.1)mc2αZ√n−k+k2−α2Z2wherenistheprincipalquantumnumber,k=j+1/2,andj=±1/2.TheDiracequationisnotnearlyasilluminatingasthePauliequation,whichistheapproximationtotheDiracequation
7、tothelowestorderinv/c.p2e2H=mc2+−+H(3.2)FS2mrThefirsttermistheelectron’srestenergy;thefollowingtwotermsarethenonrelativisticHamiltonian,andthelastterm,thefinestructureinteraction,isgivenbyp4¯h2e21¯h2e2H=−+L·S−∇2(3.3)FS8m3c22m2c2r38m2c2rTherelativisticcontribut
8、ionscanbedescribedasthekinetic,spin-orbit,andDarwinterms,Hkin,Hso,andHDar,respectively.Eachhasastraightforwardphysicalinterpretation.2M.I.T.DepartmentofPhysics,8.421–Spring200633.1.1Kineticcontr
