206 CHAPTER13. ELECTRONSPIN
whereH 0 istheHydrogenatomHamiltonianintheabsenceofanexternalfield
H 0 =
p^2
2 M
−
e^2
r
(13.6)
Goingovertoquantumtheory,H 0 andLzbecomeoperatorswhich,asithappens,
commutewitheachother:aneigenstateφnlmofH 0 isalsoaneigenstateofLz
Lzφnlm=m ̄hφnlm (13.7)
Therefore,aneigenstateofH 0 isaneigenstateofthetotalHamiltonian
Hφnlm=
(
E^0 n+mBz
e ̄h
2 Mc
)
φnlm (13.8)
withenergyeigenvalues
Enlm=En^0 +mBz
e ̄h
2 Mc
(13.9)
whereEn^0 aretheenergyeigenvaluesoftheelectronwhenthereisnoexternalfield.
Thismeansthatifweinsertacollectionofhydrogenatomsbetweenthepolesof
astrongmagnet,thenspectrallinescorresponding totransitionsbetweenprincipal
quantumnumbersn 1 andn 2 shouldsplitintomanyspectrallines,correspondingto
transitionsbetweenstateswithdifferentvaluesoftheLzquantumnumbersm 1 and
m 2 .Therearecertainselectionrulesgoverningwhichtransitionsarepossible;these
ruleswillbederivedinalaterchapter. Thissplittingofspectrallinesinanexternal
magneticfieldisknownasthe(strong-field)Zeemaneffect.
Theonlytroubleisthatwhentheexperimentisactuallydone,ananalysisofthe
spectrallinesrevealsthatinfacttherearetwiceasmanyenergylevelsasonewould
expect,foreachpairofquantumnumbersnandl. Insteadofthe 2 l+ 1 levelsone
wouldexpect(becausethereare 2 l+ 1 valuesofmforeachl),therearetwosetsof
2 l+ 1 levels,withenergies
E+nlm ≈ En^0 +(m+1)Bz
e ̄h
2 Mc
E−nlm ≈ En^0 +(m−1)Bz
e ̄h
2 Mc
(13.10)
Whatcanpossiblyaccountforthisextrasplitting?
Themostnaturalexplanationisthat,inadditiontothemagneticmomentdueto
itsorbitalangularmomentum,anelectronalsohasanintrinsicmagneticmoment,as-
sociatedwithanintrinsicspinangularmomentum.Classically,afterall,anyspinning
chargeaquiresamagneticmomentinthedirectionofthespinangularmomentum.
Soletussupposethattheintrinsicmagneticmomentoftheelectronis
μ%e=−
eg
2 Mc