QMGreensite_merged

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

S% (13.11)