207
wheregissomeconstant,knownasthegyromagneticratio. InquantumtheoryS
mustbeaHermitianoperator,andweassumethatitsatisfiesthesamecommutation
relationsasorbitalangularmomentum
[Sx,Sy] = i ̄hSz
[Sy,Sz] = i ̄hSx
[Sz,Sx] = i ̄hSy (13.12)
Fromthesecommutationrelationsalone,weknowfromthediscussioninLecture 11
thatthepossibleeigenvaluesofS^2 andSzare
S^2 = s(s+1) ̄h^2 s= 0 ,
1
2
, 1 ,
3
2
,...
Sz = sz ̄h −s≤sz≤s (13.13)
Takingtheelectronmagneticmomentintoaccount,thetotalHamiltonianisthen
H = H 0 +
e
2 Mc
BzLz−(μe)zBz
H = H 0 +
e
2 Mc
Bz(Lz+gSz) (13.14)
Theelectronspinisindependentoftheelectronpositionandmomentum,therefore
wemayassumethat
[H 0 ,Sz]=[Lz,Sz]= 0 (13.15)
andthismeansthatoperatorsH 0 , Lz, Szhaveacommonsetofeigenstates,which
wedenote|nlmsz>.Then
H|nlmsz> = Enmsz|nlmsz>
Enmsz = En^0 +
e ̄h
2 Mc
Bz(m+gsz) (13.16)
Comparisonto(13.10)showsthatwegetagreementiftheelectronhasanintrinsic
spin
s=
1
2
⇒ sz=±
1
2
(13.17)
andgyromagneticratio
g≈ 2 (13.18)
sothatE+correspondstosz=^12 ,andE−tosz=−^12.
Anindependentcheckofthedouble-valuedcharacteroftheelectronmagneticmo-
mentisprovidedbytheStern-GerlachExperiment,inwhichabeamofelectrons
issentthrougha(non-uniform)magneticfield,oriented(mainly)inthez-direction,
asshowninFig. [13.1].Classically,theforceexertedonadipoleμinanon-uniform
magneticfieldis
F%=∇(μ·B) (13.19)