QMGreensite_merged

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)