13.1. SPINWAVEFUNCTIONS 211
Incolumnvectornotation,thisiswritten
[
v′ 1
v′ 2
]
=[
M 11 M 12
M 21 M 22][
v 1
v 2]
(13.38)Afterthisbriefexcursioninvectoralgebra,letsreturntotheproblemofelectron
spin. Theideaisthatwecanexpressthespinwavefunctionasacolumnvectorof 2
components,knownasaspinor,andexpressthespinoperatorsas 2 × 2 matrices.
Tobeginwith,thebasisvectorsarechosentobethetwoeigenstatesofS^2 , Szfor
s=^12 ,
|e 1 >=|1
2
1
2
> |e 2 >=|1
2
−
1
2
> (13.39)
Inthisbasis,theeigenstatesthemselvescanbewrittenasthecolumnvectors(spinors)
|
1
2
1
2
>↔χ+≡[
1
0]
|1
2
−
1
2
>↔χ−≡[
0
1]
(13.40)andanyspin-^12 statecanbewrittenasasuperposition,inketnotation
|ψ>=ψ+|e 1 >+ψ−|e 2 > (13.41)or,incolumnvectornotation,
|ψ>↔[
ψ+
ψ−]
(13.42)Nowwe canfigureoutwhatthespinoperatorsS^2 ,Sx,Sy,Sz looklike asmatrices,
usingtherelationshipsthatwereproved,inLecture11,foranyoperatorssatisfying
theangularmomentumcommutationrelations(13.12).Writingequations(330)and
(373)fromLecture 11 inket-notation,wehave
S^2 |ssz> = s(s+1) ̄h^2 |ssz>
Sz|ssz> = szh ̄|ssz>
S−|ssz> =√
(s+sz)(s−sz+1) ̄h|s,sz− 1 >S+|ssz> =√
(s−sz)(s+sz+1) ̄h|s,sz+ 1 >Sx =1
2
(S++S−)
Sy =1
2 i(S+−S−) (13.43)
wherewehavejustusedthesymbolSinsteadofL,sinsteadofl,andszinsteadof
m. Fortheelectron,s=^12 andsz=^12 , −^12. Fromtheserelations,weeasilygetall
thematrixelements,e.g.
(Sx) 12 = <e 1 |Sx|e 2 >