212 CHAPTER13. ELECTRONSPIN
= <
1
2
1
2
|Sx|
1
2
−
1
2
>
=
1
2
[
<
1
2
1
2
|S+|
1
2
−
1
2
>+<
1
2
1
2
|S−|
1
2
−
1
2
>
]
=
1
2
[
<
1
2
1
2
| ̄h|
1
2
1
2
>+0
]
=
1
2
̄h (13.44)
Aftercomputingalltheneededcomponents,thematrixformofthespinoperators,
forspins=^12 ,are
S^2 =
3
4
̄h^2
[
1 0
0 1
]
Sz=
1
2
̄h
[
1 0
0 − 1
]
Sx =
1
2
̄h
[
0 1
1 0
]
Sy =
1
2
̄h
[
0 −i
i 0
]
(13.45)
Itisusefultoextractacommonfactorof ̄h 2 ,andtowrite
Sx=
̄h
2
σx Sy=
̄h
2
σy Sz=
̄h
2
σz (13.46)
wherethematrices
σx=
[
0 1
1 0
]
σy=
[
0 −i
i 0
]
σz=
[
1 0
0 − 1
]
(13.47)
areknownasthePauliSpinMatrices.
Exercise: Consideraparticlewithspins=1. Findthematrixrepresentationof
S^2 ,Sx,Sy,SzinthebasisofeigenstatesofS^2 ,Sz,denoted
|ssz>=| 11 >,| 10 >,| 1 − 1 >
Nextwemusttakeaccountofthespatialdegreesoffreedom. Supposethewave-
functionψ is an eigenstateof momenta andan eigenstateof Sz, witheigenvalue
sz=^12 ̄h(”spinup”).Thenwerequire,asusual,
−i ̄h∂xψ = pxψ
−ih ̄∂yψ = pyψ
−i ̄h∂zψ = pzψ (13.48)