QMGreensite_merged

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)