13.1. SPINWAVEFUNCTIONS 213
and,inaddition,
Szψ=
1
2
̄hψ (13.49)
whereSzisthematrixshownin(13.45).Itseasytoseethatthesolutionis
ψp+=ei"p·"x/ ̄h
[
1
0
]
(13.50)
Similarly,aneigenstateofmomentumandSzwithsz=−^12 (”spindown”)willbe
ψp−=ei"p·"x/ ̄h
[
0
1
]
(13.51)
Accordingtoourgeneralrules,the(normalized)superpositionoftwophysicalstates
isalsoaphysicalstate,soingeneralanelectronwavefunctionmusthavetheform
ψ =
∫
d^3 p[f+(p)ψp++f−(p)ψp−]
=
∫
d^3 p
[
f+(p)ei"p·"x
f−(p)ei"p·"x
]
=
[
ψ+(x)
ψ−(x)
]
(13.52)
whereψ+(x)andψ−(x)areanytwosquare-integrablefunctionssatisfyinganormal-
izationcondition
<ψ|ψ> =
∑
i=+,−
∫
d^3 xψi∗(%x)ψi(%x)
=
∫
d^3 x[|ψ+(%x)|^2 +|ψ−(%x)|^2 ]
= 1 (13.53)
Theinterpretationofthetwotermsinthespinorwavefunction,ψ+/−(x),iseasy
toseefromacomputationoftheSzexpectationvalue
<Sz> = <ψ|Sz|ψ>
=
∫
dxdydz[ψ+∗,ψ−∗]
̄h
2
[
1 0
0 − 1
][
ψ+
ψ−
]
=
∫
dxdydz
(
+
1
2
̄h|ψ+|^2 +(−
1
2
̄h)|ψ−|^2
)
= (
1
2
̄h)Prob(spinup)+(−
1
2
̄h)Prob(spindown) (13.54)