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)