216 CHAPTER13. ELECTRONSPIN
Notethat(i)σyisanHermitianmatrix;and(ii)theeigenvaluesarebothreal. Aswe
foundlastsemester,theeigenvaluesofanyhermitianoperatorarereal,andeigenstates
correspondingto differenteigenvaluesareorthogonal. So far,wehaveverifiedthe
realityoftheeigenvalues.Nowwesolveforthetwocorrespondingeigenstates,denoted
αy=[
a 1
a 2]
foreigenvalue λ= 1 (13.70)and
βy=[
b 1
b 2]
foreigenvalue λ=− 1 (13.71)Startbysolvingfor
[
0 −i
i 0
][
a 1
a 2]
=[
a 1
a 2][
−ia 2
ia 1]
=[
a 1
a 2]
(13.72)whichhasthesolution
a 2 =ia 1 or αy=[
a 1
ia 1]
(13.73)Finally,wedeterminea 1 bynormalization
1 = αy·αy=|a 1 |^2 +|ia 1 |^2 = 2 |a 1 |^2=⇒ a 1 =1
√
2
(13.74)
so
λ 1 =+1 αy=1
√
2
[
1
i]
(13.75)Theprocedureforλ 2 =− 1 isidentical:
[
0 −i
i 0][
b 1
b 2]
= −[
b 1
b 2][
−ib 2
ib 1]
=[
−b 1
−b 2]
(13.76)Andthistimeb 2 =−ib 1 .Normalizingtodetermineb 1 ,wefind
λ 2 =− 1 βy=1
√
2
[
1
−i]
(13.77)