13.1. SPINWAVEFUNCTIONS 209
pictureisinadequate,andinanycasetherearenosphericalharmonicswiths=^12.
So...whattodo?
Forthemomentletsforgetallaboutthex,y,z-degreesoffreedomoftheparticle,
andconcentratejustonthespin. Now,althoughwedon’thavesphericalharmonics
fors= 1 /2,wecanstillrepresenttheeigenstatesofS^2 andSzbytwoorthonormal
ketvectors
|s=1
2
,sz=1
2
> and |s=1
2
,sz=−1
2
> (13.23)
where
S^2 |
1
2
1
2
> =
3
4
̄h^2 |1
2
1
2
>
Sz|1
2
1
2
> =
1
2
̄h|1
2
1
2
>
S^2 |
1
2
−
1
2
> =
3
4
̄h^2 |1
2
−
1
2
>
Sz|1
2
−
1
2
> = −
1
2
̄h|1
2
−
1
2
> (13.24)
Then(ifwedisregardpositiondependence)anys=^12 statecanberepresentedasa
superposition
|ψ>=a|1
2
1
2
>+b|1
2
−
1
2
> (13.25)
Nowthislooksexactlylikethewaywewouldrepresentavectorinatwo-dimensional
space. Sobeforegoingon,itsworthrecallingafewfactsofvectoralgebra.
Let%e 1 and%e 2 betwoorthonormalvectors(i.e. orthogonalunitvectors)inatwo
dimensionalspace. Forexample,%e 1 and%e 2 couldbeunitvectorsalongthexandy
axes,respectively. Orthonormalitymeansthat
%e 1 ·e% 1 = 1
%e 2 ·e% 2 = 1
%e 1 ·e% 2 = 0 (13.26)IfwehaveasetofDorthonormalvectorsinaD-dimensionalspace, thosevectors
areknownasabasisforthevectorspace,andanyvectorcanbeexpressedalinear
combinationofthosebasisvectors.Inthecaseweareconsidering,D=2,anyvector
canbewrittenintheform
%v=a%e 1 +be% 2 (13.27)
Itisusefulandtraditionaltorepresenta(ket)vectorinafinitedimensionalspaceas
acolumnofnumbers,i.e.
%v↔[
a
b]
(13.28)