Atomic Structure Theory

The matrix s 2 = s 2 x + s 2 y + s 2 z is
1.2 Spin Angular Momentum 11
s2 ⎛ ⎞
200
= ⎝ 020⎠
. (1.52)
002
The three matrices sx,sy, andsz satisfy the commutation relations
[sx,sy] =isz, [sy,sz] =isx, [sz,sx] =isy. (1.53)
It follows that S =¯hs satisfies the angular momentum commutation relations
(1.4).
Eigenfunctions of S 2 and Sz satisfy the matrix equations s 2 ξµ =2ξµ and
szξµ = µξµ. The first of these equations is satisfied by an arbitrary threecomponent
vector. Solutions to the second are found by solving the corre-
sponding 3 × 3 eigenvalue problem,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 −i 0 a a
⎝ i 0 0⎠⎝b
⎠ = µ ⎝ b ⎠ . (1.54)
0 0 0 c c
The three eigenvalues of this equation are µ = −1, 0, 1 and the associated
eigenvectors are
ξ1 = − 1
⎛ ⎞ ⎛ ⎞
1
0
√ ⎝ i ⎠ , ξ0 = ⎝ 0 ⎠ , ξ−1 =
2 0
1
1
⎛ ⎞
1
√ ⎝ −i ⎠ . (1.55)
2 0
The phases of the three eigenvectors are chosen in accordance with (1.18),
which may be rewritten s+ ξµ = √ 2 ξµ+1. The vectors ξµ are called spherical
basis vectors. They satisfy the orthogonality relations
ξ † µξν = δµν.
It is, of course, possible to expand an arbitrary three-component vector v =
(vx,vy,vz) in terms of spherical basis vectors:
v =
1
µ=−1
v µ ξµ, where v µ = ξ † µv .
Using these relations, one may show, for example, that the unit vector ˆr
expressed in the spherical basis is
ˆr =
4π
3
1
µ=−1
Y ∗
1,µ(θ, φ)ξµ . (1.56)