Atomic Structure Theory

22 1 Angular Momentum
1.5 Spinor and Vector Spherical Harmonics
In constructing atomic wave functions, it is often necessary to combine the orbital
wave functions given by spherical harmonics and the spin wave functions
to obtain eigenfunctions of J = L + S. Similarly, when considering electromagnetic
interactions, it is often necessary to combine the spin and orbital angular
momentum components of the vector potential to obtain eigenfunctions
of J 2 and Jz. Combining spin 1/2 eigenfunctions with spherical harmonics
leads to spherical spinors, while combining spin 1 eigenfunctions with spherical
harmonics leads to vector spherical harmonics. These combined spin-angle
functions are discussed in the following two subsections.
1.5.1 Spherical Spinors
Spherical spinors, which are eigenstates of J 2 and Jz, are formed by combining
spherical harmonics Ylm(θ, φ), which are eigenstates of L 2 and Lz, and
spinors χµ, which are eigenstates of S 2 and Sz. We denote spherical spinors
by Ωjlm(θ, φ); they are defined by the equation
Ωjlm(θ, φ) =
C(l, 1/2,j; m − µ, µ, m)Yl,m−µ(θ, φ)χµ . (1.115)
µ
Using the explicit forms of the Clebsch-Gordan coefficients given in Table 1.1,
we obtain the following explicit formulas for spherical spinors having the two
possible values, j = l ± 1/2:
⎛
l+m+1/2
Ωl+1/2,l,m(θ, φ) = ⎝ 2l+1 Yl,m−1/2(θ, φ)
l−m+1/2
2l+1 Y ⎞
⎠ , (1.116)
l,m+1/2(θ, φ)
⎛
Ωl−1/2,l,m(θ, φ) = ⎝ −
l−m+1/2
2l+1 Yl,m−1/2(θ, φ)
l+m+1/2
Y ⎞
⎠ . (1.117)
l,m+1/2(θ, φ)
2l+1
Spherical spinors are eigenfunctions of σ·L and, therefore, of the operator
The eigenvalue equation for K is
K = −1 − σ·L.
KΩjlm(θ, φ) =κΩjlm(θ, φ) , (1.118)
where the (integer) eigenvalues are κ = −l − 1forj = l +1/2, and κ = l
for j = l − 1/2. These values can be summarized as κ = ∓(j +1/2) for
j = l ± 1/2. The value of κ determines both j and l. Consequently, the more
compact notation Ωκm ≡ Ωjlm can be used. The states associated with l and
j are designated by the spectroscopic notation s 1/2, p 1/2, p 3/2, ... where the