Atomic Structure Theory

24 1 Angular Momentum
Now, let us consider the operator σ·p. Using (1.121), it follows that
σ· [r × p ]
σ· p = σ· ˆr σ· ˆr σ· p = −i σ· ˆr iˆr · p − . (1.124)
r
In deriving this equation, we have made use of the identity in (1.50). From
(1.124), it follows that
df κ +1
σ· p f(r)Ωκm(θ, φ) =i + f Ω−κm(θ, φ) . (1.125)
dr r
This identities (1.123) and (1.125) are important in the reduction of the
central-field Dirac equation to radial form.
1.5.2 Vector Spherical Harmonics
Following the procedure used to construct spherical spinors, one combines
spherical harmonics with spherical basis vectors to form vector spherical harmonics
Y JLM(θ, φ):
Y JLM(θ, φ) =
C(L, 1,J; M − σ, σ, M)YLM−σ(θ, φ)ξσ . (1.126)
σ
The vector spherical harmonics are eigenfunctions of J 2 and Jz. The eigenvalues
of J 2 are J(J + 1), where J is an integer. For J>0, there are three
corresponding values of L: L = J ± 1andL = J. ForJ = 0, the only possible
values of L are L = 0 and L = 1. Explicit forms for the vector spherical
harmonics can be constructed with the aid of Table 1.2. Vector spherical harmonics
satisfy the orthogonality relations
2π
0
dφ
π
0
sin θdθ Y †
J ′ L ′ M ′(θ, φ) YJLM(θ, φ) =δJ ′ JδL ′ LδM ′ M . (1.127)
Vector functions, such as the electromagnetic vector potential, can be expanded
in terms of vector spherical harmonics. As an example of such an
expansion, let us consider
ˆr Ylm(θ, φ) =
aJLMY JLM(θ, φ) . (1.128)
JLM
With the aid of the orthogonality relation, this equation can be inverted to
give
aJLM =
2π
0
dφ
π
sin θdθ Y
0
†
JLM ˆr Ylm(θ, φ).
This equation can be rewritten with the aid of (1.56) as