Copyright by Kirsten Viering 2006 - Raizen Lab - The University of ...

Appendix A
Wigner-Eckart Theorem
When dealing with different angular momenta, their couplings, and transition proba-
bilities, it can be very helpful to use the Wigner-Eckart Theorem. It allows a separation
of a matrix element 〈α ′ j ′ m ′ |T(k, q)|αjm〉 into a scalar matrix element (reduced matrix
element) 〈α ′ j ′ ||T k ||αj〉, which carries the information about the dynamics of the system,
and a Wigner-3j-symbol (Clebsch-Gordan coefficient), that carries the geometrical and
symmetrical properties. The Wigner-3j-symbol therefore includes the selection rules
for radiative transitions.
We will follow the proofs given by Zare [12] and Edmonds [32] and consider
first a rotation of the state vector T(k, q)|α ′ j ′ m ′ 〉,
R [T(k, q) |α ′ j ′ m ′ 〉] = R T(k, q) R −1 R |α ′ j ′ m ′ =
〉
q ′ ,m ′′
D k q ′ q (R) Dj′
m ′′ m ′(R) [T(k, q ′ )|α ′ j ′ m ′′ 〉]
=
D k ⊗ D j′
[T(k, q ′ )|α ′ j ′ m ′′ 〉, (A.1)
q ′ ,m ′′
where Dl m ′ m (R) are the Wigner rotation matrix elements. These can be expressed by
spherical harmonics Y1,q
D 1
q,0 (r) = (−1)q
1/2
4π
Y1,q(θ(r), φ(r)). (A.2)
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