A single valence electron has orbital angular momentum and spin
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Symmetry, angular momentum and spin
The Structure of Quantum Mechanics, Ch Adding angular momenta, tensor operators and Wigner-Eckart
Problem
A single valence electron has orbital angular momentum l=2 and spin s=21, giving the fine-structure levels j=25 (six states, m=−25,…,25) and j=23 (four states, m=−23,…,23), both built from the same l,s pair. A weak magnetic field couples to the electron's magnetic moment through the factor gJ obtained from the projection theorem.
((a)) Compute gJ for the j=25 level and for the j=23 level.
((b)) A magnetic-dipole transition between the two levels is described by a rank-1 (vector) irreducible tensor operator V^q(1), q=0,±1. Of the 6×4×3=72 quantities ⟨23,m′∣V^q(1)∣25,m⟩, state how many can possibly be nonzero, using only the selection rule that follows from the Wigner-Eckart theorem, with no matrix element evaluated.
((c)) State how many independent unknown numbers are needed to specify every allowed matrix element in part (b), and name what that number is called.