A rank-1 (vector) tensor operator has spherical components
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 rank-1 (vector) tensor operator V^q has spherical components q=−1,0,1. It connects a multiplet with j=2 to a multiplet with j′=1, and its matrix elements are written ⟨j′=1,m′∣V^q∣j=2,m⟩ with m=−2,…,2 and m′=−1,0,1.
((a)) How many matrix elements ⟨1,m′∣V^q∣2,m⟩ are there in total, counting over all allowed m, m′ and q?
((b)) Using the Wigner-Eckart theorem, a matrix element vanishes unless m′=m+q. For each of the five values of m, count how many of the three values of q give an m′=m+q that actually lies in {−1,0,1}, and hence find the total number of matrix elements that can be non-zero.
((c)) The theorem also restricts j′ relative to j through the triangle rule on the Clebsch-Gordan coefficient coupling j and k=1 to j′: state the general selection rule this gives for Δj=j′−j for a rank-1 operator, including the case j=0, and confirm that j=2→j′=1 used here is consistent with it.