A rank-1 (vector) tensor operator connects the and multiplets of a fine-structure doublet
PHYSICS · Subjective · 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(1) connects the j=23 and j=21 multiplets of a fine-structure doublet. Its reduced matrix element is ⟨23∥V^(1)∥21⟩=6.0×10−2ea0, where e is the electron charge and a0 the Bohr radius.Using the Wigner-Eckart theorem in the form ⟨j′,m′∣T^q∣j,m⟩=⟨j,m;k,q∣j′,m′⟩⟨j′∥T^∥j⟩/2j′+1 and the Clebsch-Gordan coefficient ⟨21,21;1,0∣23,21⟩=2/3 (built from the ladder construction for 1⊗21), find ⟨23,21∣V^0(1)∣21,21⟩.