Use the WignerEckart theorem (Equations 6.596.61) to prove that the matrix elements of any two vector operators,

Question:

Use the Wigner–Eckart theorem (Equations 6.59–6.61) to prove that the matrix elements of any two vector operators, V and W, are proportional in a basis of angular-momentum eigenstates:

Comment: With ℓ replaced by j (the theorem holds regardless of whether the states are eigenstates of orbital, spin, or total angular momentum), V = L + 2S and W = J, this proves Equation 7.77.