Introduction to Quantum Mechanics · Problem 7.25

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · wigner_eckart_theorem

Problem

Use the Wigner–Eckart theorem (Equations 6.59–6.61) to prove that the matrix elements of any two vector operators,

\mathbf{V}

and

\mathbf{W}

, are proportional in a basis of angular-momentum eigenstates:

\langle n\,\ell'\,m' | \mathbf{V} | n\,\ell\,m \rangle = \alpha \langle n\,\ell'\,m' | \mathbf{W} | n\,\ell\,m \rangle.

Comment: With

\ell

replaced by

j

(the theorem holds regardless of whether the states are eigenstates of orbital, spin, or total angular momentum),

\mathbf{V} = \mathbf{L} + 2\mathbf{S}

and

\mathbf{W} = \mathbf{J}

, this proves Equation 7.77.