Principles of Quantum Mechanics · Problem 15.3.3

Addition of Angular Momenta — R. Shankar

Physics — Quantum Mechanics · wigner-eckart-theorem

Problem

(1) Using

\langle jj|J_1|aj\rangle = [j/(j+1)]^{1/2}

show that

\langle aj||J_1||a'j'\rangle = \delta_{aa'}\delta_{jj'}\hbar[j(j+1)]^{1/2}

(2) Using

\mathbf{J}\cdot\mathbf{A} = J_z A_z + \frac{1}{2}(J_- A_+ + J_+ A_-)

(where

A_{\pm} = A_x \pm iA_y

) argue that

\langle a'jm'|\mathbf{J}\cdot\mathbf{A}|ajm\rangle = c\langle a'j||A||aj\rangle

where

c

is a constant independent of

a

a'

and

\mathbf{A}

. Show that

c = \hbar[j(j+1)]^{1/2}\delta_{m,m'}

. (3) Using the above, show that

\langle a'jm'|A^q|ajm\rangle = \frac{\langle a'jm|\mathbf{J}\cdot\mathbf{A}|ajm\rangle}{\hbar^2 j(j+1)}\langle jm'|J^q|jm\rangle \tag{15.3.\ldots}