Quantum Mechanics: Concepts and Applications · Problem 7.25

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · wigner-eckart-theorem-spherical-tensors

Problem

Consider the tensor

A(\theta,\varphi) = \sin^2\theta\cos(2\varphi)

. (a) Calculate the reduced matrix element

\langle 2\parallel Y_2\parallel 2\rangle

. Hint: You may calculate explicitly

\langle 2,\,1\mid Y_{20}\mid 2,\,1\rangle

and then use the Wigner–Eckart theorem to calculate it again. (b) Express

A(\theta,\varphi)

in terms of the components of a spherical tensor of rank 2 (i.e., in terms of

Y_{2m}(\theta,\varphi)

). (c) Calculate

A_{m',\pm 1} = \langle 2,\,m'\mid A\mid 2,\,\pm 1\rangle

for

m' = \pm 2, \pm 1, 0

. You may need this Clebsch–Gordan coefficient:

\langle j,\,2;\,m,\,0\mid j,\,m\rangle = [3m^2 - j(j+1)]/\sqrt{(2j-1)j(j+1)(2j+3)}