Quantum Mechanics: Concepts and Applications · Problem 7.21

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · wigner-eckart-theorem-quadrupole

Problem

The quadrupole moment operator of a charge

q

is given by

\hat{Q}_{20} = q(3\hat{z}^2 - r^2)

. Write

\hat{Q}_{20}

in terms of an irreducible spherical tensor of rank 2 and then express

\langle j,\,j\mid\hat{Q}_{20}\mid j,\,j\rangle

in terms of

j

and the reduced matrix element

\langle j\parallel r^2 Y^{(2)} \parallel j\rangle

. Hint: You may use the coefficient

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