Principles of Quantum Mechanics · Problem 15.3.2

Addition of Angular Momenta — R. Shankar

Physics — Quantum Mechanics · scalar-operator-tensor

Problem

It is claimed that

\sum_q (-1)^q S_k^q T_k^{-q}

is a scalar operator. (1) For

k = 1

, verify that this is just

\mathbf{S}\cdot\mathbf{T}

. (2) Prove it in general by considering its response to a rotation. [Hint:

D^{(j)}_{m,-m'} = (-1)^{m-m'}(D^{(j)}_{m',m})^*