Quantum Mechanics: Concepts and Applications · Problem 7.16

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · rotation-matrices-addition-of-angular-momenta

Problem

Using

\hat{J} = \hat{J}_1 + \hat{J}_2

show that

d^{(j)}_{mm'}(\beta) = \sum_{m_1 m_2}\sum_{m'_1 m'_2} \langle j_1,j_2;\,m_1,m_2\mid j,\,m\rangle\langle j_1,j_2;\,m'_1,m'_2\mid j,\,m'\rangle\, d^{(j_1)}_{m_1 m'_1}(\beta)\,d^{(j_2)}_{m_2 m'_2}(\beta).