Quantum Mechanics: Concepts and Applications · Problem 7.24

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · rotation-matrices

Problem

This problem deals with another derivation of the matrix elements of

d^{(1)}(\beta)

. Use the relation

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)

for the case where

j_1 = j_2 = \frac{1}{2}

along with the matrix elements of

d^{(1/2)}(\beta)

, which are given by (7.89), to derive all the matrix elements of

d^{(1)}(\beta)