Quantum Mechanics: Concepts and Applications · Problem 7.10

Identical Particles — Nouredine Zettili

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

Problem

(a) 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 = 1

and

j_2 = \frac{1}{2}

along with the Clebsch–Gordan coefficients derived in (7.206) to (7.209), and the matrix elements of

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

and

d^{(1)}(\beta)

, which are given by (7.89) and (7.423), respectively, to find the expressions of the matrix elements of

d^{(3/2)}_{\frac{3}{2}\frac{3}{2}}(\beta)

d^{(3/2)}_{\frac{3}{2}\frac{1}{2}}(\beta)

d^{(3/2)}_{\frac{3}{2}-\frac{1}{2}}(\beta)

d^{(3/2)}_{\frac{3}{2}-\frac{3}{2}}(\beta)

d^{(3/2)}_{\frac{1}{2}\frac{1}{2}}(\beta)

, and

d^{(3/2)}_{\frac{1}{2}-\frac{1}{2}}(\beta)

. (b) Use the six expressions derived in (a) to infer the matrix of

d^{(3/2)}(\beta)