Introduction to Quantum Mechanics · Problem 6.23

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · clebsch_gordan_coefficients

Problem

The Clebsch–Gordan coefficients are defined by Equation 4.183. Adding together two states with angular momentum

j_1

and

j_2

produces a state with total angular momentum

J

according to

|J\,M\rangle = \sum_{m_1,m_2} C^{j_1 j_2 J}_{m_1 m_2 M}\,|j_1\,j_2\,m_1\,m_2\rangle. \tag{6.64}

(a) From Equation 6.64, show that the Clebsch–Gordan coefficients satisfy

C^{j_1 j_2 J}_{m_1 m_2 M} = \langle j_1\,j_2\,m_1\,m_2 | J\,M\rangle. \tag{6.65}

(b) Apply

\hat{J}_{\pm} = \hat{J}^{(1)}_{\pm} + \hat{J}^{(2)}_{\pm}

to Equation 6.64 to derive the recursion relations for Clebsch–Gordan coefficients:

A^M_J\,C^{j_1 j_2 J}_{m_1 m_2 M+1} = B^{m_1}_{j_1}\,C^{j_1 j_2 J}_{m_1-1\,m_2\,M} + B^{m_2}_{j_2}\,C^{j_1 j_2 J}_{m_1\,m_2-1\,M}

B^M_J\,C^{j_1 j_2 J}_{m_1 m_2 M-1} = A^{m_1}_{j_1}\,C^{j_1 j_2 J}_{m_1+1\,m_2\,M} + A^{m_2}_{j_2}\,C^{j_1 j_2 J}_{m_1\,m_2+1\,M} \tag{6.66}