Quantum Mechanics: Concepts and Applications · Problem 7.28

An $h$-state electron has orbital angular momentum $\ell = 5$, which couples with spin $s = \frac{1}{2}$ to produce total angular momentum $j$. Clebsch–Gordan coefficients tell you how to decompose these coupled $|j, m\rangle$ states back into uncoupled $|\ell, m_\ell\rangle \otimes |s, m_s\rangle$ basis states. The key is recognizing that $m = m_\ell + m_s$ must always hold, and using the triangle rule ($|5 - \frac{1}{2}| \leq j \leq 5 + \frac{1}{2}$) to identify which $j$ values are possible. Then apply the ladder operator method or recurrence relations to find the coefficients systematically.

An $h$-state electron has orbital angular momentum $\ell = 5$, which couples with spin $s = \frac{1}{2}$ to produce total angular momentum $j$. Clebsch–Gordan coefficients tell you how to decompose these coupled $|j, m\rangle$ states back into uncoupled $|\ell, m_\ell\rangle \otimes |s, m_s\rangle$ basis states. The key is recognizing that $m = m_\ell + m_s$ must always hold, and using the triangle rule ($|5 - \frac{1}{2}| \leq j \leq 5 + \frac{1}{2}$) to identify which $j$ values are possible. Then apply the ladder operator method or recurrence relations to find the coefficients systematically.