Chapter 14 Key Takeaways: Addition of Angular Momentum

Core Message

Two angular momenta $\hat{\mathbf{J}}_1$ and $\hat{\mathbf{J}}_2$ combine to form a total angular momentum $\hat{\mathbf{J}} = \hat{\mathbf{J}}_1 + \hat{\mathbf{J}}_2$ with allowed values $J = |j_1 - j_2|, |j_1 - j_2| + 1, \ldots, j_1 + j_2$. The Clebsch-Gordan coefficients provide the unitary transformation between the uncoupled and coupled bases, and the Wigner-Eckart theorem leverages this machinery to determine the $m$-dependence of all matrix elements of irreducible tensor operators from a single reduced matrix element.

Key Concepts

1. Two Bases for the Same Space

The tensor product space $\mathcal{H}_{j_1} \otimes \mathcal{H}_{j_2}$ can be described in either the uncoupled basis $|j_1, m_1; j_2, m_2\rangle$ (eigenstates of $\hat{J}_1^2, \hat{J}_{1z}, \hat{J}_2^2, \hat{J}_{2z}$) or the coupled basis $|J, M\rangle$ (eigenstates of $\hat{J}_1^2, \hat{J}_2^2, \hat{J}^2, \hat{J}_z$). Both are complete and orthonormal; the choice depends on which operators commute with the Hamiltonian.

2. The Triangle Rule

The allowed total angular momentum quantum numbers satisfy $|j_1 - j_2| \leq J \leq j_1 + j_2$, stepping in integer increments. The total number of coupled states equals $(2j_1 + 1)(2j_2 + 1)$, matching the uncoupled basis dimension.

3. Clebsch-Gordan Coefficients

The CG coefficients $\langle j_1, m_1; j_2, m_2 | J, M\rangle$ are the matrix elements of the unitary transformation between bases. They are real (Condon-Shortley convention), vanish unless $M = m_1 + m_2$, and are computed by the recursion/lowering operator method.

4. The Wigner-Eckart Theorem

For irreducible tensor operators of rank $k$: $\langle j', m' | \hat{T}_q^{(k)} | j, m\rangle = \frac{\langle j' \| T^{(k)} \| j \rangle}{\sqrt{2j'+1}} \langle j, m; k, q | j', m'\rangle$. All $m$-dependence lives in the CG coefficient; the physics is in the reduced matrix element.

5. Selection Rules

The CG coefficient in the Wigner-Eckart theorem immediately gives $\Delta m = q$ and $|j - k| \leq j' \leq j + k$. For electric dipole transitions ($k = 1$, odd parity): $\Delta\ell = \pm 1$, $\Delta j = 0, \pm 1$, $\Delta m = 0, \pm 1$.

Key Equations

CG Coefficients: Quick Reference

$\frac{1}{2} \otimes \frac{1}{2}$ (Two spin-1/2 particles)

Pattern: Triplet ($S = 1$) is symmetric; singlet ($S = 0$) is antisymmetric.

$1 \otimes \frac{1}{2}$ (Spin-orbit coupling, $\ell = 1$)

Decision Framework: Uncoupled vs. Coupled Basis

Common Misconceptions

Looking Ahead