Introduction: Combining Spins and Orbits
When a system has multiple sources of angular momentum (spin + orbit, or two spins), how do we describe the total? The answer involves the addition of angular momentum.
The Total Angular Momentum
\[\vec{J} = \vec{L} + \vec{S} \quad \text{or} \quad \vec{J} = \vec{J}_1 + \vec{J}_2\]Operators: \(\hat{J}^2\) and \(\hat{J}_z\) for the total.
Allowed Values of j
For two angular momenta \(j_1\) and \(j_2\):
\[j = |j_1 - j_2|, |j_1 - j_2| + 1, \ldots, j_1 + j_2\]Example: \(j_1 = 1\), \(j_2 = 1/2\) → \(j = 1/2\) or \(j = 3/2\)
Two Basis Sets
Uncoupled basis: \(|j_1, m_1; j_2, m_2\rangle\) — good for individual components
Coupled basis: \(|j, m; j_1, j_2\rangle\) — good for total angular momentum
Related by Clebsch-Gordan coefficients.
The Quantum Connection
Angular momentum addition explains fine structure (spin-orbit coupling), hyperfine structure (electron-nuclear spin), and atomic term symbols. It's essential for understanding atomic spectra and selection rules for transitions.