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Mathematics for Physics

146 Spin-Orbit Coupling

An interaction between a particle’s orbital angular momentum and its spin that is used to split levels that share the same \(n\) and \(\ell\), where spin is the particle’s intrinsic angular momentum.

The \(\mathbf{L}\cdot\mathbf{S}\) Hamiltonian. The spin-orbit energy is proportional to \(\mathbf{L}\cdot\mathbf{S}\). This principle is used to write the fine-structure perturbation.

The spin-orbit Hamiltonian is

\[ H_{\mathrm{SO}} = \xi(r)\,\mathbf{L}\cdot\mathbf{S} \]

where

  • \(H_{\mathrm{SO}}\) is the spin-orbit Hamiltonian.
  • \(\xi(r)\) is a radial coupling strength.
  • \(\mathbf{L}\) is the orbital angular momentum.
  • \(\mathbf{S}\) is the spin.

The \(J^{2}\) identity. The product \(\mathbf{L}\cdot\mathbf{S}\) is diagonal in the \(|j,m_{j}\rangle\) basis. This principle is used to replace \(\mathbf{L}\cdot\mathbf{S}\) by a combination of \(J^{2}\), \(L^{2}\), and \(S^{2}\).

The identity for \(\mathbf{L}\cdot\mathbf{S}\) is

\[ \mathbf{L}\cdot\mathbf{S} = \dfrac{1}{2}\bigl(J^{2}-L^{2}-S^{2}\bigr) \]

where

  • \(\mathbf{J}=\mathbf{L}+\mathbf{S}\) is the total angular momentum.

The fine-structure splitting. For one electron the two \(j\) values \(\ell\pm 1/2\) are split by the spin-orbit term. This principle is used to explain fine-structure doublets.

146.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — spin-orbit coupling.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — \(\mathbf{L}\cdot\mathbf{S}\) term.