= Solution
<Spin-orbit coupling> is a relativistic interaction between a particle's <spin> and its <orbital angular momentum> or, more generally, its motion through an <electric field>. A moving <Electron> experiences a transformed <magnetic field>, which couples to its <magnetic dipole moment>. For an electron bound in a central electrostatic potential energy $U(r)$, the leading interaction, including the <Thomas precession> correction, is
$$
\boxed{H_{\rm so}=\frac1{2m_e^2c^2r}\frac{dU}{dr}\,\mathbf L\cdot\mathbf S.}
$$
Here $\mathbf L$ is <orbital angular momentum> and $\mathbf S$ is <spin>. The factor of one-half relative to the simplest rest-field argument comes from <Thomas precession> of the accelerated <Electron> frame.<Topics in Quantum Mechanics by David Tong>[§3.1.3, printed pp. 71–75: rest-frame magnetic interaction and Thomas correction]{ref}
For example, in the <hydrogen atom>, the $2p$ states with total <angular momentum> $j=1/2$ and $j=3/2$ have different $\mathbf L\cdot\mathbf S$ <eigenvalues>. Their splitting is part of the atomic <fine structure>. The interaction redistributes and splits spin-dependent energies; it is not the same physical effect as the spin-independent convective magnetic force between charged filaments.
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