Who first wrote down the spin-orbit coupling term in the atom? Was it Dirac himself? The term should be derived from the Dirac equation.


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We can attribute the term to L. H. Thomas's The Kinematics of an Electron with an Axis (1927), derived a year before the Dirac equation was discovered. He writes

$\mathbf{K}=m[\mathbf{r}\times\mathbf{v}]$ is the angular momentum of revolution, and can be supposed nearly constant during one revolution, so the secular change in the direction of the spin axis is $$\frac{e}{mc}\mathbf{H}+\frac{1}{2}\frac{e^2}{mc^2}\frac{Z}{r^3}\mathbf{K}\tag{6.2}$$

This is for the case of, as Thomas puts it, a "Coulomb field"; $\mathbf{H}$ is an external magnetic field, and should not be confused with the Hamiltonian. The precession in this case is then $$\mathbf{\sigma}=\frac{1}{2}\frac{e^2}{mc^2}\frac{Z}{r^3}\mathbf{K}$$ The change in the Hamiltonian from the precession is then just $$\Delta H_T=\mathbf{\sigma}\cdot\mathbf{S}$$ although we could also apply this to $\text{(6.2)}$ in its entirety to get the full $\Delta H$.


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