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The electron of a hydrogen atom or any hydrogenic ion sees an internal magnetic field $\vec{B}_{\rm int}$ due to the proton or the nucleus in relative motion w.r.t the electron. The spin magnetic moment $\vec{\mu}_s$ of the electron couples to $\vec{B}_{\rm int}$ to give rise to an interaction of the form $f(r)\vec{S}\cdot\vec{L}$. However, spin-orbit interaction also exists in complicated many-electron atoms as well as in the nuclei. How can I explain someone the physical picture of the origin of this interaction in these situations?

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A moving electron, or other spinning particle, appears to have an electric dipole moment as well as a magnetic one. This experiences the potential of the nucleus, other electrons or charged particles.

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  • $\begingroup$ Is your answer about many-electron atoms? $\endgroup$ Feb 18, 2020 at 20:06
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    $\begingroup$ It is about any particle with a magnetic dipole moment. $\endgroup$
    – my2cts
    Feb 18, 2020 at 20:33
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As long as there is finite $L$ for the electron then there is a magnetic field felt by it. The origin of the interaction (magnetic field) is the same as before. The orbital angular momentum and the coulomb potential. Only that now $l$ may no longer be a good quantum number and we might have to use a different basis (if the electron-electron interaction is at least if the order of the spin-orbit coupling).

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