On the classical analysis of spin-orbit coupling in hydrogenlike atoms

Abstract

We reanalyze the usual classical derivation of spin-orbit coupling in hydrogenlike atoms. We point out the presence of an additional force exerted on a spinning electron due to the appearance of its electric dipole moment in the rest frame of the nucleus. This force has been ignored, although its inclusion in the electron’s equation of motion influences the energy level of an orbiting electron on an equal footing with other effects in the usual analysis of spin-orbit coupling. A fortuitous cancellation between two terms leaves the overall energy level unaffected, which explains in part why this effect has been overlooked. An account of this effect in the Bohr model produces the usual expression for the spin-orbit coupling but with different radii of the electron’s orbit for different spatial orientations of the electron’s spin. This result is in qualitative agreement with the solution of the Dirac–Coulomb equation for hydrogenlike atoms.