Comment on "On the classical analysis of spin-orbit coupling in hydrogenlike atoms," [Am. J. Phys. 78 (4) 428-432, April 2010]

TL;DR

This paper critiques a recent classical analysis of spin-orbit coupling in hydrogenlike atoms, emphasizing the importance of hidden momentum for conservation laws and accurate energy calculations, which leads to discrepancies with experimental results.

Contribution

It highlights the necessity of including hidden momentum in classical models of spin-orbit coupling to ensure momentum conservation and accurate energy predictions.

Findings

Omission of hidden momentum leads to inconsistent force calculations.

Including hidden momentum results in predictions that disagree with experiments.

Classical analysis must account for hidden momentum to be physically consistent.

Abstract

In their recent paper, Kholmetskii, Missevitch, and Yarman "reanalyze the usual classical derivation of spin-orbit coupling in hydrogenlike atoms" and find a result "in qualitative agreement with the solution of the Dirac-Coulomb equation for hydrogenlike atoms." However, the authors' result is based on an equation of translational motion of the electron that omits any contribution due to the existence of "hidden" momentum of the electron intrinsic magnetic dipole moment in the electric field of the nucleus. Accounting for hidden momentum is necessary to obtaining conservation of linear momentum in the interaction of a magnetic dipole with a point charge. If hidden momentum is omitted from the description, the force on the nucleus due to the electron will differ from the force on the electron due to the nucleus. Thus, omitting the hidden momentum contribution, the binding energy including the spin-orbit coupling cannot be consistently calculated, while including hidden momentum, the value obtained is in disagreement with experiment.