On the influence of hidden momentum and hidden energy in the classical analysis of spin-orbit coupling in hydrogenlike atoms

TL;DR

This paper investigates how hidden momentum and hidden energy influence the classical analysis of spin-orbit coupling in hydrogen-like atoms, revealing that including hidden energy affects agreement with experimental data.

Contribution

It introduces the concept that hidden energy must be considered alongside hidden momentum for a relativistically consistent classical analysis of spin-orbit coupling.

Findings

Hidden energy impacts the magnitude of spin-orbit coupling.

Reinterpreting the Bohr postulate to include hidden momentum aligns theory with experiments.

Including hidden energy resolves discrepancies in classical models.

Abstract

In a recent article, Kholmetskii, Missevitch and Yarmin ["On the classical analysis of spin-orbit coupling in hydrogenlike atoms," Am. J. Phys. 78(4), April 2010 (428-432)] examine in detail the spin-orbit coupling in the semiclassical hydrogenic atom, and identify a need to account for non-Coulomb forces not included in the standard analysis. Kholmetskii, et al., showed that the experimentally-measured coupling continues to be obtained when the new forces are incorporated in the analysis. This requires that the change in orbital radius due to non-Coulomb forces is also properly accounted for. In response to a comment, Kholmetskii, et al., showed that the experimentally-measured coupling continues to be obtained when so-called "hidden momentum" forces are also included. However it has been postulated that when hidden momentum is nonvanishing, a corresponding hidden energy must also be present, that was not included in the total energy by Kholmetskii, et al. Hidden energy is postulated necessary to obtaining a relativistically covariant description. Inclusion of hidden energy leads to disagreement with the experimentally-determined spin-orbit coupling magnitude if the Bohr postulate, that orbital angular momentum is quantized in whole multiples of the reduced Planck constant, is assumed to apply to kinetic momentum alone. The empirical result may be recovered in the semiclassical picture and using the general approach developed by Kholmetskii, et al., if the Bohr postulate is reinterpreted to apply to orbital angular momentum that consists of hidden as well as kinetic momentum.