Relativistic effects on information measures for hydrogen-like atoms

TL;DR

This paper investigates how relativistic effects influence various information measures in hydrogen-like atoms, revealing fundamental issues with traditional uncertainty measures and exploring alternative entropy-based metrics across different relativistic regimes.

Contribution

It provides a detailed analysis of relativistic effects on information measures, highlighting shortcomings of conventional variances and evaluating alternative entropy measures for hydrogen-like atoms.

Findings

Relativistic position density decays exponentially at large r

Momentum density decays as an inverse power of p

Some information measures diverge away from the Shannon entropy

Abstract

Position and momentum information measures are evaluated for the ground state of the \emph{relativistic} hydrogen-like atoms. Consequences of the fact that the radial momentum operator is not self-adjoint are explicitly studied, exhibiting fundamental shortcomings of the conventional uncertainty measures in terms of the radial position and momentum variances. The Shannon and R\'enyi entropies, the Fisher information measure, as well as several related information measures, are considered as viable alternatives. Detailed results on the onset of relativistic effects for low nuclear charges, and on the extreme relativistic limit, are presented. The relativistic position density decays exponentially at large $r$, but is singular at the origin. Correspondingly, the momentum density decays as an inverse power of $p$. Both features yield divergent R\'enyi entropies away from a finite vicinity of the Shannon entropy. While the position space information measures can be evaluated analytically for both the nonrelativistic and the relativistic hydrogen atom, this is not the case for the relativistic momentum space. Some of the results allow interesting insight into the significance of recently evaluated Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral atoms.