Entropic uncertainty measures for large dimensional hydrogenic systems

TL;DR

This paper rigorously derives the leading term of Rénnyi entropies for large-dimensional hydrogenic atoms, providing insights into entropic uncertainty measures and their saturation of known uncertainty relations.

Contribution

It offers the first rigorous determination of the leading term of Rénnyi entropies for large D-dimensional hydrogenic systems, connecting to uncertainty relations.

Findings

Leading term of Rénnyi entropies determined for large D

Results saturate known entropic uncertainty relations

Provides a mathematical foundation for large-dimensional atomic systems

Abstract

The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system, but also the entropic uncertainty measures of R\'enyi type which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in $1/D$ in similar systems with a non-standard dimensionality $D$; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-$D$ limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The $D$-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work we rigorously determine the leading term of the R\'enyi entropies of the $D$-dimensional hydrogenic atom at the limit of large $D$. As a byproduct, we show that our results saturate the known position-momentum R\'enyi-entropy-based uncertainty relations.