Complexity of D-dimensional hydrogenic systems in position and momentum spaces

TL;DR

This paper analyzes the internal disorder of D-dimensional hydrogenic systems in position and momentum spaces using shape complexity, providing explicit formulas and numerical discussions for various states and dimensions.

Contribution

It introduces explicit expressions for shape complexity in D-dimensional hydrogenic systems and explores their dependence on quantum numbers and dimensionality.

Findings

Shape complexity is explicitly calculated for ground and circular states.

Complexity does not depend on Coulomb potential strength in stationary states.

Numerical analysis covers various states, dimensions, and energy limits.

Abstract

The internal disorder of a D-dimensional hydrogenic system, which is strongly associated to the non-uniformity of the quantum-mechanical density of its physical states, is investigated by means of the shape complexity in the two reciprocal spaces. This quantity, which is the product of the disequilibrium or averaging density and the Shannon entropic power, is mathematically expressed for both ground and excited stationary states in terms of certain entropic functionals of Laguerre and Gegenbauer (or ultraspherical) polynomials. We emphasize the ground and circular states, where the complexity is explicitly calculated and discussed by means of the quantum numbers and dimensionality. Finally, the position and momentum shape complexities are numerically discussed for various physical states and dimensionalities, and the dimensional and Rydberg energy limits as well as their associated uncertainty products are explicitly given. As a byproduct, it is shown that the shape complexity of the system in a stationary state does not depend on the strength of the Coulomb potential involved.