Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D

TL;DR

This paper derives large-dimensional asymptotic expressions for position and momentum expectation values in D-dimensional hydrogenic systems, enabling the formulation of generalized uncertainty relations and bounds on entropic measures.

Contribution

It provides the first large-D limit analysis of expectation values and uncertainty bounds for all hydrogenic states using hypergeometric function asymptotics.

Findings

Large-D asymptotic expressions for expectation values derived

Generalized Heisenberg-like uncertainty relations established

Bounds on entropic uncertainty measures obtained

Abstract

The radial expectation values of the probability density of a quantum system in position and momentum spaces allow one to describe numerous physical quantities of the system as well as to find generalized Heisenberg-like uncertainty relations and to bound entropic uncertainty measures. It is known that the position and momentum expectation values of the main prototype of the $D$-dimensional Coulomb systems, the $D$-dimensional hydrogenic system, can be expressed in terms of some generalized hypergeometric functions of the type $_{p+1}F_p(z)$ evaluated at unity with $p=2$ and $p=3$, respectively. In this work we determine the position and momentum expectation values in the limit of large $D$ for all hydrogenic states from ground to very excited (Rydberg) ones in terms of the spatial dimensionality and the hyperquantum numbers of the state under consideration. This is done by means of two different approaches to calculate the leading term of the special functions $_{3}F_2\left(1\right)$ and $_{5}F_4\left(1\right)$ involved in the large $D$ limit of the position and momentum quantities. Then, these quantities are used to obtain the generalized Heisenberg-like and logarithmic uncertainty relations, and some upper and lower bounds to the entropic uncertainty measures (Shannon, R\'enyi, Tsallis) of the $D$-dimensional hydrogenic system.