R\'enyi entropies for multidimensional hydrogenic systems in position and momentum spaces

TL;DR

This paper analytically computes the Re9nyi entropies in position and momentum spaces for all discrete states of multidimensional hydrogenic systems, advancing understanding of quantum uncertainty in these Coulomb systems.

Contribution

It provides explicit formulas for Re9nyi entropies of all stationary states in multidimensional hydrogenic systems, a problem previously unsolved except for special cases.

Findings

Derived formulas for Re9nyi entropies in position space.

Derived formulas for Re9nyi entropies in momentum space.

Applicable to all discrete states based on hyperquantum numbers.

Abstract

The R\'enyi entropies of Coulomb systems $R_{p}[\rho], 0 < p < \infty$ are logarithms of power functionals of the electron density $\rho(\vec{r})$ which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, its analytical determination is a hard issue not yet solved except for the first lowest-lying energetic states of some specific systems. This is so even for the $D$-dimensional hydrogenic system, which is the main prototype of the multidimensional Coulomb many-body systems. Recently, the R\'enyi entropies of this system have been found in the two extreme high-energy (Rydberg) and high-dimensional (pseudo-classical) cases. In this work we determine the position and momentum R\'enyi entropies (with integer $p$ greater than 1) for all the discrete stationary states of the multidimensional hydrogenic system directly in terms of the hyperquantum numbers which characterize the states, the nuclear charge and the space dimensionality. We have used a methodology based on linearization formulas for powers of the orthogonal Laguerre and Gegenbauer polynomials which control the hydrogenic states.