Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces

Abstract

The Rényi entropies of Coulomb systems Rp[ρ], 0 < p < ∞ are logarithms of power functionals of the electron density ρ( r) which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, their 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 Rényi 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 Rényi 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, nuclear charge and 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.