Entropic measures of Rydberg-like harmonic states

TL;DR

This paper analytically computes the angular and radial Shannon and Rènyi entropies for Rydberg-like states in spherically symmetric potentials, revealing how these measures depend on quantum numbers and potential form.

Contribution

It introduces analytical procedures to calculate angular entropy for any central potential and derives the dominant radial entropy term for highly excited Rydberg states.

Findings

Explicit formulas for angular entropy in terms of quantum numbers.

Analytical expressions for the radial entropy of Rydberg states.

Application of recent information theory results to oscillator-like wavefunctions.

Abstract

The Shannon entropy, the desequilibrium and their generalizations (R\'enyi and Tsallis entropies) of the three-dimensional single-particle systems in a spherically-symmetric potential $V(r)$ can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this paper we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillator-like systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator-like wavefunctions.