Entropy and complexity analysis of hydrogenic Rydberg atoms

TL;DR

This paper analyzes the disorder and complexity of hydrogenic Rydberg atoms using information-theoretic measures in position and momentum spaces, providing asymptotic calculations and uncertainty relations for various states.

Contribution

It introduces a rigorous asymptotic analysis of information-theoretic quantities and complexity measures for Rydberg atoms, extending understanding of their quantum disorder.

Findings

Explicit asymptotic expressions for entropy and complexity measures.

Derivation of generalized uncertainty relations.

Application to specific Rydberg states like linear, circular, and quasicircular.

Abstract

The internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information-theoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy and the Fisher information. As well, the complexity measures of Cr\'amer-Rao, Fisher-Shannon and LMC types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberg-like, logarithmic and entropic uncertainty relations are also given. Finally, application to linear ($l=0$), circular ($l=n-1$) and quasicircular ($l=n-2$) states is explicitly done.