R\'enyi, Shannon and Tsallis entropies of Rydberg hydrogenic systems

TL;DR

This paper analytically calculates Rènyi, Shannon, and Tsallis entropies for Rydberg hydrogenic states, revealing how these quantum information measures depend on quantum numbers and nuclear charge, especially in high-energy states.

Contribution

It introduces a method linking entropies to Laguerre polynomial norms and derives their asymptotic behavior for highly excited hydrogenic states.

Findings

Explicit formulas for Rènyi entropies of Rydberg states.

Asymptotic analysis of entropies as quantum number n increases.

Numerical examination of entropy dependence on quantum numbers and charge.

Abstract

The R\'enyi entropies $R_{p}[\rho], 0<p<\infty$ of the probability density $\rho_{n,l,m}(\vec{r})$ of a physical system completely characterize the chemical and physical properties of the quantum state described by the three integer quantum numbers $(n,l,m)$. The analytical determination of these quantities is practically impossible up until now, even for the very few systems where their Schr\"odinger equation is exactly solved. In this work, the R\'enyi entropies of Rydberg (highly-excited) hydrogenic states are explicitly calculated in terms of the quantum numbers and the parameter $p$. To do that we use a methodology which first connects these quantities to the $\mathcal{L}_{p}$-norms $N_{n,l}(p)$ of the Laguerre polynomials which characterize the state's wavefunction. Then, the R\'enyi, Shannon and Tsallis entropies of the Rydberg states are determined by calculating the asymptotics ($n\rightarrow\infty$) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge.