R\'enyi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

TL;DR

This paper analytically computes the Rényi entropies for highly-excited states of multidimensional harmonic oscillators using strong Laguerre polynomial asymptotics, revealing their dependence on quantum numbers, order, and dimensionality.

Contribution

It introduces a novel analytical method to determine Rényi entropies of Rydberg states via strong Laguerre asymptotics, connecting entropies to polynomial norms and quantum parameters.

Findings

Rényi entropy decreases monotonically with increasing order p.

Disequilibrium exhibits a quasi-Gaussian behavior in relation to dimension D.

Explicit formulas for Rényi entropies in terms of quantum numbers, p, and D are derived.

Abstract

The R\'enyi entropies $R_{p}[\rho]$, $p>0,\neq 1$ of the highly-excited quantum states of the $D$-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydberg wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the asymptotics $n\to\infty$ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the R\'enyi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers ($n,l$), the parameter order $p$ and the universe dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the R\'enyi entropy power decreases monotonically as the order $p$ is increasing and (b) the disequilibrium (closely related to the second order R\'enyi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of $D$.