Asymptotics of $L_p$-norms of Hermite polynomials and R\'enyi entropy of Rydberg oscillator states

TL;DR

This paper derives the asymptotic behavior of weighted $L_p$-norms of Hermite polynomials, linking it to the Rényi entropy of quantum oscillator states, especially for highly excited Rydberg states, revealing new asymptotic insights.

Contribution

It provides the first detailed asymptotic analysis of $L_p$-norms of Hermite polynomials and applies these results to Rydberg oscillator states in quantum mechanics.

Findings

Asymptotic formulas for weighted $L_p$-norms of Hermite polynomials for large degree.

Explicit calculation of Rényi entropy for Rydberg oscillator states.

Insights into the entropy behavior of highly excited quantum states.

Abstract

The asymptotics of the weighted $L_{p}$-norms of Hermite polynomials, which describes the R\'enyi entropy of order $p$ of the associated quantum oscillator probability density, is determined for $n\to\infty$ and $p>0$. Then, it is applied to the calculation of the R\'enyi entropy of the quantum-mechanical probability density of the highly-excited (Rydberg) states of the isotropic oscillator.