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