Dispersion and entropy-like measures of multidimensional harmonic systems. Application to Rydberg states and high-dimensional oscillators

TL;DR

This paper analytically examines the spreading and entropy-like measures of quantum multidimensional harmonic oscillator states, focusing on highly-excited Rydberg and high-dimensional states, providing explicit formulas involving system parameters.

Contribution

It introduces a methodology for deriving explicit expressions of dispersion and entropy measures for multidimensional harmonic oscillator states using algebraic properties of orthogonal polynomials.

Findings

Explicit formulas for dispersion and entropy measures in terms of system parameters.

Analysis of highly-excited Rydberg states and high-dimensional oscillator states.

Methodology based on algebraic and asymptotic properties of Laguerre and Gegenbauer polynomials.

Abstract

The spreading properties of the stationary states of the quantum multidimensional harmonic oscillator are analytically discussed by means of the main dispersion measures (radial expectation values) and the fundamental entropy-like quantities (Fisher information, Shannon and R\'enyi entropies, disequilibrium) of its quantum probability distribution together with their associated uncertainty relations. They are explicitly given, at times in a closed compact form, by means of the potential parameters (oscillator strength, dimensionality, $D$) and the hyperquantum numbers $(n_r,\mu_1,\mu_2,\ldots,\mu_{D-1})$ which characterize the state. Emphasis is placed on the highly-excited Rydberg (high radial hyperquantum number $n_r$, fixed $D$) and the high-dimensional (high $D$, fixed hyperquantum numbers) states. We have used a methodology where the theoretical determination of the integral functionals of the Laguerre and Gegenbauer polynomials, which describe the spreading quantities, leans heavily on the algebraic properties and asymptotical behavior of some weighted $\mathfrak{L}_{q}$-norms of these orthogonal functions.