Eur. Phys. J. Plus (2020) 135: 721
https://doi.org/10.1140/epjp/s13360-020-00736-7

Review

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

¹ Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071, Granada, Spain
² Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071, Granada, Spain
³ Departamento de Matemática Aplicada, Universidad Rey Juan Carlos, 28933, Madrid, Spain

^a dehesa@ugr.es

Received: 14 July 2020
Accepted: 30 August 2020
Published online: 12 September 2020

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 Rényi 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 $(n_r,{\mu }_1,{\mu }_2,\dots ,{\mu }_{D-1})$ 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.