Eur. Phys. J. B (2016) 89: 85
https://doi.org/10.1140/epjb/e2016-60860-9

Regular Article

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

¹ Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, 125047 Moscow, Russia
² Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain
³ Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, 18071 Granada, Spain

^a e-mail: dehesa@ugr.es

Received: 27 October 2015
Received in final form: 10 February 2016
Published online: 28 March 2016

Abstract

The Rényi entropies R_p [ ρ ], p> 0, ≠ 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 ρ_n(r) density of the Rydberg wavefunctions Ψ_{n,l, { μ }}(r); so, to the ℒ_p-norms of the associated Laguerre polynomials. Then, we determine the asymptotics n → ∞ 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 Rényi 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 ≥ 1. We find that (a) the Rényi entropy power decreases monotonically as the order p is increasing and (b) the disequilibrium (closely related to the second order Rényi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of D.