Dispersion and entropy-like measures of multidimensional harmonic systems: application to Rydberg states and high-dimens
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Dispersion and entropy-like measures of multidimensional harmonic systems: application to Rydberg states and high-dimensional oscillators J. S. Dehesa1,2,a
, I. V. Toranzo3
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
Received: 14 July 2020 / Accepted: 30 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 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 (nr , μ1 , μ2 , . . . , μ D−1 ) which characterize the state. Emphasis is placed on the highly excited Rydberg (high radial hyperquantum number nr , 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 Lq -norms of these orthogonal functions.
1 Introduction The spatial localization/delocalization of quantum systems with high precision and its quantification play a central role in quantum mechanics. Presently, the modern quantum technologies are making possible the actual realization of such experiments. The investigations have been driven by the possibility of practical applications including laser cooling and trapping, nanolithography and many other areas of atomic and molecular physics as well as classical and quantum information. While this uncertainty quantification was originally formulated in terms of variances and their generalization (the radial expectation values), they have later been successfully expressed with entropies of Fisher and Shannon types, and their generalization (particularly the Rényi entropies) in a much more appropriate manner. The determination of these dispersion and entropy-like quantities for arbitrary quantum states is a formidable numerical task and certainly an impossible analytical task, except for a few solvable quantum-mechanical potentials which allow to model the mean field of numerous many-body systems. The spatial delocalization of physical systems with solvable
a e-mail: [email protected] (corresponding author)
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