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

Alexander Ivanovich Aptekarev, Dmitry Nikolaevich Tulyakov, Irene Valero Toranzo and Jesús Sanchez Dehesa ()
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Alexander Ivanovich Aptekarev: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
Dmitry Nikolaevich Tulyakov: Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
Irene Valero Toranzo: Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada
Jesús Sanchez Dehesa: Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada

The European Physical Journal B: Condensed Matter and Complex Systems, 2016, vol. 89, issue 3, 1-12

Abstract: 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.

Keywords: Statistical; and; Nonlinear; Physics (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (1)

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DOI: 10.1140/epjb/e2016-60860-9

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