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

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.

Entropic measures of Rydberg‐like harmonic states

“…Then, the problem of determination of the radial Rényi entropy of a general oscillator‐like state boils down to the study of the asymptotics ( ) of the Laguerre norm . The latter problem can be solved by means of the recent methodology of Aptekarev et al, which takes explicitly into account the different asymptotical representations for the Laguerre polynomials at different regions of the real half‐line.…”

“…Then, we determine both the angular contribution to these entropies for all the quantum‐mechanically allowed states of the central potential V(r) and the radial entropy of the highly energetic (i.e., Rydberg) states of the (three‐dimensional) harmonic oscillator in an analytical way. The latter is done by using some recent powerful results of the information theory of Laguerre and Gegenbauer polynomials (see also Refs. [ ]).…”

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

“…Then, uncertainty measures of entropic character [26][27][28][29][30][31][32][33][34] have been considered; they are much more appropriate because, contrary to the Heisenberg-like ones, they do not depend on any specific point of the system. Recently these studies have been extended by calculating the dominant term of the Heisenberglike and Rényi-entropy-based uncertainty measures for both the D-dimensional hydrogenic and harmonic systems at the quassiclassical border in the two conjugated position and momentum spaces [35,36].…”

Exact Shannon entropies for the multidimensional harmonic states

“…In addition, the Shannon entropy for the highdimensional (i.e., pseudoclassical) harmonic states has been already conjectured [59]. We should also mention here that the other basic uncertainty measures (namely the Heisenberg-like measures, the Fisher information and the Rényi entropies) have been recently calculated for all stationary states of the multidimensional harmonic system in [60][61][62], [63] and [59] respectively, for the highdimensional harmonic states in [64] and for the Rydberg harmonic states in [47,57]. See also [65] for the values of the Rényi entropies for the Rydberg states of the one-dimensional harmonic oscillator.…”