The Shannon entropy of high‐dimensional hydrogenic and harmonic systems

Abstract

There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an informationtheoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D ! ∞. K E Y W O R D S high-dimensional harmonic systems, high-dimensional hydrogenic systems, high-dimensional quantum systems, logarithmic integrals of orthogonal polynomials-asymptotics, Shannon entropy of high-dimensional quantum systems 1 | INTRODUCTION Multidimensional quantum systems consisting of many particles are a major challenge in Quantum Chemistry and Physics, as their behavior can be determined only with immense computational power. The physical solutions of their associated wave equations, and consequently all the chemical and physical properties of these systems, crucially depend on the space dimensionality D. Scientists are trying to discover elegant notions and techniques to simplify the problem (see, eg, ref. [1]). The 1986 Nobel Prize in Chemistry winner Dudley Robert Herschbach et al have developed [2][3][4][5] a very useful strategy to study the atomic and molecular systems, the D-dimensional scaling method, where the D is the basic variable. The pseudoclassical or highdimensional (D ! ∞) limit is the starting point of this strategy. Indeed, this method requires to solve a finite many-electron problem in the (D ! ∞)-limit, and then perturbation theory in 1D is used to have an approximate result for the standard dimension (D = 3), obtaining at times a quantitative accuracy comparable to the self-consistent Hartree-Fock calculations. Most important here is that the electronic structure for the (D ! ∞)-limit is beguilingly simple and exactly computable for any atom and molecule. For finite D but very large, the electrons are confined to harmonic oscillations about the fixed positions attained in the (D ! ∞)-limit. Indeed, at this high-dimensional limit, the electrons of a many-electron system assume fixed positions relative to

Coulomb correlation and information entropies in confined helium-like atoms

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Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

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Analytical Shannon information entropies for all discrete multidimensional hydrogenic states

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