Entropic uncertainty measures for large dimensional hydrogenic systems

Abstract

The entropic moments of the probability density of a quantum system in position and momentum spaces describe not only some fundamental and/or experimentally accessible quantities of the system, but also the entropic uncertainty measures of Rényi type which allow one to find the most relevant mathematical formalizations of the position-momentum Heisenberg's uncertainty principle, the entropic uncertainty relations. It is known that the solution of difficult three-dimensional problems can be very well approximated by a series development in $1/D$ in similar systems with a non-standard dimensionality $D$; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the large-$D$ limit comparable to the corresponding ones provided by the three-dimensional numerical self-consistent field methods. The $D$-dimensional hydrogenic atom is the main prototype of the physics of multidimensional many-electron systems. In this work we rigorously determine the leading term of the Rényi entropies of the $D$-dimensional hydrogenic atom at the limit of large $D$. As a byproduct, we show that our results saturate the known position-momentum Rényi-entropy-based uncertainty relations.

Accepted in J. Math. Phys

References in corpus (6)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• Some extensions of the uncertainty principle
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Entropic properties of $D$-dimensional Rydberg systems
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Cited by in corpus (4)

• Information-entropic measures for non-zero l states of confined hydrogen-like ions
• Quantum mechanical virial-like theorem for confined quantum systems
• Information-Entropic Measures in Confined Isotropic Harmonic Oscillator
• Linearization and Krein-like functionals of hypergeometric orthogonal polynomials