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 (11)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• A class of Rényi information estimators for multidimensional densities
• Universal geometric approach to uncertainty, entropy and information
• Some extensions of the uncertainty principle
• On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
• Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems
• Entropic properties of $D$-dimensional Rydberg systems
• Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
• Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D

Cited by in corpus (16)

• Information-entropic measures in free and confined hydrogen atom
• Exact Shannon entropies for the multidimensional harmonic states
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• Information-entropic measures for non-zero l states of confined hydrogen-like ions
• Quantum mechanical virial-like theorem for confined quantum systems
• Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
• Exact Rényi entropies of $D$-dimensional harmonic systems
• Shell-confined atom and plasma: incidental degeneracy, metallic character and information entropy
• Multidimensional hydrogenic states: Position and momentum expectation values
• Information and quantum theories: an analysis in one-dimensional systems
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Information-Entropic Measures in Confined Isotropic Harmonic Oscillator
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• Linearization and Krein-like functionals of hypergeometric orthogonal polynomials
• Complexity-like properties and parameter asymptotics of ${\mathfrak{L}}_q$-norms of Laguerre and Gegenbauer polynomials