Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces

Abstract

The Rényi entropies of Coulomb systems $R_p[\rho ],0<p<\infty$ are logarithms of power functionals of the electron density $\rho (\vec{r})$ which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, its analytical determination is a hard issue not yet solved except for the first lowest-lying energetic states of some specific systems. This is so even for the $D$-dimensional hydrogenic system, which is the main prototype of the multidimensional Coulomb many-body systems. Recently, the Rényi entropies of this system have been found in the two extreme high-energy (Rydberg) and high-dimensional (pseudo-classical) cases. In this work we determine the position and momentum Rényi entropies (with integer $p$ greater than 1) for all the discrete stationary states of the multidimensional hydrogenic system directly in terms of the hyperquantum numbers which characterize the states, the nuclear charge and the space dimensionality. We have used a methodology based on linearization formulas for powers of the orthogonal Laguerre and Gegenbauer polynomials which control the hydrogenic states.

arXiv commentAccepted in Journal of Statistical Mechanics: Theory and Experiment

References in corpus (23)

• Formulation of the uncertainty relations in terms of the Renyi entropies
• A class of Rényi information estimators for multidimensional densities
• Entropic Uncertainty Relations in Quantum Physics
• Qubits with electrons on liquid helium
• Some extensions of the uncertainty principle
• On observability of Renyi's entropy
• A closed-form expression for the Sharma-Mittal entropy of exponential families
• On Rényi and Tsallis entropies and divergences for exponential families
• On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles
• Electrons above a Helium Surface and the One-Dimensional Rydberg Atom
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Entropic measures of Rydberg-like harmonic states
• Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
• Note on the Existence of Hydrogen Atoms in Higher Dimensional Euclidean Spaces
• Radial Coulomb and Oscillator Systems in Arbitrary Dimensions
• Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems
• Entropic properties of $D$-dimensional Rydberg systems
• Entropic uncertainty measures for large dimensional hydrogenic systems
• Renyi entropy and improved equilibration rates to self-similarity for nonlinear diffusion equations
• Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters
• Renyi entropy of the infinite well potential in momentum space and Dirichlet-like trigonometric functionals
• The world according to Renyi: Thermodynamics of multifractal systems
• Entropy and complexity analysis of the D-dimensional rigid rotator and hyperspherical harmonics

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• Rényi and Tsallis entropies: three analytic examples
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Multidimensional hydrogenic states: Position and momentum expectation values
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots