Examinando por Autor "Toranzo, I. V."

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Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
(Wiley, 2019-11-15) Toranzo, I. V.; Puertas-Centeno, D.; Sobrino, N.; Dehesa, J. S.
The entropic uncertainty measures of the multidimensional hydrogenic states quantify the multiple facets of the spatial delocalization of the electronic probability density of the system. The Shannon entropy is the most adequate uncertainty measure to quantify the electronic spreading and to mathematically formalize the Heisenberg uncertainty principle, partially because it does not depend on any specific point of their multidimensional domain of definition. In this work, the radial and angular parts of the Shannon entropies for all the discrete stationary states of the multidimensional hydrogenic systems are obtained from first principles; that is, they are given in terms of the states' principal and magnetic hyperquantum numbers (n, μ1, μ2, …, μD−1), the system's dimensionality D and the nuclear charge Z in an analytical, compact form. Explicit expressions for the total Shannon entropies are given for the quasi-spherical states, which conform to a relevant class of specific states of the D-dimensional hydrogenic system characterized by the hyperquantum numbers μ1 = μ2 … = μD−1 = n − 1, including the ground state.
Biparametric complexities and generalized Planck radiation law
(IOP Publishing, 2017-10-25) Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
Complexity theory embodies some of the hardest, most fundamental and most challenging open problems in modern science. The very term complexity is very elusive, so the main goal of this theory is to find meaningful quantifiers for it. In fact, we need various measures to take into account the multiple facets of this term. Here, some biparametric Crámer–Rao and Heisenberg–Rényi measures of complexity of continuous probability distributions are defined and discussed. Then, they are applied to blackbody radiation at temperature T in a d-dimensional universe. It is found that these dimensionless quantities do not depend on T nor on any physical constants. So, they have a universal character in the sense that they only depend on spatial dimensionality. To determine these complexity quantifiers, we have calculated their dispersion (typical deviations) and entropy (Rényi entropies and the generalized Fisher information) constituents. They are found to have a temperature-dependent behavior similar to the celebrated Wien’s displacement law of the dominant frequency νmax at which the spectrum reaches its maximum. Moreover, they allow us to gain insights into new aspects of the d-dimensional blackbody spectrum and the quantification of quantum effects associated with space dimensionality.
Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems
(IOP Publishing, 2017-08-14) Sobrino-Coll, N.; Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same firstorder values for all the quantum states of the D-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical (D → ∞) limit but a similar phenomenon also happens for both the Fisher-information-based uncertainty product and the Shannon-entropy-based uncertainty sum, as well as for the Crámer–Rao and Fisher–Shannon complexities. Moreover, we show that the López–Ruiz–Mancini–Calvet (LMC) and LMC-Rényi complexity measures capture the hydrogenic-harmonic dierence in the high dimensional limit already at first order.
Entropic measures of Rydberg-like harmonic states
(Wiley Periodicals, Inc., 2016-10-13) Dehesa, J. S.; Toranzo, I. V.; Puertas-Centeno, D.
The Shannon entropy, the desequilibrium and their generalizations (Renyi and Tsallis entropies) of the three-dimensional single-particle systems in a spherically symmetric potential V(r) can be decomposed into angular and radial parts. The radial part depends on the analytical form of the potential, but the angular part does not. In this article, we first calculate the angular entropy of any central potential by means of two analytical procedures. Then, we explicitly find the dominant term of the radial entropy for the highly energetic (i.e., Rydberg) stationary states of the oscillatorlike systems. The angular and radial contributions to these entropic measures are analytically expressed in terms of the quantum numbers which characterize the corresponding quantum states and, for the radial part, the oscillator strength. In the latter case, we use some recent powerful results of the information theory of the Laguerre polynomials and spherical harmonics which control the oscillator-like wavefunctions.
Entropic properties of D-dimensional Rydberg systems
(Elsevier, 2016-07-04) Toranzo, I. V.; Puertas-Centeno, D.; Dehesa, J. S.
The fundamental information-theoretic measures (the Rényi Rp[ρ] and Tsallis Tp[ρ] entropies, p > 0) of the highly-excited (Rydberg) quantum states of the D-dimensional (D > 1) hydrogenic systems, which include the Shannon entropy (p → 1) and the disequilibrium (p = 2), are analytically determined by use of the strong asymptotics of the Laguerre orthogonal polynomials which control the wavefunctions of these states. We first realize that these quantities are derived from the entropic moments of the quantum-mechanical probability ρ(⃗r) densities associated to the Rydberg hydrogenic wavefunctions Ψn,l,{µ}(⃗r), which are closely connected to the Lp-norms of the associated Laguerre polynomials. Then, we determine the (n → ∞)-asymptotics of these norms in terms of the basic parameters of our system (the dimensionality D, the nuclear charge and the hyperquantum numbers (n, l, {µ}) of the state) by use of recent techniques of approximation theory. Finally, these three entropic quantities are analytically and numerically discussed in terms of the basic parameters of the system for various particular states.
Entropic uncertainty measures for large dimensional hydrogenic systems
(AIP Publishing, 2017-09-25) Puertas-Centeno, D.; Temme, N. M.; Toranzo, I. V.; Dehesa, J. S.
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 Renyi 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 nonstandard dimensionality D; moreover, several physical quantities of numerous atomic and molecular systems have been numerically shown to have values in the largeD 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 Renyi 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 Renyi-entropy-based uncertainty relations.
Exact Rényi entropies of D-dimensional harmonic systems
(Springer, 2018-09-28) Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue per se and because these measures, which are functionals of the single-particle probability density of the systems, describe numerous fundamental and experimentally accessible physical quantities. However, it is a formidable task (not yet solved, except possibly for the ground and a few lowestlying energetic states) even for the small bunch of elementary quantum potentials which are used to approximate the mean-field potential of the physical systems. Recently, the dominant term of the Heisenberg and Rényi measures of the multidimensional harmonic system (i.e., a particle moving under the action of a D-dimensional quadratic potential, D > 1) has been analytically calculated in the high-energy (i.e., Rydberg) and the high-dimensional (i.e., pseudoclassical) limits. In this work we determine the exact values of the R´enyi uncertainty measures of the D-dimensional harmonic system for all ground and excited quantum states directly in terms of D, the potential strength and the hyperquantum numbers.
Heisenberg and Entropic Uncertainty Measures for Large-Dimensional Harmonic Systems
(MDPI, 2017-04-09) Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical (D → ∞) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.
Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
(IOP Publishing, 2018-07-19) Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
The Rényi entropies of Coulomb systems Rp[ρ], 0 < p < ∞ are logarithms of power functionals of the electron density ρ( r) which quantify most appropriately the electron uncertainty and describe numerous physical observables. However, their 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, nuclear charge and 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.
The biparametric Fisher–Rényi complexity measure and its application to the multidimensional blackbody radiation
(IOP Publishing, 2017-04-24) Puertas-Centeno, D.; Toranzo, I. V.; Dehesa, J. S.
We introduce a biparametric Fisher–Rényi complexity measure for general probability distributions and we discuss its properties. This notion, which is composed of two entropy-like components (the Rényi entropy and the biparametric Fisher information), generalizes the basic Fisher–Shannon measure and the previous complexity quantifiers of Fisher–Rényi type. Then, we illustrate the usefulness of this notion by carrying out a informationtheoretical analysis of the spectral energy density of a d-dimensional blackbody at temperature T. It is shown that the biparametric Fisher–Rényi measure of this quantum system has a universal character in the sense that it does not depend on temperature nor on any physical constant (e.g. Planck constant, speed of light, Boltzmann constant), but only on the space dimensionality d. Moreover, it decreases when d is increasing, but exhibits a non trivial behavior for a fixed d and a varying parameter, which somehow brings up a non standard structure of the blackbody d-dimensional density distribution.