Examinando por Autor "Puertas-Centeno, D."

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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.
Crámer-Rao complexity of the confined two-dimensional hydrogen
(Wiley, 2020-09-10) Estañón, C. R.; Aquino, N.; Puertas-Centeno, D.; Dehesa, J. S.
The internal disorder of the confined two-dimensional hydrogenic atom is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the statistical Crámer-Rao complexity measure. First, the confinement dependence of the variance and the Fisher information of the position and momentum spreading of its electron distribution are computed and discussed. Then, the Crámer-Rao complexity measure (which quantifies the combined balance of the charge concentration around the mean value and the gradient content of the electron distribution) is investigated in position and momentum spaces. We found that confinement does distinguish complexity of the system for all quantum states by means of these two component measures.
Differential-escort transformations and the monotonicity of the LMC-Rényi complexity measure
(Elsevier, 2018-03-15) Puertas-Centeno, D.
Escort distributions have been shown to be very useful in a great variety of fields ranging from information theory, nonextensive statistical mechanics till coding theory, chaos and multifractals. In this work we give the notion and the properties of a novel type of escort density, the differential-escort densities, which have various advantages with respect to the standard ones. We highlight the behavior of the differential Shannon, Rényi and Tsallis entropies of these distributions. Then, we illustrate their utility to prove the monotonicity property of the LMC-Rényi complexity measure and to study the behavior of general distributions in the two extreme cases of minimal and very high LMC-Rényi complexity. Finally, this transformation allows us to obtain the Tsallis q-exponential densities as the differential-escort transformation of the exponential density.
Entropic Analysis of the Quantum Oscillator with a Minimal Length
(MDPI, 2019-11-19) Puertas-Centeno, D.; Portesi, M.
The well-known Heisenberg–Robertson uncertainty relation for a pair of noncommuting observables, is expressed in terms of the product of variances and the commutator among the operators, computed for the quantum state of a system. Different modified commutation relations have been considered in the last years with the purpose of taking into account the effect of quantum gravity. Indeed it can be seen that letting [X, P] = i hbar (1 + bP^2) implies the existence of a minimal length proportional to Sqrt(b). The Bialynicki-Birula–Mycielski entropic uncertainty relation in terms of Shannon entropies is also seen to be deformed in the presence of a minimal length, corresponding to a strictly positive deformation parameter b. Generalized entropies can be implemented. Indeed, results for the sum of position and (auxiliary) momentum Rényi entropies with conjugated indices have been provided recently for the ground and first excited state. We present numerical findings for conjugated pairs of entropic indices, for the lowest lying levels of the deformed harmonic oscillator system in 1D, taking into account the position distribution for the wavefunction and the actual momentum.
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.
Multidimensional hydrogenic states: Position and momentum expectation values
(IOP Publishing, 2021-04-20) Dehesa, J. S.; Puertas-Centeno, D.
The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values ⟨rα⟩ and , respectively. These quantities, which describe and/or are closely related to various fundamental properties of realistic systems, have not been calculated in an analytical and effective manner up until now except for a number of three-dimensional hydrogenic states. In this work we explicitly show these expectation values for all discrete stationary D-dimensional hydrogenic states in terms of the dimensionality D, the strength of the Coulomb potential (i.e. the nuclear charge) and the D state's hyperquantum numbers. Emphasis is placed on the momentum expectation values (mostly unknown, specially the ones with odd order) which are obtained in a closed compact form. Applications are made to circular, S-wave, high-energy (Rydberg) and high-dimensional (pseudo-classical) states of three- and multidimensional hydrogenic atoms. This has been possible because of the analytical algebraic and asymptotical properties of the special functions (orthogonal polynomials, hyperspherical harmonics) which control the states' wavefunctions. Finally, some Heisenberg-like uncertainty inequalities satisfied by these dispersion quantities are also given and discussed.
On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures
(MDPI, 2017-09-14) Zozor, S.; Puertas-Centeno, D.; Dehesa, J. S.
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, . . . ) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named (p, b, l)-Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function r characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of r, and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the (p, b, l)-Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.
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.
Two-dimensional confined hydrogen: An entropy and complexity approach
(Wiley, 2020-02-19) Estañón, C. R.; Aquino, N.; Puertas-Centeno, D.; Dehesa, J. S.
The position and momentum spreading of the electron distribution of the two-dimensional confined hydrogenic atom, which is a basic prototype of the general multidimensional confined quantum systems, is numerically studied in terms of the confinement radius for the 1s, 2s, 2p, and 3d quantum states by means of the main entropy and complexity information-theoretical measures. First, the Shannon entropy and the Fisher information, as well as the associated uncertainty relations, are computed and discussed. Then, the Fisher-Shannon, LopezRuiz-Mancini-Calvet, and LMC-Rényi complexity measures are examined and mutually compared. We have found that these entropy and complexity quantities reflect the rich properties of the electron confinement extent in the two conjugated spaces.