arXiv subjects

I. V. Toranzo

Publications and source records attributed to I. V. Toranzo.

At least 19 recordsLinked to original sources

arXiv2020

Dispersion and entropy-like measures of multidimensional harmonic systems. Application to Rydberg states and high-dimensional oscillators

The spreading properties of the stationary states of the quantum multidimensional harmonic oscillator are analytically discussed by means of the main dispersion measures (radial expectation values) and the fundamental entropy-like quantities (Fisher information, Shannon and Rényi entropies, disequilibrium) of its quantum probability distribution together with their associated uncertainty relations. They are explicitly given, at times in a closed compact form, by means of the potential parameters (oscillator strength, dimensionality, $D$) and the hyperquantum numbers $(n_r,μ_1,μ_2,\ldots,μ_{D-1})$ which characterize the state. Emphasis is placed on the highly-excited Rydberg (high radial hyperquantum number $n_r$, fixed $D$) and the high-dimensional (high $D$, fixed hyperquantum numbers) states. We have used a methodology where the theoretical determination of the integral functionals of the Laguerre and Gegenbauer polynomials, which describe the spreading quantities, leans heavily on the algebraic properties and asymptotical behavior of some weighted $\mathfrak{L}_{q}$-norms of these orthogonal functions.

quant-ph↗

arXiv2018

Linearization and Krein-like functionals of hypergeometric orthogonal polynomials

The Krein-like $r$-functionals of the hypergeometric orthogonal polynomials $\{p_{n}(x) \}$ with kernel of the form $x^{s}[ω(x)]^βp_{m_{1}}(x)\ldots p_{m_{r}}(x)$, being $ω(x)$ the weight function on the interval $Δ\in\mathbb{R}$, are determined by means of the Srivastava linearization method. The particular $2$-functionals, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments and the novel Krein-like moments. Moreover, various related types of exponential and logarithmic functionals are also investigated.

math-ph↗

arXiv2018

Exact Shannon entropies for the multidimensional harmonic states

In this work we determine and discuss the entropic uncertainty measures of Shannon type for all the discrete stationary states of the multidimensional harmonic systems directly in terms of the states' hyperquantum numbers, the dimensionality and the oscillator strength. We have found that these entropies have a monotonically increasing behavior when both the dimensionality and the population of the states are increasing

quant-ph↗

arXiv2018

Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces

The Rényi entropies of Coulomb systems $R_{p}[ρ], 0 < p < \infty$ are logarithms of power functionals of the electron density $ρ(\vec{r})$ 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.

quant-ph↗

arXiv2018

Exact Rényi entropies of $D$-dimensional harmonic systems

The determination of the uncertainty measures of multidimensional quantum systems is a relevant issue \textit{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 lowest-lying 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 Rényi 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.

math-ph↗

arXiv2017

Biparametric complexities and the generalized Planck radiation law

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 that 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 the 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 an universal character in the sense that they only depend on the 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 $ν_{max}$ at which the spectrum reaches its maximum. Moreover, they allow us to gain insights into new aspects of the d-dimensional blackbody spectrum and about the quantification of quantum effects associated with space dimensionality.

cond-mat.stat-mech↗

arXiv2017

Entropic uncertainty measures for large dimensional hydrogenic systems

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.

quant-ph↗

arXiv2017

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

In this work we find that not only the Heisenberg-like uncertainty products and the Rényi-entropy-based uncertainty sum have the same first-order values for all the quantum states of the $D$-dimensional hydrogenic and oscillator-like systems, respectively, in the pseudoclassical ($D \to \infty$) 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 LMC (López-Ruiz-Mancini-Calvet) and LMC-Rényi complexity measures capture the hydrogenic-harmonic difference in the high dimensional limit already at first order.

quant-ph↗

arXiv2017

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the $D$-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre ($\mathcal{L}_{m}^{(α)}(x)$) and Gegenbauer ($\mathcal{C}^{(α)}_{m}(x)$) polynomials in both position and momentum spaces, where the parameter $α$ linearly depends on $D$. In this work we study the asymptotic behavior as $α\to \infty$ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.

math-ph↗

arXiv2017

The biparametric Fisher-Rényi complexity measure and its application to the multidimensional blackbody radiation

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 information-theoretical 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.

quant-ph↗

arXiv2017

Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems

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 the $D$-dimensional harmonic oscillator states in the pseudoclassical ($D \to \infty$) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.

quant-ph↗

arXiv2016

Entropic measures of Rydberg-like harmonic states

The Shannon entropy, the desequilibrium and their generalizations (Rényi 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 paper 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 oscillator-like 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.

quant-ph↗

arXiv2016

Entropic properties of $D$-dimensional Rydberg systems

The fundamental information-theoretic measures (the Rényi $R_{p}[ρ]$ and Tsallis $T_{p}[ρ]$ entropies, $p>0$) of the highly-excited (Rydberg) quantum states of the $D$-dimensional ($D>1$) hydrogenic systems, which include the Shannon entropy ($p \to 1$) 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 $ρ(\vec{r})$ densities associated to the Rydberg hydrogenic wavefunctions $Ψ_{n,l,\{μ\}}(\vec{r})$, which are closely connected to the $\mathfrak{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the ($n\to\infty$)-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.

quant-ph↗

arXiv2016

Heisenberg-like uncertainty measures for $D$-dimensional hydrogenic systems at large D

The radial expectation values of the probability density of a quantum system in position and momentum spaces allow one to describe numerous physical quantities of the system as well as to find generalized Heisenberg-like uncertainty relations and to bound entropic uncertainty measures. It is known that the position and momentum expectation values of the main prototype of the $D$-dimensional Coulomb systems, the $D$-dimensional hydrogenic system, can be expressed in terms of some generalized hypergeometric functions of the type $_{p+1}F_p(z)$ evaluated at unity with $p=2$ and $p=3$, respectively. In this work we determine the position and momentum expectation values in the limit of large $D$ for all hydrogenic states from ground to very excited (Rydberg) ones in terms of the spatial dimensionality and the hyperquantum numbers of the state under consideration. This is done by means of two different approaches to calculate the leading term of the special functions $_{3}F_2\left(1\right)$ and $_{5}F_4\left(1\right)$ involved in the large $D$ limit of the position and momentum quantities. Then, these quantities are used to obtain the generalized Heisenberg-like and logarithmic uncertainty relations, and some upper and lower bounds to the entropic uncertainty measures (Shannon, Rényi, Tsallis) of the $D$-dimensional hydrogenic system.

quant-ph↗

arXiv2016

Rényi, Shannon and Tsallis entropies of Rydberg hydrogenic systems

The Rényi entropies $R_{p}[ρ], 0<p<\infty$ of the probability density $ρ_{n,l,m}(\vec{r})$ of a physical system completely characterize the chemical and physical properties of the quantum state described by the three integer quantum numbers $(n,l,m)$. The analytical determination of these quantities is practically impossible up until now, even for the very few systems where their Schrödinger equation is exactly solved. In this work, the Rényi entropies of Rydberg (highly-excited) hydrogenic states are explicitly calculated in terms of the quantum numbers and the parameter $p$. To do that we use a methodology which first connects these quantities to the $\mathcal{L}_{p}$-norms $N_{n,l}(p)$ of the Laguerre polynomials which characterize the state's wavefunction. Then, the Rényi, Shannon and Tsallis entropies of the Rydberg states are determined by calculating the asymptotics ($n\rightarrow\infty$) of these Laguerre norms. Finally, these quantities are numerically examined in terms of the quantum numbers and the nuclear charge.

quant-ph↗

arXiv2016

Rényi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics

The Rényi entropies $R_{p}[ρ]$, $p>0,\neq 1$ of the highly-excited quantum states of the $D$-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability $ρ_n(\vec{r})$ density of the Rydberg wavefunctions $Ψ_{n,l,\{μ\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the asymptotics $n\to\infty$ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the Rényi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers ($n,l$), the parameter order $p$ and the universe dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the Rényi entropy power decreases monotonically as the order $p$ is increasing and (b) the disequilibrium (closely related to the second order Rényi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of $D$.

math-ph↗

arXiv2015

Extremum-entropy-based Heisenberg-like uncertainty relations

In this work we use the extremization method of various information-theoretic measures (Fisher information, Shannon entropy, Tsallis entropy) for $d$-dimensional quantum systems, which complementary describe the spreading of the quantum states of natural systems. Under some given constraints, usually one or two radial expectation values, this variational method allows us to determine an extremum-entropy distribution, which is the \textit{least-biased} one to characterize the state among all those compatible with the known data. Then we use it, together with the spin-dependent uncertainty-like relations of Daubechies-Thakkar type, as a tool to obtain relationships between the position and momentum radial expectation values of the type $\langle r^α\rangle^{\frac{k}α}\langle p^k\rangle\geq f(k,α,q,N), q=2s+1$, for $d$-dimensional systems of $N$ fermions with spin $s$. The resulting uncertainty-like products, which take into account both spatial and spin degrees of freedom of the fermionic constituents of the system, are shown to often improve the best corresponding relationships existing in the literature.

quant-ph↗

arXiv2015

Quantum Entanglement in $(d-1)$-Spherium

There are very few systems of interacting particles (with continuous variables) for which the entanglement of the concomitant eigenfunctions can be computed in an exact, analytical way. Here we present analytical calculations of the amount of entanglement exhibited by $s$-states of \emph{spherium}. This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a $(d-1)$-sphere (that is, to the surface of a $d$-dimensional ball). We investigate the dependence of entanglement on the radius $R$ of the system, on the spatial dimensionality $d$, and on energy. We find that entanglement increases monotonically with $R$, decreases with $d$, and also tends to increase with the energy of the eigenstates. These trends are discussed and compared with those observed in other two-electron atomic-like models where entanglement has been investigated.

quant-ph↗