R. J. Yáñez

Fisher information of D-dimensional hydrogenic systems in position and momentum spaces

The spreading of the quantum-mechanical probability distribution density of D-dimensional hydrogenic orbitals is quantitatively determined by means of the local information-theoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.

Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials

The Boltzmann–Shannon information entropy of quantum‐mechanical systems in central potentials can be expressed in terms of the entropy Sn of the classical orthogonal polynomials. Here, an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained. It is shown that this entropy is intimately related to the relative entropy I (ρ0,ρ) of the equilibrium measure ρ0(x) and the weight function ρ(x) of the polynomials. To do so, the theory of strong asymptotics of orthogonal polynomials on compact sets is used.

Entropy of orthogonal polynomials with Freud weights and information entropies of the harmonic oscillator potential

The information entropy of the harmonic oscillator potential V(x)=1/2λx2 in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity Sn(H):= −∫−∞+∞H2n(x)log H2n(x) e−x2dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−‖x‖m), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by Sρ and Sγ, respectively. Briefly, it is shown that, for large values of n, Sρ+1/2logλ≂log(π√2n/e)+o(1) and Sγ−1/2log λ≂log(π√2n/e)+o(1), so that Sρ+Sγ≂log(2π2n/e2)+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. M...

Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Abstract Starting from the second-order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials ∗p n ∗ , we find the analytical expressions of the expansion coefficients of any polynomial rm(x) and of the product rm(x)qj(x) in series of the set ∗p n ∗ . These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric polynomials. Here qj(x) denotes an arbitrary discrete hypergeometric polynomial of degree j. The particular cases in which ∗r m ∗ corresponds to the non-orthogonal families ∗x m ∗ , the rising factorials or Pochhammer polynomials ∗(x) m ∗ and the falling factorial or Stirling polynomials ∗x [m] ∗ are considered in detail. The connection problem between discrete hypergeometric polynomials, which here corresponds to the product case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hypergeometric (CDOH) polynomials is given. Also, the inversion problems of CDOH polynomials associated to the three aforementioned nonorthogonal families are solved.

Functionals of Gegenbauer polynomials and D-dimensional hydrogenic momentum expectation values

The system of Gegenbauer or ultraspherical polynomials {Cnλ(x);n=0,1,…} is a classical family of polynomials orthogonal with respect to the weight function ωλ(x)=(1−x2)λ−1/2 on the support interval [−1,+1]. Integral functionals of Gegenbauer polynomials with integrand f(x)[Cnλ(x)]2ωλ(x), where f(x) is an arbitrary function which does not depend on n or λ, are considered in this paper. First, a general recursion formula for these functionals is obtained. Then, the explicit expression for some specific functionals of this type is found in a closed and compact form; namely, for the functionals with f(x) equal to (1−x)α(1+x)β, log(1−x2), and (1+x)log(1+x), which appear in numerous physico-mathematical problems. Finally, these functionals are used in the explicit evaluation of the momentum expectation values 〈pα〉 and 〈log p〉 of the D-dimensional hydrogenic atom with nuclear charge Z⩾1. The power expectation values 〈pα〉 are given by means of a terminating 5F4 hypergeometric function with unit argument, which is...

Spreading lengths of Hermite polynomials

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments, mentioned previously, are posed.

Fisher information of special functions and second-order differential equations

We investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a non-negative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic D-dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometric-type functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others.

Information-theoretic measures of hyperspherical harmonics

The multidimensional spreading of the hyperspherical harmonics can be measured in a different and complementary manner by means of the following information-theoretic quantities: the Fisher information, the average density or first-order entropic moment, and the Shannon entropy. They give measures of the volume anisotropy of the eigenfunctions of any central potential in the hyperspace. Contrary to the Fisher information, which is a local measure because of its gradient-functional form, the other two quantities have a global character because they are powerlike (average density) and logarithmic (Shannon’s entropy) functionals of the hyperspherical harmonics. In this paper we obtain the explicit expression of the first two measures and a lower bound to the Shannon entropy in terms of the labeling indices of the hyperspherical harmonics.

Four-term recurrence relations of hypergeometric-type polynomials

SummaryThe structural properties of the hypergeometric-type polynomials are, still today, poorly known, except those of the classical orthogonal polynomials (i. e. hypergeometric-type polynomials with Favards orthogonality) in spite of their great usefulness in Mathematical Physics. Here, we study in detail the four-term recurrence and differential-difference relations of the hypergeometric-type polynomials in terms of the coefficients of its second-order differential equation. In so doing, some results of several authors (Tricomi, Marcellán and others) are considerably extended.

Quantum expectation values of D-dimensional Rydberg hydrogenic states by use of Laguerre and Gegenbauer asymptotics

The radial position and momentum (pβ, β ( − 1, 3)) expectation values of the D-dimensional Rydberg hydrogenic states (i.e. states where the electron has a large hyperquantum number n) are rigorously determined by means of powerful tools of the modern approximation theory relative to the asymptotics of the varying orthogonal Laguerre and Gegenbauer polynomials which control the corresponding wavefunctions in position and momentum spaces.