The spreading of the quantum-mechanical probability cloud for the ground state of the Morse and modified Poschl–Teller potentials, which controls the chemical and physical properties of some molecu...

Information-theoretic measures for Morse and Pöschl-Teller potentials

Information-theoretic measures for Morse and Poschl-Teller potentials J. S. DEHESAy, A. MARTINEZ-FINKELSHTEINz

Bounds and inequalities involving Jacobian elliptic functions are obtained. The main results are established with the aid of new bounds for the incomplete elliptic integral R F . It is also shown that some known inequalities for the trigonometric and hyperbolic functions are the limiting cases of the results proven in this paper.

One- and two-sided inequalities for Jacobian elliptic functions and related results

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.

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Fisher information of special functions and second-order differential equations

Based on general discussion of series expansions of Carlson’s symmetric elliptic integrals, we developed fifteen kinds of them including eleven new ones by utilizing the symmetric nature of the integrals. Thanks to the special addition formulas of the integrals, we also obtained their complementary series expansions. By considering the balance between the speed of convergence and the amount of computational labor, we chose four of them as the best series expansions. Practical evaluation of the integrals is conducted by the most suitable one among these four series expansions. Its selection rule was analytically specified in terms of the numerical values of given parameters. As a by-product, we obtained an efficient asymptotic expansion of the integrals around their logarithmic singularities. Numerical experiments confirmed the effectiveness of these new series expansions.

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Series expansions of symmetric elliptic integrals

The spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann–Shannon information entropy. This problem boils down to the calculation of the asymptotics of the entropy-like integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number n is given in detail. It is shown that the semiclassical (WKB) position–space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single-particle systems with any power-type potential of the form V(x)=x2k, x∈R and k∈N. The momentum–space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form −ln ln n, in strong contrast with the...

V. N. Sorokin

Papers

Asymptotics of information entropies of some Toda-like potentials