Entropy and complexity analysis of the $$D$$ D -dimensional rigid rotator and hyperspherical harmonics

In this research work, the quantum information‐theoretic analysis of the static screened Coulomb potential has been carried out by studying both analytically and numerically the entropic measures, Fisher information as well as the Onicescu information energy of its wave function. Explicit expressions of these information‐theoretic measures were obtained. Using the Srivastava–Daoust linearization formula in terms of the multivariate Lauricella hypergeometric function, the Rényi entropy, Tsallis entropy, Onicescu information energy were analytically obtained. From the results obtained, it is observed that some of the Shannon entropies are negative, indicating that, negative entropies exists for the probability densities that are highly localized. The trends in the variation of the information‐theoretic measures with the potential screening parameter a for this atomic model are discussed. The Bialynicki‐Birula, Mycielski inequality (BBM), and the Fisher information based uncertainty relation are also verified.

“…The explicit evaluation of the Rényi entropy, Tsallis entropy, and Onicescu information energy for the SCP shall be shown in this section. The Rényi entropy from Equation is expressed as

R_{q}^{SCP} true[{normalρ}_{n} true] = \frac{1}{1 - q} \ln W_{q}^{SCP} true[{normalρ}_{n} true],

where the entropic moment

W_{q}^{SCP} true[{normalρ}_{n} true]

is expressed as

W_{q}^{SCP} true[{normalρ}_{n} true] = \int_{0}^{\infty} {[normalρ false(r false)]}^{q} d r = \frac{N_{n l}^{2 q}}{4 a} \int_{- 1}^{1} {(\frac{1 - z}{2})}^{2 ϵ q - 1} {true(\frac{1 + z}{2} true)}^{2 ℓ q + 2 q} {true[P_{n}^{(2 ϵ, 2 ℓ + 1)} (z) true]}^{2 q} d z, z = 1 - 2 s .

…”

Section: Information‐theoretic Measures For the Scpmentioning

confidence: 99%

“…The term

{true[P_{n}^{(2 ϵ, 2 ℓ + 1)} (z) true]}^{2 q}

in Equation in the integral is linearized by means of Srivastava Linearization formula, which gives:

{true[P_{n}^{(2 ϵ, 2 ℓ + 1)} (z) true]}^{2 q} = \sum_{i = 0}^{\infty} \overset{true\sim}{c_{i}} (0, 2 q, n, 2 ϵ, 2 ℓ + 1, 2 ϵ q, 2 ℓ q - 2 q) P_{i}^{(2 ϵ q, 2 ℓ + 1)} (z) .

…”

Section: Information‐theoretic Measures For the Scpmentioning

confidence: 99%

Quantum information‐theoretic measures for the static screened Coulomb potential

Isonguyo

Oyewumi

Oyun

2018

“…The computational determination of these quantities 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. [1][2][3][4][5] The harmonic oscillator is both a pervasive concept in science and technology and a fundamental building block in our system of knowledge of the physical universe. [6,7] Indeed it has been applied from the physics of quarks to quantum cosmology.…”

mentioning

confidence: 99%

“…Then, the position probability density of the isotropic harmonic oscillator has the form (5) where the radial part is given by…”

mentioning

confidence: 99%

Entropic measures of Rydberg‐like harmonic states

Dehesa

Toranzo

Puertas‐Centeno

2016

The Shannon entropy, the desequilibrium and their generalizations (R\'enyi 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.Comment: Accepted in International Journal of Quantum Chemistr

“…[9] The modified LMC complexity, that is, the shape LMC is the product of the Shannon length and the disequilibrium [2,4,5] and have been studied in different contexts. [4,[10][11][12][13][14] Another measure of complexity is the Fisher-Shannon (FS) complexity. [15][16][17][18] It is defined as the product of the Fisher information [19] and Shannon entropic power.…”

mentioning

confidence: 99%

Complexity analysis of two families of orthogonal functions

Ghosh¹,

Nath

2019

We study the shape LMC (López-Ruiz, Mancini and Calvet), Fisher-Shannon (FS) andCramér-Rao (CR) complexities of two families of orthogonal functions associated with the solutions of isospectral deformations of the Pöschl-Teller and Harmonic oscillator potentials. We have compared the behavior of these complexities for the orthogonal functions with the complexities associated with Pöschl-Teller and Harmonic oscillator potentials whose solutions are given in terms of the classical orthogonal polynomials. All these complexities are discussed in terms of the quantum number n and isospectrality parameter λ. K E Y W O R D SCramér-Rao complexity, Fisher-Shannon complexity, LMC complexity 1 | INTRODUCTION Application of statistical measures in physical as well as in social sciences has a role of growing importance. There exists a vast literature on the use of various measures of complexity in different contexts for example, dynamical systems, cellular automata, neural networks, social sciences, complex molecules, geophysical and astrophysical processes etc. In particular, López-Ruiz, Mancini, and Calvet (LMC) complexity [1][2][3][4][5] has been computed in position and momentum spaces [6] for the density functions of the hydrogen-like atoms and the isotropic harmonic oscillator. [7,8] The LMC complexity is defined as a product of two factors-one of which is a measure of the disequilibrium, that is, it quantifies the departure of the probability density from uniformity while the other one is the Shannon entropy which is a measure of uncertainty or randomness. [9] The modified LMC complexity, that is, the shape LMC is the product of the Shannon length and the disequilibrium [2,4,5] and have been studied in different contexts. [4,[10][11][12][13][14] Another measure of complexity is the Fisher-Shannon (FS) complexity. [15][16][17][18] It is defined as the product of the Fisher information [19] and Shannon entropic power. The LMC and the FS complexity measures have been applied in different fields of physics such as multi electron systems in position and momentum spaces, [20,21] analysis of signals, [15] electron correlation, [16] atomic systems and ionization processes [18,22] and in quantum mechanics. [7,8,23] The third complexity measure that we shall study is the Cramér-Rao (CR) complexity which is defined as the product of the Fisher information and the variance of the density function measuring the degree of deviation from the mean value. [20,22,24,25] The three complexities mentioned above share a set of characteristics, namely, they are (a) dimensionless (b) bounded below by unity, and (c) minimum for two extreme distributions which correspond to perfect order and maximum disorder. Also they are invariant under replication, translation and scaling transformation. [4,[26][27][28][29][30] It is to be noted that various information theoretic measures of uncertainty and complexity have been studied in great detail for the wellknown classical orthogonal polynomials which generate solutions of problems like the Harm...