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Information and complexity measures for the ring-shaped modified Kratzer potential †

W. A. Yahya, K. J. Oyewumi, K. D. Sen

Published Oct 1, 2014• 7 citations
OpenAlex

Abstract

There has been an interest in studies on informationtheoretic and complexity measures for quantum model systems in the recent years. The information entropy for the trigonometric Rosen-Morse potential, harmonic oscillator, hydrogenic atoms, Morse potential, Yukawa and Hulthen potentials have been investigated. The Kratzer potential plays an important role in atomic and molecular physics and quantum chemistry. Cheng and Dai proposed a new potential consisting of the modified Kratzer potential plus a new proposed ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. In this work, we shall obtain the Fisher information, Fisher length, Shannon entropy, Shannon length, Renyi entropy, Renyi length, Tsallis entropy and the Fisher-Shannon complexity for the ring-shaped modified Kratzer potential. In this section, we present the definition of the information theoretical measures used in this work which are based on the 1-normalized electron density in the position and momentum space. The Shannon information entropy in the position space is given by Eq. (1). ( ) ( ) log ( )d S ρ ρ ρ = −∫ r r r ... (1)

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There has been an interest in studies on informationtheoretic and complexity measures for quantum model systems in the recent years. The information entropy for the trigonometric Rosen-Morse potential, harmonic oscillator, hydrogenic atoms, Morse potential, Yukawa and Hulthen potentials have been investigated. The Kratzer potential plays an important role in atomic and molecular physics and quantum chemistry. Cheng and Dai proposed a new potential consisting of the modified Kratzer potential plus a new proposed ring-shaped potential which has possible applications to ring-shaped organic molecules like cyclic polyenes and benzene. In this work, we shall obtain the Fisher information, Fisher length, Shannon entropy, Shannon length, Renyi entropy, Renyi length, Tsallis entropy and the Fisher-Shannon complexity for the ring-shaped modified Kratzer potential. In this section, we present the definition of the information theoretical measures used in this work which are based on the 1-normalized electron density in the position and momentum space. The Shannon information entropy in the position space is given by Eq. (1). ( ) ( ) log ( )d S ρ ρ ρ = −∫ r r r ... (1)
The approximate solutions of the Schrödinger equation with the central generalized Hulthén and Yukawa potential are investigated within the framework of the functional method. The obtained wave function and the energy levels are used to study the Shannon entropy, Renyi entropy, Fisher information, Shannon Fisher complexity, Shannon power and Fisher-Shannon product in both position and momentum spaces for the ground and first excited states. The theoretic information theories such as Shannon entropies Sr,Sp, Fisher information Ip,Ip and Renyi entropies Rβ as well as the information entropy probability densities ρ(r),ϕ(p) are illustrated graphically. The Bialynicki-Birula – Mycielski (BBM) inequality for the Shannon entropies and the Cramer-Rao inequality for the Fisher information are satisfied for the potential model.
Information entropies, such as Shannon, Fisher and Rényi entropies, with their associated quantities, are calculated and discussed for the Morse potential in the position ( r ) and momentum ( p ) spaces using the J-matrix method. The entropies, for the s-wave, are numerically obtained for the diatomic molecules H 2 and LiH. It is the first information entropy calculation for real diatomic molecules using the Morse potential. Interesting characteristic features of the entropy densities are shown, where the position and momentum space information entropies satisfy the uncertainty relations.
We calculate information theoretic spreading measures of the position and momentum wave functions of the symmetric trigonometric Rose–Morse potential for the states . The position space Renyi entropy, Fisher entropy and their corresponding entropic lengths are presented analytically for . The momentum space Renyi entropy and the Renyi lengths are calculated analytically for n = 0,1 and are calculated numerically for . We also calculated the Fisher entropy and the Fisher lengths in the momentum space numerically for .
In this research article, the influence of magnetic and Aharonov-Bohm flux fields on Shannon entropy with improved Kratzer potential for Scandium fluoride (ScF) and hydrogen (H2) molecules are studied analytically and numerically for three low-lying states. The wave function and probability density which controls the chemical and physical properties of atomic and molecular systems have been obtained by solving the Schrodinger wave equation coupled with charged particle Hamiltonian via the Nikiforov-Uvarov functional analysis method. The total normalized wave function was expressed in terms of the hypergeometric function of Jacobi polynomials and then utilized to evaluate the Shannon entropy for two diatomic molecules in position and momentum spaces. The wave function and electronic densities were graphically analyzed. Our numerical results of Shannon entropies sum as applied to ScF and H2 were verified to obey the Shannon-entropy-based uncertainty relation. Observations from our results also reveal negative values of Shannon entropies in the position coordinates depicting high localization of particles and stability. In Addition, the effects of the external fields on the quantum systems were observed to influence the quantum information-theoretic measure by small decrements in the Shannon entropy values when compared to the case of the absence of the external fields, indicating a further improvement in the localization of particles and system's stability. This work has applications in atomic and molecular dynamic, molecular drug design, nanostructure processes, information technology, amongst others.
The identification on probabilities of nonrelativistic particles moved in doubled-ring shaped Coulomb oscillator (DRSCO) potential has been analyzed by the definition on Shannon entropy information to get the quantify of the quantum entanglement and its correlations. We proposed the new modified potential called DRSCO potential that have been solved in recent research paper and the solution of ground state wave function in axial part was used to get its entropy information. The position-space and momentum-space information entropy were investigated by numerical results by using GNU Octave 4.2.2 and have been shown graphicaly.
In the recent years, information theory of quantum-mechanical systems have aroused the interest of many Theoretical Physicist. This due to the fact that it provides a deeper insight into the internal structure of the systems. Also, It is the strongest support of the modern quantum computation and information, which is basic for numerous technological developments. This study report the any ℓ−\ell-ℓ−state solution of the radial Schrödinger equation with the Mie-type ring shaped diatomic molecular potential. Rotational-vibration of some few selected diatomic molecules are given. The probability distribution density of the system which gives the probability density for observing the electron in the state characterized by the quantum numbers (n,l,m)(n, l, m)(n,l,m) in the Mie-type ring shaped diatomic molecular potential is obtained. Finally, we analyze this distribution via a complementary information measures of a probability distribution called as the Fisher's information entropy.