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arXiv:1408.3535

Analysis of quantum-mechanical states of the Mie-type ring shaped model via the Fisher's information entropy

B. J. Falaye, K. J. Oyewumi, Sameer M. Ikhdair, M. Hamzavi

Published Aug 15, 2014• 0 citations
arXivHTMLPDF

Abstract

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 $\ell-$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 $(n, l, m)$ 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.

DOI
10.48550/arxiv.1408.3535

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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.
Recently, the information theory of quantum-mechanical systems has aroused the interest of many theoretical physicists. This is due to the fact that it provides a deeper insight into the internal structure of the system. Also, it is the strongest support of the modern quantum computation and information, which is a basic theory for numerous technological developments. This study reports the solution of Schrödinger equation with the ring-shaped Mie-type potential. The rotational-vibrational spectroscopic study 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) in the ring-shaped Mie-type potential is obtained. Finally, the analysis for this distribution via a complementary information measures of a probability distribution known as the Fisher's information have been presented.
In this study, approximate analytical solution of Schrödinger, Klein-Gordon and Dirac equations under the Tietz-Wei (TW) diatomic molecular potential are represented by using an approximation for the centrifugal term. We have applied three types of eigensolution techniques; the functional analysis approach (FAA), supersymmetry quantum mechanics (SUSYQM) and asymptotic iteration method (AIM) to solve Klein-Gordon Dirac and Schrödinger equations, respectively. The energy eigenvalues and the corresponding eigenfunctions for these three wave equations are obtained and some numerical results and figures are reported. It has been shown that these techniques yielded exactly same results. some expectation values of the TW diatomic molecular potential within the framework of the Hellmann-Feynman theorem (HFT) have been presented. The probability distributions which characterize the quantum-mechanical states of TW diatomic molecular potential are analysed by means of complementary information measures of a probability distribution called the Fishers information entropy. This distribution has been described in terms of Jacobi polynomials, whose characteristics are controlled by the quantum numbers.
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 entropic perspective on the molecular electronic structure is investigated. Information-theoretic description of electron probabilities is extended to cover the complex amplitudes (wave functions) of quantum mechanics. This analysis emphasizes the entropic concepts due to the phase part of electronic states, which generates the probability current density, thus allowing one to distinguish the information content of states generating the same electron density and differing in their current densities. The classical information measures of Fisher and Shannon, due to the probability/density distributions themselves, are supplemented by the nonclassical terms generated by the wave-function phase or the associated probability current. A complementary character of the Fisher and Shannon information measures is explored and the relationship between these classical information densities is derived. It is postulated to characterize also their nonclassical (phase/current-dependent) contributions. The continuity equations of the generalized information densities are examined and the associated nonclassical information sources are identified. The variational rules involving the quantum-generalized Shannon entropy, which generate the stationary and time-dependent Schrödinger equations from the relevant maximum entropy principles, are discussed and their implications for the system “thermodynamic” equilibrium states are examined. It is demonstrated that the lowest, stationary “thermodynamic” state differs from the true ground state of the system, by exhibiting the space-dependent phase, linked to the modulus part of the wave function, and hence also a nonvanishing probability current.
In this paper, Fisher and Shannon information entropies in two-dimensional spaces were analysed for hydrogen diatomic molecules using a non-central exponential-type potential, both in the presence and absence of magnetic and Aharanov-Bohm (AB) flux fields. By solving the two-dimensional Schrödinger equation in the presence of charged Hamiltonian using the parametric Nikiforov-Uvarov (NU) method, we obtained numerical solutions for different potential depths for Fisher and Shannon information entropies. These results adhere to Bialynicki-Birula and Mycielski (BBM) and Fisher-based inequalities. The findings have significant applications in chemical physics, aiding in the analysis of molecular electronic properties, stability and reactivity under varying external influences. Additionally, the study contributes to quantum information by offering insights into the localisation and delocalisation phenomena, with direct applications to quantum computation, entanglement, and communication technologies.
In this study, the Shannon entropy and the Fisher information is investigated with molecular Deng-Fan-Eckart potential for the diatomic molecules and ScF in position and momentum spaces in three dimensions for the ground and the excited states. The results were numerically obtained for diatomic molecules. Localization is observed for Shannon entropy and delocalization for Fisher information for both molecules in the position and momentum spaces. The uncertainty relations for the selected diatomic molecules were satisfied accordingly.