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arXiv:1406.1716 · Physica Scripta

Eigensolution techniques, their applications and Fisherʼs information entropy of the Tietz–Wei diatomic molecular model

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

Published Oct 21, 2014• 87 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0031-8949/89/11/115204

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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.
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Using the supersymmetric quantum mechanics (SUSY) approach, we obtained analytical approximate solutions to the Dirac equation with an improved Tietz diatomic potential function under spin and pseudospin symmetries. The energy eigenvalues equation and the corresponding wave spinor for the two symmetries have been explicitly obtained. The non relativistic limit of the spin symmetry has been obtained. The solution of Morse, Deng-Fan, and Rosen-Morse were obtained by changing the numerical value of q We have also obtained the thermodynamic properties of a system under the improved Tietz potential function.
The eigensolutions of the Schrödinger equation under the Varshni potential function are studied with two eigensolution techniques such as the Nikiforov–Uvarov and the semi-classical WKB approximation methods. We extended the work to investigate the analytical and numerical Fisher information measure of complexities and also the expectations values using the Hellmann–Feynman theorem. We determined the transmission coefficient using the WKB method. The WKB energy levels and mean values fluctuate with the potential range compared with the NU derived values. Our results for the Fisher information measure obey the uncertainty relation I(ρ)I(γ)≥36 and the Cramer–Rao inequality for position space (I(ρ)⟨r2⟩≥9). The mean values conform to the ones reported in existing literature.
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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.