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Polynomial Solutions of the Schroé dinger Equation for the ëëDeformedíí Hyperbolic Potentials by NikiforovñUvarov Method

Harun Egë

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

In this study, the bound state energy eigenvalues and the corresponding wave functions of the deformed Poe schlTeller and Hyperbolic Kratzer-like potentials have been obtained by the NikiforovUvarov method using the deformed hyperbolic functions (sinh q (x) 4 1 (ex ( qe~x), cosh q (x) 4 and that 1 (ex ) qe~x), sech q (x) 4 1/cosh q (x) tanh q (x) 4 sinh q (x)/cosh q (x)) were introduced for the —rst time by Arai (J. Math. Anal. Appl. 158, 63 (1991)). It is also observed that, the energy eigenvalues of these iinewˇˇ type deformed hyperbolic potentials bring up the criteria for the shape invari- ance of the potentials according to the deformation parameter q.

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In this study, the bound state energy eigenvalues and the corresponding wave functions of the deformed Poe schlTeller and Hyperbolic Kratzer-like potentials have been obtained by the NikiforovUvarov method using the deformed hyperbolic functions (sinh q (x) 4 1 (ex ( qex), cosh q (x) 4 and that 1 (ex ) qex), sech q (x) 4 1/cosh q (x) tanh q (x) 4 sinh q (x)/cosh q (x)) were introduced for the —rst time by Arai (J. Math. Anal. Appl. 158, 63 (1991)). It is also observed that, the energy eigenvalues of these iinewˇˇ type deformed hyperbolic potentials bring up the criteria for the shape invari- ance of the potentials according to the deformation parameter q.
In this paper, we introduce the exact solution for the wave function in the presence of potential energy, consisting of combination between q-deformed hyperbolic and exponential function with different argument. The functions we have used in the present communication can be regarded as a generalization of the Arai q-deformed function (modified q-deformed Morse potential). In this context, we have restricted our discussion for some particular cases of the q-deformed hyperbolic functions. This is due to the difficulty for dealing with most of the arguments included in the potential functions. For the most particular cases, the energy eigenfunctions are obtained, and the behavior is also discussed. It has been shown that the wave functions are sensitive to the variation in the value of q-deformed parameter as well as the strength of the potential parameter λ. Furthermore, the energy eigenvalues are also considered for some particular cases where the argument of the exponential function plays a strong role effecting its value.
The arbitrary l -wave solutions to the Schrödinger equation for the deformed hyperbolic Manning-Rosen potential is investigated analytically by using the Nikiforov-Uvarov method, the centrifugal term is treated with an improved Greene and Aldrich’s approximation scheme. The wavefunctions depend on the deformation parameter q , which is expressed in terms of the Jocobi polynomial or the hypergeometric function. The bound state energy is obtained, and the discrete spectrum is shown to be independent of the deformation parameter q .
In this work, we propose a new potential called the "q-deformed Woods-Saxon plus hyperbolic tangent potential." We derive the generalized Schrödinger equation for quantum mechanical systems with position-dependent masses under these potentials using the Nikiforov-Uvarov method, with the mass relationship defined as m(x)=m_1⁄((1+qe^(-2λx))). The solutions to this equation, expressed in terms of hypergeometric functions and Jacobi polynomials, offer insights into the quantum behavior of particles. The energy eigenvalues depend on system parameters such as the deformation parameter q, potential parameters, and quantum numbers. We analyzed the effect of the deformation parameter q numerically and visually using different values of these parameters.
In this study, an approximate solution of the Schrödinger equation for the q-deformed Hulthen-quadratic exponential-type potential model within the framework of the Nikiforov–Uvarov method was obtained. The bound state energy equation and the corresponding eigenfunction was obtained. The energy spectrum is applied to study H2, HCl, CO and LiH diatomic molecules. The effect of the deformation parameters and other potential parameters on the energy spectra of the system were graphically and numerically analyzed in detail. Special cases were considered when the potential parameters were altered, resulting in deformed Hulthen potential, Hulthen potential, deformed quadratic exponential-type potential and quadratic exponential-type potential. The energy eigenvalues expressions agreed with what obtained in literature. Finally, the results can find many applications in quantum chemistry, atomic and molecular physics.
We study the nonrelativistic Schrödinger equation for q-deformed Hulthen plus modified inversely quadratic Yukawa potential using the generalized parametric form of NIkiforov-Uvarov method. The energy eigenvalues and the corresponding normalized wave functions expressed in terms of hypergeometric function are obtained. We have also discussed four (4) special cases of this potential, i.e. Woods-Saxon, Hulthen, Morse and inversely quadratic potentials. Keywords: Nonrelativistic Schrödinger equation, q-deformed Hulthen potential, modified inversely quadratic Yukawa potential, parametric Nikiforov-Uvarov method.