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International Journal of Modern Physics A

Approximate solutions and determination of the thermodynamic properties of the generalized Yukawa potential using extended Nikiforov–Uvarov method

Haifa I. Alrebdi, A. N. Ikot, I. B. Okon, U. S. Okorie, R. Horchani

Published Jun 20, 2025• 1 citation
Publisher

Abstract

In this study, the approximate analytical bound state solutions with a newly proposed generalized Yukawa potential are solved using the extended Nikiforov–Uvarov method with the help of Greene–Aldrich approximation to the centrifugal term. The corresponding wave function is expressed in terms of the confluent Heun function. The numerical bound state solutions are reported for various values of the screening parameter and orbital angular quantum number. Furthermore, the bound state energy spectrum is used to study the thermodynamic properties of the system comprehensively, and we have discussed the influence of inverse temperature and maximum quantum number on the thermodynamic properties.

DOI
10.1142/s0217751x25501106

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The approximate analytical solutions of the non-relativistic Schrӧdinger equation for the Attractive potential model with the centrifugal term are investigated using the elegant methodology of the parametric Nikiforov-Uvarov. The energy equation and the corresponding un-normalized radial wave functions are obtained in a close and compact form after a proper Greene-Aldrich approximation scheme is applied. By changing the numerical values of some potential strengths, special cases of the Attractive potential are investigated in detail. The effects of the potential strengths and potential range respectively on the energy are also studied. The energy is found to be very sensitive to each of the potential parameters. Some theoretic quantities such as information energy, Rẻnyi entropy and Tsallis entropy.
Abstract: In this study, the Schrödinger equation was solved with a superposition of the Hulthen potential and generalized inverse quadratic Yukawa potential model using the Nikiforov-Uvarov (NU) method. For completeness, we also calculated the wave function. To validate our results, the numerical bound state energy eigenvalues was computed for various principal n and angular momentum l quantum numbers. With the aid of the Hellmann-Feynman theorem, the expressions for the expectation values of the square of inverse of position, r^(-2), inverse of position, r^(-1), kinetic energy, T ̂ and square of momentum, p ̂ are calculated. By adjusting the potential parameters, special cases of the potential were considered, resulting in Generalized Inverse Quadratic Yukawa potential, Hulthen potential, Coulomb potential, Kratzer potential, Inversely Quadratic Yukawa potential and Coulomb plus inverse square potential, respectively. Their energy eigenvalue expressions and numerical computations agreed with the literature. Keywords: Schrödinger equation, Hulthen potential (HP), Generalized inverse quadratic Yukawa potential (GIQYP), Nikiforov-Uvarov method. PACS: 03.65.−w, 03.65.Ca, 03.65.Ge.
In this study, we employ the extended Nikiforov–Uvarov method to solve the d-dimensional Schrödinger equation with exponential pseudo-harmonic potential. With the help of Greene–Aldrich approximation scheme, analytical expression of the approximate energy eigenvalues and wave function was obtained in closed form and in terms of confluent Heun function, respectively. In addition, expressions of partition function and other thermodynamic functions for the exponential pseudo-harmonic potential were obtained, as they relate with temperature, screening parameter and dimensionality. Numerical and graphical variations of the energies and thermodynamic functions have been presented for various quantum states, temperature, screening parameter and dimensionality values. It is observed that energy eigenvalues and thermodynamic functions of the exponential pseudo-harmonic potential are significantly affected by the screening parameters, dimensionality, temperature and quantum state values considered. Special results for pseudo-harmonic and harmonic potentials are realized as the screening parameter approaches zero. Our results agree with some physical phenomena in condensed matter and atomic physics.
In this work, we obtained an approximate bound state solution to Schrodinger with Hulthen plus exponential Coulombic potential with centrifugal potential barrier using parametric Nikiforov-Uvarov method.We obtained both the eigen energy and the wave functions to non -relativistic wave equations.We implement Matlab algorithm to obtained the numerical bound state energies for various values of adjustable screening parameter at various quantum state..The developed potential reduces to Hulthen potential and the numerical bound state energy conform to that of existing literature.