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Karbala International Journal of Modern Science

Bound state solutions of D-dimensional Klein–Gordon equation with hyperbolic potential

C. A. Onate, A. N. Ikot, M. C. Onyeaju, M. E. Udoh

Published Dec 27, 2016• 20 citations
Publisher

Abstract

By using the basic supersymmetric quantum mechanics concepts and formalism, the energy eigenvalue equation and the corresponding wave function of the Klein–Gordon equation with vector and scalar potentials for an arbitrary dimensions are obtained together with hyperbolic potential using a suitable approximation scheme to the orbital centrifugal term. The non-relativistic limit is obtained and the numerical values for various values of D, n, α and ℓ are obtained.

DOI
10.1016/j.kijoms.2016.12.001

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By using the basic supersymmetric quantum mechanics concepts and formalism, the energy eigenvalue equation and the corresponding wave function of the Klein–Gordon equation with vector and scalar potentials for an arbitrary dimensions are obtained together with hyperbolic potential using a suitable approximation scheme to the orbital centrifugal term. The non-relativistic limit is obtained and the numerical values for various values of D, n, α and ℓ are obtained.
Abstract In this study, the bound state solutions of the Klein-Fock-Gordon equation are examined for the sum of Manning-Rosen and Yukawa potential by using a recent improved scheme to deal with the centrifugal term. For any l ≠ 0, the energy eigenvalues and corresponding radial wave functions are determined under the condition of equal scalar and vector potentials. In order to obtain bound state solutions, we use two different methods called supersymmetric quantum mechanics (SUSYQM) and Nikiforov-Uvarov (NU) methods. The identical expressions for the energy eigenvalues are obtained, and the expression of radial wave functions transformations to each other is revealed via both methods. For arbitrary l states, the energy levels and the corresponding normalized eigenfunctions are given in terms of the Jacobi polynomials. A closed form of the normalized wave function is also obtained. It is seen that the energy eigenvalues and eigenfunctions are sensitive to nr radial and l orbital quantum numbers.
The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.
In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.
In this paper, the analytical solutions of the [Formula: see text]-dimensional hyper-radial Schrödinger equation are studied in great detail for the Hulthén potential. Within the framework, a novel improved scheme to surmount centrifugal term, the energy eigenvalues and corresponding radial wave functions are found for any [Formula: see text] orbital angular momentum case within the context of the Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. In this way, based on these methods, the same expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transforming each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are obtained in terms of orthogonal polynomials for arbitrary [Formula: see text] states for [Formula: see text]-dimensional space.