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Physica Scripta

The Manning–Rosen potential studied by a new approximate scheme to the centrifugal term

Wen-Chao Qiang, Shi‐Hai Dong

Published Mar 31, 2009• 78 citations
Publisher

Abstract

We present a new approximate scheme to the centrifugal term and then apply this new approach to solve the Schr?dinger equation with the Manning?Rosen potential. The bound state energy levels are obtained. A closed form of normalization constant of the wavefunctions is also found. It is shown that the present results are much better than those obtained previously and are in good agreement with the accurate numerical results obtained by a MATHEMATICA package.

DOI
10.1088/0031-8949/79/04/045004

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We present a new approximate scheme to the centrifugal term and then apply this new approach to solve the Schr?dinger equation with the Manning?Rosen potential. The bound state energy levels are obtained. A closed form of normalization constant of the wavefunctions is also found. It is shown that the present results are much better than those obtained previously and are in good agreement with the accurate numerical results obtained by a MATHEMATICA package.
In this paper, the bound-state solution of the modified radial Schrödinger equation is obtained for the Manning–Rosen plus Hulthén potential by using new developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial wave functions are defined for any [Formula: see text] angular momentum case via the Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. Thanks to both methods, equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is presented. 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 the present work, an analytical solution for bound states of the modified Schrödinger equation is found for the new supposed combined Manning-Rosen potential plus the Yukawa class. To overcome the difficulties arising in the case l ≠ 0 in the centrifugal part of the Manning-Rosen potential plus the Yukawa class for bound states, we applied the developed approximation. Analytical expressions for the energy eigenvalue and the corresponding radial wave functions for an arbitrary value l ≠ 0 of the orbital quantum number are obtained. And also obtained eigenfunctions expressed in terms of hypergeometric functions. It is shown that energy levels and eigenfunctions are very sensitive to the choice of potential parameters.
In this paper, the bound state solution of the modified radial Schrödinger equation is obtained for the Manning-Rosen plus Hulthén potential by implementing the novel improved scheme to surmount the centrifugal term. The energy eigenvalues and corresponding radial wave functions are defined for any l ≠ 0 angular momentum case via the Nikiforov-Uvarov (NU) and supersymmetric quantum mechanics (SUSYQM) methods. By using these two different methods, equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is demonstrated. The energy levels are worked out and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary l 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 nr radial and l orbital quantum numbers.
The Schrödinger equation in D-dimensions for the Manning-Rosen potential with the centrifugal term is solved approximately to obtain bound states eigensolutions (eigenvalues and eigenfunctions).The Nikiforov-Uvarov (NU) method is used in the calculations.We present numerical calculations of energy eigenvalues to two-and four-dimensional systems for arbitrary quantum numbers n and l with three different values of the potential parameter α.It is shown that because of the interdimensional degeneracy of eigenvalues, we can also reproduce eigenvalues of a upper/lower dimensional sytem from the well-known eigenvalues of a lower/upper dimensional system by means of the transformation (n, l, D) → (n, l ± 1, D ∓ 2)..This solution reduces to the Hulthén potential case.
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 .
We comment on several incorrect results obtained by Dong and Garcia-Ravelo (2007 Phys. Scr. 75 307) in the discussion of the bound-state solutions of the radial Schrodinger equation for the Manning–Rosen potential. We have identified the inconsistency in their reasoning on the allowed values of the parameter α and we have constructed the correct energy spectrum together with the normalized wavefunctions. Furthermore, we have shown that the wavefunction normalization constant can be obtained in closed form.