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Physica Scripta

Comment on ‘Exact solutions of thes-wave Schrödinger equation with Manning–Rosen potential’

Farid Benamira, Larbi Guechi, A. Zouache

Published Jun 12, 2009• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/80/01/017001

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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.
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.
We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.
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.
We present solutions of the Schrodinger equation with superposition of Manning-Rosen plus inversely Mobius square plus quadratic Yukawa potentials using parametric Nikiforov Uvarov method along with an approximation to the centrifugal term. The bound state energy eigenvalues for any angular momentum quantum number l and the corresponding un-normalized wave functions are calculated. The mixed potential which in some particular cases gives the solutions for different potentials: the Manning-Rosen, the Mobius square, the inversely quadratic Yukawa and the Hulthén potentials along with their bound state energies are obtained.