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Progress of Theoretical and Experimental Physics

Ring potential generated from the central hyperbolic Manning-Rosen potential using the transformation method

Arup Bharali

Published Mar 1, 2013• 0 citations
Publisher

Abstract

An exactly solved ring potential is generated from the central hyperbolic Manning–Rosen potential using the transformation method in the framework of non-relativistic quantum mechanics. The basis of the method is coordinate redesignation and extended transformation, comprising a coordinate transformation supplemented by a functional transformation. Application of the coordinate transformation entailing redesignation of the radial coordinate with polar angle to the Schrödinger radial equation for the exactly solved central hyperbolic Manning–Rosen potential in 3D Euclidean space yields a second-order homogeneous angular differential equation. The functional transformation is indispensable in molding the angular differential equation to the Schrödinger angular equation form and, in the process of retrieving the standard Schrödinger angular equation by invoking a plausible ansatz, a new exactly solved ring potential is generated. The transformed angular wave functions for the generated ring potential are normalizable.

DOI
10.1093/ptep/ptt001

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An exactly solved ring potential is generated from the central hyperbolic Manning–Rosen potential using the transformation method in the framework of non-relativistic quantum mechanics. The basis of the method is coordinate redesignation and extended transformation, comprising a coordinate transformation supplemented by a functional transformation. Application of the coordinate transformation entailing redesignation of the radial coordinate with polar angle to the Schrödinger radial equation for the exactly solved central hyperbolic Manning–Rosen potential in 3D Euclidean space yields a second-order homogeneous angular differential equation. The functional transformation is indispensable in molding the angular differential equation to the Schrödinger angular equation form and, in the process of retrieving the standard Schrödinger angular equation by invoking a plausible ansatz, a new exactly solved ring potential is generated. The transformed angular wave functions for the generated ring potential are normalizable.
We apply the extended transformation method, comprising a coordinate transformation supplemented by a functional transformation, to generate (regenerate) exactly solvable ring-shaped potentials from already known exactly solvable central potentials for physical quantum systems in a non-relativistic regime. The angular wave functions corresponding to the generated potentials are transformed from the radial wave functions of the original central potentials in a very straightforward way and they are also shown to be normalizable.
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 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 Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.
We present the solution of the Duffin–Kemmer–Petiau equation for Manning–Rosen potential plus a ring-shaped-like potential in (1+3)-dimensional space–time for spin-one particles within the framework of an exponential approximation for the centrifugal term. We have used the Nikiforov–Uvarov method in our calculations. The radial wavefunction and the angular wavefunctions are expressed in terms of Jacobi polynomials. We have also represented some numerical results for the Manning–Rosen potential plus a ring-shaped-like potential.
In this paper, we study the Schrödinger equation with non-central modified Killingbeck potential plus a ring-shaped-like potential problem, which is not spherically symmetric. The factorization method is used to solve the hypergeometric equation types which lead to solutions with the associate Laguerre function for the radial part and Jacobi polynomial for the polar part. We introduce the raising and lowering operators to calculate the energies eigenvalues, which show that the lack of spherical symmetry removes the degeneracy of second quantum number m which is completely expected. These obtained energies are better to explain the superposition of the energy levels of the atoms in the crystalline structure of molecules.