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Physica Scripta

Systematic search of exactly solvable ring-shaped potential using the transformation method

Arup Bharali

Published Aug 22, 2013• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/88/03/035009

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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.
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.
A transformation method is presented which consist of a coordinate transformation and a functional transformation that allow generation of exact analytic bound state solutions of the Schrodinger Green's function equation. The method proposed here, which involves the generation of an exact analytic solution from solvable decemvirate power potential within the framework of Green's functions technique, makes it possible to generate a number of solved quantum systems for original quantum systems with multi-term potentials. The generated quantum systems are in general energy dependent with a single normalized eigenstate. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential.
We present a general procedure by which solvable non-central potentials can be obtained in 2 and 3 dimensions by the separation of the angular and radial variables. The method is applied to generate solvable non-central PT-symmetric potentials in polar coordinates. General considerations are presented concerning the PT transformation properties of the eigenfunctions, their pseudo-norm and the nature of the energy eigenvalues. It is shown that within the present framework the spontaneous breakdown of PT symmetry can be implemented only in two dimensions.
This is the first in a series of papers where we succeed in enlarging the class of exactly solvable potentials in one and three dimensions by obtaining solutions for new relativistic and nonrelativistic problems. This is accomplished by constructing a matrix representation of the wave operator in a complete square integrable basis that makes it tridiagonal. Expanding the wave function in this basis makes the wave equation equivalent to a three-term recursion relation for the expansion coefficients. Consequently, finding solutions of the recursion relation is equivalent to solving the original problem. Doing so results in a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction from the larger class to the conventional class of solvable potentials, giving the well-known energy spectra and the corresponding wave functions. Moreover, some of the new solvable problems show evidence of a Klauder-like phenomenon. In the present work, we give an exact solution for the infinite potential well with a bottom that has a sinusoidal shape.
The general solutions of the Schrödinger equation for a non-central potential are obtained by using the Nikiforov–Uvarov method. The Schrödinger equation with general non-central potential is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived analytically. By making special selections, the non-central potential is reduced to Coulomb and Hartmann ring-shaped potentials, and the obtained results are compared with the solutions of Coulomb and Hartmann potentials given in the literature.