Journal of Modern Physics, 2013, 4, 463-467

http://dx.doi.org/10.4236/jmp.2013.44065 Published Online April 2013 (http://www.scirp.org/journal/jmp)

Construction of Exactly Solvable

Ring-Shaped Potentials

Arup Bharali1*, Ngangkham Nimai Singh2

1Bajali College, Pathsala-781325, Assam, India

2Department of Physics, Gauhati University, Guwahati-781014, Assam, India

Email: *arup.brp@gmail.com

Received January 2, 2013; revised February 1, 2013; accepted February 10, 2013

Copyright © 2013 Arup Bharali, Ngangkham Nimai Singh. This is an open access article distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

ABSTRACT

We propose a method for construction of exactly solvable ring-shaped potentials where the linear homogeneous second-

order differential equation satisfied by special function is subjected to the extended transformation comprising a coor-

dinate transformation and a functional transformation to retrieve the standard Schrödinger polar angle equation form in

non-relativistic quantum mechanics. By invoking plausible ansatze, exactly solvable ring-shaped potentials and corre-

sponding angular wave functions are constructed. The method is illustrated using Jacobi and hypergeometric polynomi-

als and the wave functions for the constructed ring-shaped potentials are normalized.

Keywords: Schrödinger Equation; Extended Transformation; Special Function; Ring-Shaped Potential; Normalization

1. Introduction formation and a functional transformation on second-

order differential equation for a particular special function

to mould the differential equation to the standard Schröd-

inger polar angle equation form. The coordinate transfor-

mation is the basic transformation required to change the

characteristics of the differential equation, while the func-

tional transformation is essential to retrieve the Schröd-

inger polar angle equation form. By invoking suitable

ansatze, exactly solvable ring-shaped potentials and their

angular wave functions are constructed. The special func-

tions are found to be the multiplicative factors in the an-

gular wave functions and the wave functions are normal-

ized. In standard literature, the Schrödinger equation for a

specific exactly solvable ring-shaped potential is solved

by transforming it to a differential equation satisfied by a

particular special function [1,2,9], but the ethics of the

present work is not to solve Schrödinger equation to ob-

tain wave functions for a particular ring-shaped potential,

but to construct exactly solvable ring-shaped potentials as

well as their angular wave functions starting from second-

order differential equations satisfied by special functions.

The fundamental nature of the adopted method for trans-

forming a solvable differential equation to a particular type

bearing some physical significance would have a wide

range of applicability not only in quantum mechanics but

in other branches of science also.

Generation/construction of exactly solvable quantum me-

chanical potentials is an important topic of fundamental

research; as such type of research always incorporates

new ideas and/or mathematical techniques to quantum

mechanics. Again, exactly solvable potentials are essen-

tial for the successful implementation of approximate

methods in the study of practical quantum systems. The

study of quantum systems with non-central potentials is

an upcoming field of research for the theoreticians. The

quantum systems with non-central potentials have been

studied extensively in quantum chemistry and nuclear

physics in the context of organic molecules and deformed

nuclei respectively in non-relativistic regime. Different

methods applied by various authors to obtain exact solu-

tions of Schrödinger equation for bound states with non-

central potentials, are the factorization method [1], the

standard approach [2], the path integral representation [3],

the Nikiforov-Uvarov method [4-6], the supersymmetric

approach [7], etc. Here, we apply a method for construc-

tion of exactly solvable ring-shaped potentials starting from

the linear homogeneous second-order differential equa-

tions satisfied by the special functions. We perform the ex-

tended transformation [8] consisting of a coordinate trans-

The plan of the paper is as follows: in Section 2, for-

*Corresponding author.

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