A transformation method is applied to the second order ordinary differential equation satisfied by orthogonal polynomials to construct a family of exactly solvable quantum systems in any arbitrary dimensional space. Using the properties of orthogonal polynomials, the method transforms polynomial differential equation to D-dimensional radial Schrodinger equation which facilitates construction of exactly solvable quantum systems. The method is also applied using associated Laguerre and Hypergeometric polynomials. The quantum systems generated from other polynomials are also briefly highlighted.
We apply a simple transformation method to construct a set of new exactly solvable potentials (ESP) which gives rise to bound state solution of D-dimensional Schrödinger equation. The important property of such exactly solvable quantum systems is that their normalized eigenfunctions can be written in terms of recently introduced exceptional orthogonal polynomials (EOP).
Abstract. In this paper, we apply a simple transformation method to construct exactly solvable potentials of Schrödinger equation in any arbitrary dimensional Euclidean space. The normalized wave functions of the constructed potentials are obtained in terms of Romanovski polynomials.
IntroductionExact solution of Schrödinger equation with a physical potential is of utmost importance in nonrelativistic quantum mechanics. Successful solution of Schrödinger equation provides analytical form of the normalized wave function and quantized energy eigenvalues. However, a very few quantum systems yield exact solutions for potentials of physical interest. Along the years, many authors have tried different methods to obtain the exact solution of the Schrödinger equation [1][2][3][4][5][6]. To get the bound state wave functions in terms of Romanovski polynomials, we apply a simple transformation method [6] which comprises of a co-ordinate transformation followed by a functional transformation. By applying this method, we transform the second order ordinary differential equation satisfied by Romanovski polynomials to standard radial Schrödinger equation in D-dimensional Euclidean space and thus try to construct exactly solvable potentials. The motivation for doing this work comes from the fact that exact solutions of Schrödinger equation in terms of Romanovski polynomials is not so widespread [7,8] like other orthogonal polynomials [1-5, 9, 10].The paper is organized as follows. In section 2, the formalism of the theory is given. In section 3, exact solution of Schrödinger equation in terms of Romanovski polynomials is discussed. The conclusions are discussed in section 4.
Abstract:By applying Extended Transformation method we have generated exact solution of D-dimensional radial Schrödinger equation for a set of power-law multi-term potentials taking singular potentials V (r) = ar3 , V (r) = ar +br −1 +cr 2 and V (r) = ar 2 +br −2 +cr −4 +dr −6 as input reference. The restriction on the parameters of the given potentials and angular momentum quantum number are obtained. The multiplet structure of the generated exactly solvable potentials are also shown.
PACS
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.