A Transformation Method to Construct Family of Exactly Solvable Potentials in Quantum Mechanics

Abstract

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.

Exactly Solvable Extended Potentials in Arbitrary Dimensions

“…We have to choose one or more than one terms containing the function g(r) in expression (10) and put it equal to a constant to get the energy eigenvalues E n . In our recent paper [21], we have identified Q(g) as one of the COPs and constructed many new ESPs. In this paper, we identify Q(g) as extended Laguerre polynomial Lα n (x) and also as extended Jacobi polynomial P(α,β) n (x) and try to construct ESPs associated with them.…”

“…To get the correct form of centrifugal barrier term in D-dimensional Euclidean space, we have to identify the coefficient of 1 r 2 in potential term (19) to be ℓ(ℓ + D − 2) [21], which fixes the value of α as…”

“…So, V (r) is an extended Morse potential with the same energy spectrum (29) as that of Morse potential. It is interesting to note that the potential given by expression (30) is non-power law and as our formalism suggests, it has an inverse square potential term in spaces where the dimensionality is other than 1 and 3 [21]. The wavefunctions of the quantum system can be written as…”

“…The use of these polynomials in constructing new ESPs [19,20] has already drawn attention in quantum mechanics. In this paper, we use a simple transformation method [21] to construct four new exactly solvable extended potentials using the properties of EOPs. These potentials are "extended" in the sense that they can be expressed in terms of some well-known exactly solvable potentials and a few additional rational terms.…”

New Method for Mapping of Exact Analytic S-wave Solutions for Regenerated Central Hyperbolic Potentials

“…Again, the extended transformation (ET) [6] is applied successfully by Ahmed et al and others for the generation of exactly solvable central potentials (ESCPs) in an Euclidean space of any desired dimension from already known ESCPs (power law and non-power law) [7][8][9][10][11][12]. The ET includes a coordinate transformation (CT) required to modify the spatial character of an already known ESCP to generate a new ESCP and a functional transformation (FT) for manipulation of the dimensionality of the space to which the known QS gets transformed.…”