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

TL;DR

This paper introduces a transformation technique leveraging orthogonal polynomials to systematically construct a broad family of exactly solvable quantum potentials across arbitrary dimensions.

Contribution

It presents a novel method that transforms polynomial differential equations into D-dimensional radial Schrödinger equations, enabling the creation of new exactly solvable quantum systems.

Findings

Constructed new exactly solvable quantum potentials in arbitrary dimensions.

Applied the method to Laguerre and Hypergeometric polynomials.

Highlighted potential extensions to other polynomial families.

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.