Riccati equation, Factorization Method and Shape Invariance

TL;DR

This paper reviews factorizable problems and shape invariant potentials in quantum mechanics, analyzing their relation to the Riccati equation and generalizing the classical Factorization Method by Infeld and Hull.

Contribution

It provides a detailed analysis connecting shape invariance with Riccati equations and offers a simplified generalization of the Infeld-Hull factorization method.

Findings

Generalized solutions to the Riccati equation for quantum problems

Unified framework linking shape invariance and factorization methods

Simplified approach to classical factorization techniques

Abstract

The basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of Shape Invariant potentials are reviewed. The relation of this last theory with a generalization of the classical Factorization Method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are greatly generalized in a simple way.