A universal Laplace-transform approach to solving Schrödinger equations for all known solvable models

Abstract

Solvable models of the Schrödinger equation are important models of quantum systems because they are idealistic approximations of real quantum systems and much insight into real quantum systems can be gained from the exact solutions of the solvable models. In this paper we show that a universal Laplace transform scheme can be used to solve the Schrödinger equations in closed form for all known solvable models. The work demonstrates how to apply the Laplace transform to differential equations with non-constant coefficients, which is useful in many branches of physics in addition to quantum mechanics. The advantages of the Laplace transform over the power expansion method and its connection with the methods of supersymmetry shape-invariant potentials and quantum canonical transformation, which also give closed-form solutions for solvable models, are elucidated.