Revisiting the Laplace transform in quantum mechanics: correcting a flawed approach for the stationary Schr\"odinger equation

TL;DR

This paper critically examines and corrects previous flawed methods of applying Laplace transforms to the stationary Schrödinger equation, providing a sound framework for their proper use in quantum mechanics.

Contribution

It identifies and rectifies methodological errors in prior Laplace transform applications to quantum mechanics, establishing correct practices for boundary conditions and singularities.

Findings

Corrected the application of boundary conditions in Laplace transform methods.

Clarified the proper handling of singularities and the residue theorem.

Provided a consistent framework for using Laplace transforms in quantum mechanics.

Abstract

The Laplace transform is a valuable tool in physics, particularly in solving differential equations with initial or boundary conditions. A 2014 study by Tsaur and Wang (2014 \emph{Eur.~J.~Phys.} \textbf{35} 015006) introduced a Laplace-transform-based method to solve the stationary Schr\"odinger equation for various potentials. However, their approach contains critical methodological flaws: the authors disregard essential boundary conditions and apply the residue theorem incorrectly in the inverse transformation process. These errors ultimately cancel out, leading to correct results despite a flawed derivation. In this paper, we revisit the use of the Laplace transform for the one-dimensional Schr\"odinger equation, clarifying correct practices in handling boundary conditions and singularities. This analysis offers a sound and consistent framework for the application of Laplace transforms in stationary quantum mechanics, underscoring their educational utility in quantum mechanics coursework.