Frustating use of the Laplace transform for the quantum states of a particle in a box

Abstract

In a recent paper diffused in the literature, the quantum problem of a particle in a infinite square well potential was claimed to be solved via Laplace transform. Surprisingly, the correct solutions were found for an eigenvalue problem with variable defined on a finite range. As a matter of fact, the authors have misused the Laplace transform of the time-independent Schrödinger equation, and they have also used incorrectly the inversion process.In the present paper it is pointed out carefully and clearly those mistakes that might occur with the use of inappropriate methods for a given eigenvalue problem.

A Brief Review on Mathematical Tools Applicable to Quantum Computing for Modelling and Optimization Problems in Engineering

“…Laplace transform [40,[45][46][47] Definition of quantum Laplace transform and its properties [38][39][40][41][42][43][44][45][46][47][48] Using the Laplace transform to solve engineering problems 48 Report an incorrect use of the Laplace transform to solve a specific type of the Schrödinger equation Englefield (1968) used the Laplace transform method to examine the problem of the one-dimensional quantum harmonic oscillator [37]. Further, it has been shown that the wave function which defines a quantum mechanical system can be represented by the Laplace transform of a certain distribution and the result of a certain subclass [38].…”

“…Correlations among Aleph functions and the above-mentioned integral transforms in quantum calculus have been derived. Finally, de Castro (2020) reported and addressed an incorrect use of Laplace transforms of the time-independent Schrödinger equation as well as its inverse [48]. The author established a clear and detailed list of errors that can occur when an inappropriate method is used for a specific eigenvalue problem [48].…”