Revisiting the quantum harmonic oscillator via unilateral Fourier transforms

TL;DR

This paper critically revisits the quantum harmonic oscillator problem, demonstrating that previous solutions based on exponential Fourier methods are flawed, and provides a correct solution using Fourier sine and cosine transforms with proper boundary conditions.

Contribution

It introduces a revised approach using Fourier sine and cosine transforms to accurately determine stationary states of the quantum harmonic oscillator.

Findings

Previous exponential Fourier solutions are flawed

Stationary states are correctly derived using sine and cosine transforms

Proper boundary conditions yield accurate eigenfunctions

Abstract

The literature on the exponential Fourier approach to the one-dimensional quantum harmonic oscillator problem is revised and criticized. It is shown that the solution of this problem has been built on faulty premises. The problem is revisited via the Fourier sine and cosine transform method and the stationary states are properly determined by requiring definite parity and square-integrable eigenfunctions.