On approximating the free harmonic oscillator by a particle in a box

TL;DR

This paper demonstrates how the variational method using particle-in-a-box wavefunctions can effectively approximate the eigenenergies and eigenfunctions of a one-dimensional free harmonic oscillator.

Contribution

It introduces a novel application of the variational technique with particle-in-a-box trial functions to approximate quantum harmonic oscillator states.

Findings

Approximate eigenenergies closely match exact solutions.

Eigenfunctions obtained resemble true harmonic oscillator states.

Highlights the effectiveness of variational methods in quantum systems.

Abstract

The main purpose of this paper is to demonstrate and illustrate, once again, the potency of the variational technique as an approximation procedure for the quantization of quantum mechanical systems. By choosing particle-in-a-box wavefunctions as trial wavefunctions, with the size of the box as the variation parameter, approximate eigenenergies and the corresponding eigenfunctions are obtained for the one dimensional free harmonic oscillator.