The Harmonic Oscillator in Quantum Mechanics: A Third Way

TL;DR

This paper presents a new pedagogical approach to teaching the quantum harmonic oscillator by framing it as a matrix diagonalization problem using familiar basis functions, enhancing undergraduate understanding.

Contribution

It introduces a matrix-based method for solving the harmonic oscillator, making the problem more accessible and applicable for undergraduate students learning quantum mechanics.

Findings

Students can obtain low-lying bound states using the matrix approach.

The method generalizes to other short-range potentials.

Enhances conceptual understanding of quantum states.

Abstract

Courses on undergraduate quantum mechanics usually focus on solutions of the Schr\"odinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced only academic examples are used, such as the matrix representation of Dirac's raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.