A completely algebraic solution of the simple harmonic oscillator

TL;DR

This paper provides a fully algebraic derivation of the simple harmonic oscillator wavefunctions, emphasizing an abstract approach that can be easily integrated into physics education, and revisits a historically overlooked method.

Contribution

It introduces a novel algebraic derivation of harmonic oscillator wavefunctions using translation operators, enhancing pedagogical approaches and historical understanding.

Findings

Derivation is algebraic, avoiding derivatives.

Method is suitable for undergraduate and graduate teaching.

Highlights historical neglect of this algebraic approach.

Abstract

We present a full algebraic derivation of the wavefunctions of the simple harmonic oscillator in coordinate and momentum space. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the representation-dependent wavefunctions from the representation-independent energy eigenfunctions. It is simple to incorporate into the undergraduate and graduate curricula. This new derivation begins with the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schroedinger factorization method), and then supplements it by employing the translation (or boost) operator to determine the wavefunctions algebraically, without any derivatives. In addition, we provide a summary of the history of this approach, which seems to have been neglected by most historians of quantum mechanics, until now.