A Heuristic Derivation of the Linear Harmonic Oscillator Energy Levels

Michael Messina
Department of Chemistry, University of North Carolina at Wilmington, Wilmington, NC 28403-3297
J. Chem. Educ., 2000, 77 (4), p 492
DOI: 10.1021/ed077p492
Publication Date (Web): April 1, 2000
History, Education, and Documentation

Abstract

In this work we present a heuristic derivation for the energy levels of a linear harmonic oscillator that involves only algebra. This derivation offers the instructor a way to rationalize the energy level spectrum of a harmonic oscillator without having to solve the complicated differential equation prescribed by the Schrödinger equation for the harmonic oscillator. This derivation treats the harmonic oscillator as a quasiparticle in a box and yields qualitative results, these being the correct scaling of the energy with quantum number and vibrational frequency, but does not yield the exact result. By comparing the exact result with the result of the heuristic derivation students are introduced to the idea of tunneling in quantum mechanics and how tunneling can effect the energy levels of quantum mechanical systems.

Keywords (Audience):

Upper-Division Undergraduate

Keywords (Domain):

Physical Chemistry

Keywords (Subject):

Computational Chemistry
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  • Received: August 03, 2009

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