Variational Method Applied to the Harmonic Oscillator©
S. Keith Dunn
Centre College
Danville, KY 40422
United States
mail to: dunnk@centre.edu

This laboratory exercise is a simple, one-dimensional problem that introduces undergraduate physical chemistry students to the variational method. Students find approximate solutions to the Schrodinger equation for the quantum harmonic oscillator using the sinusoidal particle-in-a-box wavefunctions as a basis set. The details of the variational method, including constructing trial wavefunctions from linear combinations of an orthonormal basis set, are explored without the mathematical complexities involved with even the simplest Molecular Orbital (MO) theory calculations. The one-dimensional basis functions and trial wavefunctions are plotted on simple XY graphs, so that the important concept of additivity of basis functions is clearly demonstrated. Symmetry characteristics of the basis functions and the hamiltonian are used to discover why odd basis functions contribute to one set of trial wavefunctions, while even basis functions contribute to another. Additionally, since the harmonic oscillator wavefunctions are known, the students compare their trial wavefunctions and energies with actual ones. By comparison then, students can see the value and limitations of approximate methods of solving the Schrodinger equation. This exercise is an effective way to introduce students to the basics of MO theory before they are asked to use software packages like Spartan or Hyperchem to solve chemically relevant problems.
Editor's Commentary
Audiences: Upper-Division Undergraduate
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: MO Theory, Quantum Chemistry, Mathematics / Symbolic Mathematics
File NameDescriptionSoftware TypeSoftware Version
vari_harm2001.mcd JCE Mathcad 2001 version
vari_harm2001.pdf Read-Only Document mathcad Mathcad 2001
vari_harm8.mcd JCE Mathcad 8 version. Mathcad 8
JCE JCE Subscribers only: name and password or institutional IP number access required.
Dunn, S. K. J. Chem. Educ. 2002, 79, 1378
Comments to: S. Keith Dunn at dunnk@centre.edu
©Copyright 2002 Journal of Chemical Education