This laboratory exercise is a simple, one-dimensional problem that introduces physical chemistry students to the variational method. It is suitable for advanced undergraduate and graduate students. Students find approximate solutions to the Schrödinger 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 Schrödinger 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. The exercise requires about 20 minutes of prelab work, 3–5 hours of laboratory time, and approximately 1–2 hours of analysis, and preparation of a laboratory report.

Figure. A linear combination of particle-in-a-box wavefunctions is used with the variational method to create a trial function, g0(x), for the v=0 level of the quantum mechanical harmonic oscillator wavefunction, y0(x).