JCE
Variational Methods Applied to the Particle in the Box©
W. Tandy Grubbs
Department of Chemistry
Stetson University
DeLand, FL 32720
United States
mail to: william.grubbs@stetson,edu



Abstract
A Mathcad instructional document is presented that allows students to explore the variational method, an approximate method (based upon the variation theorem) that is commonly used in commercial software packages to estimate the energies of real chemical systems. The use of this method involves supplying an initial guess about the form of a wavefunction for a particular system, and then calculating the energy based upon this trial function. Excellent estimates for energy can be obtained if the selected trial function is suitably flexible. The variational method is used here to estimate the energy levels of a well-known system, the particle in a one-dimensional box, where exact expressions are available for the energies and wavefunctions. By practicing the variational method on this simple system, students can compare their estimated energies with the known results. Consequently, the students can explore the factors that govern the accuracy of the estimated energy and will subsequently understand the computational trade-off between the complexity of the selected trial function and the accuracy of the energy estimate. By working with a known system, students will gain an appreciation and confidence in the variational method that is difficult to gain in any other fashion. Exercises are included throughout the document that allow students to practice the numerical methods and permit students to try the variational method using a wide range trial functions. These documents are suitable for use in a physical chemistry course.
Commentary
Editor's Commentary
Keywords
Audiences: Upper-Division Undergraduate
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: Mathematics / Symbolic Mathematics, Quantum Chemistry, Theoretical Chemistry
Documents
File NameDescriptionSoftware TypeSoftware Version
variational6.mcd JCE Computational Document for Mathcad 6 and Higher Mathcad 6
variational.pdf Read-Only Document
variational_nb.pdf Read-Only Document for Mathematica Mathematica
variational.nb JCE Mathematica Computational Document. Mathematica
variational2000.mcd JCE Mathcad 2000 Computational Document. Mathcad 2000
JCE JCE Subscribers only: name and password or institutional IP number access required.
Citations
Grubbs, W. T. J. Chem. Educ. 2001, 78, 1557.
Comments to: W. Tandy Grubbs at william.grubbs@stetson.edu.
©Copyright 2001 Journal of Chemical Education