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Variational Methods Applied to the Particle in a Box |
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W. Tandy Grubbs
Department of Chemistry, Stetson University, DeLand, FL 32720
| Abstract |
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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 wave function for a particular system and then calculating the energy on the basis of 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 model system, the particle in a one-dimensional box, where exact expressions are available for the energies and wave functions. By practicing the variational method on a known system, students explore the factors that govern the accuracy of the estimated energy and thereby gain an appreciation and confidence in the variational method that is difficult to obtain in any other fashion. Exercises are included throughout the document that allow students to practice the numerical methods and to try the variational method using a wide range of trial functions. These documents are suitable for use in a physical chemistry course.
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Comparison of a trial function with the ground state particle-in-a-box wave function prior to application of the variation method.
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| Mathcad documents (fully interactive with Mathcad) |
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variational2000.mcd (Mathcad 2000)
variational6.mcd (Mathcad 6)
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| PDF version (non-interactive) |
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variational.pdf
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| Commentary |
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Editor's Commentary
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| Keywords |
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Computational Chemistry, Quantum Chemistry, Theoretical Chemistry, and Numerical Methods
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| JCE Citation |
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Grubbs, W. T. J. Chem. Educ. 2001, 78, 1557. |
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