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Applications of the Variation Method
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Abstract
The variation method is applied to two examples selected for illustration of fundamental principles of the method along with ease of calculation. The first example applies the linear version of the variation method to the particle in a box model, using a basis with explicit parity symmetry, Phik(t) = N (1-t2)tk, where t = 2x/L -1 and N is the normalization constant. Convergence of ground and excited state energies and wavefunctions with increasing length of the expansion basis is shown. The second example employs an exponential variational wavefunction to describe the harmonic oscillator model, using the nonlinear variation method. The ground state of even parity is a special case, due to the quantum requirement that the derivative of the wavefunction be continuous everywhere; a (fixed) linear combination of two exponential functions Phi(x) = exp(-Y|x|) - exp(-ß|x|)/ß is used to enforce this. In this variational wavefunction Y is the variational parameter, and ß is any positive constant. The numerical portions of each example can be treated by spreadsheet calculations.
Keywords (Audience):
Upper-Division UndergraduateKeywords (Domain):
Physical ChemistryKeywords (Subject):
Learning TheoriesCiting Articles
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This article has been cited by 1 ACS Journal articles (1 most recent appear below).
Solution of the Schrödinger Equation for One-Dimensional Anharmonic Potentials: An Undergraduate Computational Experiment
Godfrey S. BeddardJournal of Chemical Education2011 88 (7), 929-931Solution of the Schrödinger Equation for One-Dimensional Anharmonic Potentials: An Undergraduate Computational Experiment
Godfrey S. BeddardJournal of Chemical Education2011 88 (7), 929-931A method of solving the Schrödinger equation using a basis set expansion is described and used to calculate energy levels and wavefunctions of the hindered rotation of ethane and the ring puckering of cyclopentene. The calculations were performed using a ...
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- Received: August 03, 2009
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