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The Journal of Chemical Physics

A comparison of solutions to the time dependent Schrödinger equation

Chi-Ming Chang

Published Oct 15, 1977• 0 citations
Publisher

Abstract

Various methods providing approximate solutions to the time dependent Schrödinger equation proposed in a previous paper are compared using two examples for which the exact analytic solutions are known. Results show that the new method proposed in the previous paper is better than the traditional method for time dependent parameters.
DOI
10.1063/1.435340

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A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed.
Approximate solutions to the time-dependent Schr\"odinger equation can be sought in terms of time-dependent parameters. In this work, a reasonable definition of the "best" values for these parameters is given by requiring that the time integral of the deviation of the approximate solution from the exact solution be minimized. A procedure for specifically evaluating the parameters to optimize the approximate solution is given in terms of equations analogous to Hamilton's equations of classical mechanics, which produce a minimum-deviation integral in analogy to Hamilton's principle. A comparison of this technique with previous approaches is made, both in terms of general principles, and in terms of a simple hypothetical example of a system subjected to a time-dependent perturbation.