Optimized Approximate Solutions to the Time Dependent Schrödinger Equation

Hildegard Schultheis, Rainer Schultheis, Anatole B. Volkov

Arthur B. Weglein

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.

José M. Maidagan, Rubén Darío Piacentini

In a previous work [A. B. Weglein, Phys. Rev. A 17, 1810 (1978)], an optimization method to approximate solutions to the time-dependent Schr\"odinger equation proposed by Chang and Rapp was further developed. We show that, if correct in a mathematical sense, it leads to results which are physically inadequate.

Patrick L. Nash, Liao Yuan Chen

Seogjoo Jang

Arthur B. Weglein

A method for determining optimal approximate solutions of the time-dependent Schr\"odinger equation was previously presented by Weglein [Phys. Rev. A 17, 1810 (1978)]. Maidagan and Piacentini have written a comment (preceding Comment) on that work. Their conclusion is simply incorrect. We show precisely where the error in their reasoning occurred.