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Physical Review A

Comment on ‘‘Optimization of approximate solutions to the time-dependent Schrödinger equation’’

José M. Maidagan, Rubén Darío Piacentini

Published Jun 1, 1985• 3 citations
Publisher

Abstract

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.
DOI
10.1103/physreva.31.4023

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Reply to ‘‘Comment on ‘Optimization of approximate solutions to the time-dependent Schrödinger equation’ ’’

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.

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Further comment on ‘‘Optimization of approximate solutions to the time-dependent Schrödinger equation’’

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We show that the Reply of Weglein [Phys. Rev. A 31, 4025 (1985)] to our previous Comment [Phys. Rev. A 31, 4023 (1985)] is not valid.Received 18 September 1985DOI:https://doi.org/10.1103/PhysRevA.35.933©1987 American Physical Society

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Optimization of approximate solutions to the time-dependent Schrödinger equation

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.

Physical Review A•Jun 1, 1978•9 citations
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.