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Physical Review A

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

Arthur B. Weglein

Published Jun 1, 1985• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.31.4025

Related papers

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

José M. Maidagan, Rubén Darío Piacentini

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

Physical Review A•Jan 1, 1987•0 citations

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

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.

Physical Review A•Jun 1, 1985•3 citations

Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

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In this reply I discuss further the misconception in my paper [Phys. Rev. A 63, 034102 (2001)] related to the solution of the Schr\"odinger equation for a particle in a time-dependent linear potential.

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Reply to “Comment on ‘Family of modified contracted Schrödinger equations’ ”

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The preceding Comment by Yasuda [Phys. Rev. A 79, 016101 (2009)] on our earlier paper [Phys. Rev. A 64, 062105 (2001)] raises several objections about the modified contracted Schr\"odinger equations (MCSEs) to which we reply here. In his comment, Yasuda also questions the possibility of obtaining the exact solution of the fourth-order MCSE due to its indeterminacy. We maintain the opposite, at least from a theoretical point of view.

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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
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.
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.