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Optimized Approximate Solutions to the Time Dependent Schrödinger Equation

Arthur B. Weglein

Jan 1, 19780 citations

Schroedinger equation and classical physics

M. V. Lokajíček

Any time-dependent solution of Schrödinger equation may be always correlated to a solution of Hamilton equations or to a statistical combination of their solutions; only the set of corresponding solutions is somewhat smaller (due to existence of quantization). There is not any reason to the physical interpretation according to Copenhagen alternative as Bell's inequalities are valid in the classical physics only (and not in any alternative based on Schrödinger equation). The advantage of Schrödinger equation consists then in that it enables to represent directly the time evolution of a statistical distribution of classical initial states (which is usual in collision experiments). The Schrödinger equation (without assumptions added by Bohr) may then represent the common physical theory for microscopic as well as macroscopic physical systems. However, together with the last possibility the solutions of Schrödinger equation may be helpful also in analyzing the influence of other statistically distributed properties (e.g., spin orientations or space structures) of individual matter objects forming a corresponding physical system, which goes in principle beyond the classical physics. In any case, the contemporary quantum theory represents the phenomenological approximative description of some matter characteristics only, without providing any insight into quantum mechanism emergence. In such a case it is necessary to take into account more detailed properties at least of some involved objects.Any time-dependent solution of Schrödinger equation may be always correlated to a solution of Hamilton equations or to a statistical combination of their solutions; only the set of corresponding solutions is somewhat smaller (due to existence of quantization). There is not any reason to the physical interpretation according to Copenhagen alternative as Bell's inequalities are valid in the classical physics only (and not in any alternative based on Schrödinger equation). The advantage of Schrödinger equation consists then in that it enables to represent directly the time evolution of a statistical distribution of classical initial states (which is usual in collision experiments). The Schrödinger equation (without assumptions added by Bohr) may then represent the common physical theory for microscopic as well as macroscopic physical systems. However, together with the last possibility the solutions of Schrödinger equation may be helpful also in analyzing the influence of other statistically distributed properties (e.g., spin orientations or space structures) of individual matter objects forming a corresponding physical system, which goes in principle beyond the classical physics. In any case, the contemporary quantum theory represents the phenomenological approximative description of some matter characteristics only, without providing any insight into quantum mechanism emergence. In such a case it is necessary to take into account more detailed properties at least of some involved objects.

arXiv:1205.6720May 30, 20122 citations

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.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 AJun 1, 19789 citations

Time-dependent Schrodinger equation

Luca Salasnich

Springer eBooksJan 1, 20200 citations

Time-Dependent Schrödinger Equation

MASSIMO RUDAN

Springer eBooksSep 27, 20170 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 AJun 1, 19853 citations

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