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Journal of Physics A Mathematical and General

The principle of minimal quantum fluctuations for the time-dependent Schrödinger equation

Y C Lee, Wuming Zhu

Published Jan 1, 1999• 2 citations
Publisher

Abstract

It is shown that the optics-mechanics analogy which originally led Schrödinger to his famous equation stops at the time-independent level. For potentials involving time explicitly, the time-dependent Schrödinger equation cannot be deduced in a similar manner. Instead, a variational principle for the time-dependent Schrödinger equation is established. It minimizes the total quantum fluctuations of a newly defined Hamilton-Jacobi function about its limiting classical value over space-time, thereby demonstrating the existence of a deeper relationship between classical and quantum mechanics beyond the simple optics-mechanics analogy. This principle is similar in spirit to Feynman's space-time approach to quantum mechanics.

DOI
10.1088/0305-4470/32/17/305

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It is shown that the optics-mechanics analogy which originally led Schrödinger to his famous equation stops at the time-independent level. For potentials involving time explicitly, the time-dependent Schrödinger equation cannot be deduced in a similar manner. Instead, a variational principle for the time-dependent Schrödinger equation is established. It minimizes the total quantum fluctuations of a newly defined Hamilton-Jacobi function about its limiting classical value over space-time, thereby demonstrating the existence of a deeper relationship between classical and quantum mechanics beyond the simple optics-mechanics analogy. This principle is similar in spirit to Feynman's space-time approach to quantum mechanics.
In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: ``"Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum mechanics for a function ψ through an exact procedure. This element is the linear independence of the functions ψ and ψ* (their complex conjugate). Their independence is demonstrated for physical systems where the acting physical potential does not explicitly depend on time. Proceeding in reverse, from the Schrödinger equation, we obtain the Hamilton-Jacobi equation, exactly, without additional terms.
It is shown how the time-dependent Schrödinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton–Jacobi equation of classical mechanics. Schrödinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton–Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an a priori assumption equivalent to Feynman's dynamical postulate. de Broglie's concepts of ‘matter waves’ and their phase and group velocities are also critically discussed.
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.
In his first paper on wave mechanics, Schrödinger presented a heuristic argument which led from the Hamilton–Jacobi equation through the quantum variational principle to his famous wave equation. In his second paper, Schrödinger withdrew this heuristic argument as “incomprehensible.” We show by using a recently generalized form of Maupertuis’ principle that Schrödinger’s original heuristic argument can be made more logical. Aside from pedagogical interest, this path is useful as a method of quantization of general mechanical systems.
Beginning with the principle that a closed mechanical composite system is timeless, time can be defined by the regular changes in a suitable position coordinate (clock) in the observing part, when one part of the closed composite observes another part. Translating this scenario into both classical and quantum mechanics allows a transition to be made from a time-independent mechanics for the closed composite to a time-dependent description of the observed part alone. The use of Hamilton-Jacobi theory yields a very close parallel between the derivations in classical and quantum mechanics. The time-dependent equations, Hamilton-Jacobi or Schr\"odinger, appear as approximations since no observed system is truly closed. The quantum case has an additional feature in the condition that the observing environment must become classical in order to define a real classical time variable. This condition leads to a removal of entanglement engendered by the interaction between the observed system and the observing environment. Comparison is made to the similar emergence of time in quantum gravity theory.