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Physical Review A

Stochastic mechanics according to E. Schrödinger

Jean-Claude Zambrini

Published Mar 1, 1986• 96 citations
Publisher

Abstract

In 1931 Schr\"odinger tried unsuccessfully to probabilistically derive his wave equation. We show that, after some improvements, his program yields two versions of stochastic mechanics. The first one coincides with Nelson stochastic mechanics, the second with the imaginary-time version of this theory, whose existence was unknown. The construction is founded on a variational approach inspired by Yasue.

DOI
10.1103/physreva.33.1532

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In 1931 Schr"odinger tried unsuccessfully to probabilistically derive his wave equation. We show that, after some improvements, his program yields two versions of stochastic mechanics. The first one coincides with Nelson stochastic mechanics, the second with the imaginary-time version of this theory, whose existence was unknown. The construction is founded on a variational approach inspired by Yasue.
The dynamical structure of any reasonable stochastic version of classical mechanics is investigated, including the version created by Nelson [E. Nelson, Quantum Fluctuations (Princeton U.P., Princeton, NJ, 1985); Phys. Rev. 150, 1079 (1966)] for the description of quantum phenomena. Two different theories result from this common structure. One of them is the imaginary time version of Nelson’s theory, whose existence was unknown, and yields a radically new probabilistic interpretation of the heat equation. The existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of Yasue [K. Yasue, J. Math. Phys. 22, 1010 (1981)].
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
Stochastic mechanics develops a formulation of quantum physics by assuming that a classical particle is also subjected to Brownian-like motion. Because this presumes that a particle's path is non-differentiable, making this transformation often involves implicitly replacing time-derivatives in classical mechanics by finite differences. In this paper, we will develop a stochastic mechanics by explicitly replacing time-derivatives in classical mechanics by finite differences, deriving the resulting dynamics and then introducing stochasticity. Like Nelson's original version of stochastic mechanics, our stochastic mechanics is also consistent with the Schrödinger Equation.