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Progress of Theoretical Physics

A Diffusion Model for the Schrödinger Equation

Leon Bess

Published Jun 1, 1973• 9 citations
Publisher

Abstract

A diffusion process has been formulated to have a mass conservation equation and a momentum conservation equation that together are equivalent to the Schrödinger equation of quantum mechanics. It is suggested that this diffusion process be used as a physical model to help describe the microphysical processes that underlie the Schrodinger equation.

DOI
10.1143/ptp.49.1889

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Journal of Mathematical Physics•Feb 1, 1996•24 citations
A diffusion process has been formulated to have a mass conservation equation and a momentum conservation equation that together are equivalent to the Schrödinger equation of quantum mechanics. It is suggested that this diffusion process be used as a physical model to help describe the microphysical processes that underlie the Schrodinger equation.
Schrödinger Equations and Diffusion Theory addresses the question "What is the Schrödinger equation?" in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tell us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.
Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schrodinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schrodinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schrodinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.
We have formulated non-relativistic quantum mechanics as a diffusion theory in the preceding chapters, but ignored to some extent the mathematical problems that arise in connection with the singularity of coefficients of diffusion equations, since we have concentrated on clarifying fundamental concepts and mathematical structures. We shall see that our diffusion processes, Schrödinger diffusion processes, must be treated more carefully. We will analyze some typical stationary cases to see the movement of quantum particles.
We examine the hypothesis that every particle of mass mmm is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an external field is expressed by means of Newton's law F=ma\mathbf{F}=m\mathbf{a}F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownian motion with friction. The hypothesis leads in a natural way to the Schr\"odinger equation, but the physical interpretation is entirely classical. Particles have continuous trajectories and the wave function is not a complete description of the state. Despite this opposition to quantum mechanics, an examination of the measurement process suggests that, within a limited framework, the two theories are equivalent.
The dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.