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Relation between Schrödinger equation and diffusion equation

Mojmír Poprocký

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

The Schrodinger's equation and the diffusion equation can be written in a similar form. While the Schrodinger's equation is complex, the diffusion equation is real. In this thesis we wish to outline similarities of both equations. We present a way to treat a problem in non-relativistic quantum mechanics as a diffusion process. This shows that under certain circumstances both theories are equivalent. We demonstrate this on a few simple examples.

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The Schrodinger's equation and the diffusion equation can be written in a similar form. While the Schrodinger's equation is complex, the diffusion equation is real. In this thesis we wish to outline similarities of both equations. We present a way to treat a problem in non-relativistic quantum mechanics as a diffusion process. This shows that under certain circumstances both theories are equivalent. We demonstrate this on a few simple examples.
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.
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.
In this paper, we derive and compare the solutions of the real and complex diffusion equations obtained by the self-similar solution to each other. Additionally, possible quantum mechanical aspects are analyzed as well. In the second part of the study, a complex reaction–diffusion equation is investigated and compared to the quantum mechanical solutions of the corresponding Schrödinger equation with a power-law-type potential. We show that the same parameter ratio emerges in the quantum mechanical and the self-similar solutions as well, which is a remarkable property. For both investigated equations, the complex diffusion and complex reaction–diffusion equations, we found some solutions for which the absolute value squared has a convergent finite numerical integral. This is not a rigorous [Formula: see text] integrability condition but a good hint for that, which is a key property in quantum mechanics.
According to Schrödinger–Fuerth quantum diffusion theory the equations of quantum theory can be derived by applying the Fuerth transformation to the equations of diffusion theory. By introducing a complex flux whose real and imaginary parts are the x and y components of the flux, a complex form of the two-dimensional diffusion equation is obtained and the equation is then generalized to three dimensions. Application of the Fuerth transformation to the generalized complex diffusion equation yields a quantum equation, which is shown to be equivalent to the Dirac equation.
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.