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Journal of Mathematical Physics

Schrödinger–Fuerth quantum diffusion theory: Generalized complex diffusion equation

Morton D. Kostin

Published Jan 1, 1992• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.529950

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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.
Schrödinger's equation can be considered as a diffusion equation with a diffusion coefficient β2=ℏ/2m. In this article, we explore the implications of this view. Rewriting the wave function in a polar form and transforming Schrödinger's equation into two real equations, we show that one of them reduces to the continuity equation, and the other, a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of stationary states of a particle in a rigid box and in harmonic potential; quasi-classical Gaussian probability densities of a free particle; and coherent states and squeezed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the meaning of the diffusion coefficient β2 for each quantum system using a density plot.
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.
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.
In this paper, we introduce (at least formally) a diffusion effect that is based on an axiom postulated by Werner Heisenberg in the early days of quantum mechanics. His statement was that—in quantum mechanics—kinematical quantities such as velocity must be treated as complex matrices. In the hydrodynamic formulation of quantum mechanics according to Madelung, the complex Schrödinger equation is rewritten in terms of two real equations—a continuity equation and a modified Hamilton–Jacobi equation. Considering seriously Heisenberg’s axiom, the velocity occurring in the continuity equation should be replaced by a complex one, thus introducing a diffusion term with an imaginary diffusion coefficient. Therefore, in quantum mechanics, there should be a diffusion effect showing up in the dynamics. This is the case in the time evolution of the free-motion wave packet under time reversal. The maximum returns to the initial position; however, the width of the wave packet does not shrink to its initial width. This effect is obvious but—as far as we know—it is not mentioned in any textbook. In our paper, we discuss this effect in detail and show the connection with a complex version of quantum hydrodynamics.