Schrödinger’s Equation as an Energy Diffusion Equation

Shu Hotta

Kwok Sau Fa

The description of a particle in the quantum system is probabilistic. In the ordinary quantum mechanics the total probability of finding the particle is conserved, i.e. the probability is normalized for all the times. To find a non-constant total probability an imaginary term should be added to the potential energy which is not physical. Recently, generalizations of the ordinary Schrödinger equation have been proposed by using the Feynman path integral and analogy between the Schrödinger equation and diffusion equation. In this work, an integro-differential Schrödinger equation is proposed by using analogy between the Schrödinger equation and diffusion equation. The equation is obtained from the continuous time random walk model with diverging jump length variance and generic waiting time probability density. The equation generalizes the ordinary and fractional Schrödinger equations. One can show that the integro-differential Schrödinger equation can describe a non-constant total probability for a free particle, and it includes the exponential decay which is fundamental for the description of radioactive decay.

Mojmír Poprocký

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.

Shu Hotta

Masao Nagasawa

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.

Shu Hotta