Schrödinger Equations and Diffusion Theory

Authors: 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 tells 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.

Table of contents

Front Matter
Pages i-xii

Introduction and Motivation
Pages 1-12

Diffusion Processes and their Transformations
Pages 13-54

Duality and Time Reversal of Diffusion Processes
Pages 55-88

Equivalence of Diffusion and Schrödinger Equations
Pages 89-114


Variational Principle
Pages 115-138


Diffusion Processes in q-Representation
Pages 139-162

Segregation of a Population
Pages 163-206

The Schrödinger Equation can be a Boltzmann Equation
Pages 207-222

Applications of the Statistical Model for Schrödinger Equations
Pages 223-238

Relative Entropy and Csiszar’s Projection
Pages 239-252

Large Deviations
Pages 253-260

Non-Linearity Induced by the Branching Property
Pages 261-280

Back Matter
Pages 281-319