Schrödinger Equations and Diffusion Theory

✍ Scribed by Masao Nagasawa (auth.)

• Publisher: Birkhäuser Basel
• Year: 1993
• Tongue: English
• Leaves: 334
• Series: Monographs in Mathematics 86
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ 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
Variation 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 281-323

✦ Subjects

Probability Theory and Stochastic Processes