Schrödinger equations and diffusion theory
著者
書誌事項
Schrödinger equations and diffusion theory
（Modern Birkhäuser classics）
Birkhäuser, c1993
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注記
"Reprint of the 1993 edition"
Includes bibliographical references (p. 295-310) and index
内容説明・目次
内容説明
Schroedinger Equations and Diffusion Theory addresses the question "What is the Schroedinger equation?" in terms of diffusion processes, and shows that the Schroedinger equation and diffusion equations in duality are equivalent. In turn, Schroedinger's conjecture of 1931 is solved. The theory of diffusion processes for the Schroedinger 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 Schroedinger equations.
The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schroedinger equation, namely, quantum mechanics.
The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level.
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This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be regarded as an introduction to the theory of diffusion processes and applications written with the physicists in mind. Interesting topics present themselves as the chapters proceed. (...) this book is an excellent addition to the literature of mathematical sciences with a flavour different from an ordinary textbook in probability theory because of the author's great contributions in this direction. Readers will certainly enjoy the topics and appreciate the profound mathematical properties of diffusion processes.
(Mathematical Reviews)
目次
Preface.- I Introduction and Motivation.- II Diffusion Processes and their Transformations.- III Duality and Time Reversal of Diffusion Processes.- IV Equivalence of Diffusion and Schroedinger Equations.- V Variational Principle.- VI Diffusion Processes in q-Representation.- VII Segregation of a Population.- VIII The Schroedinger Equation can be a Boltzmann Equation.- IX Applications of the Statistical Model for Schroedinger Equations.- X Relative Entropy and Csiszar's Projection.- XI Large Deviations.- XII Non-Linearity Induced by the Branching Property.- Appendix.- References.- Index.
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