This book discusses quantum theory as the theory of random (Brownian) motion of small particles (electrons etc.) under external forces. Implying that the Schrödinger equation is a complex-valued evolution equation and the Schrödinger function is a complex-valued evolution function, important applications are given.
Readers will learn about new mathematical methods (theory of stochastic processes) in solving problems of quantum phenomena. Readers will also learn how to handle stochastic processes in analyzing physical phenomena.
Author(s): Masao Nagasawa
Series: Monographs in Mathematics 109
Edition: 1
Publisher: Birkhäuser
Year: 2021
Language: English
Pages: 339
Tags: Random Motion, Markov Processes, Quantum Theory
Preface
Contents
Chapter 1 Mechanics of Random Motion
1.1 Smooth Motion and Random Motion
1.2 On Stochastic Processes
1.3 Itô's Path Analysis
1.4 Equation of Motion for a Stochastic Process
1.5 Kinematics of Random Motion
1.6 Free Random Motion of a Particle
1.7 Hooke's Force
1.8 Hooke's Force and an Additional Potential
1.9 Complex Evolution Functions
1.10 Superposition Principle
1.11 Entangled Quantum Bit
1.12 Light Emission from a Silicon Semiconductor
1.13 The Double-Slit Problem
1.14 Double-Slit Experiment with Photons
1.15 Theory of Photons
1.16 Principle of Least Action
1.17 Transformation of Probability Measures
1.18 Schrödinger Equation and Path Equation
Chapter 2 Applications
2.1 Motion induced by the Coulomb Potential
2.2 Charged Particle in a Magnetic Field
2.3 Aharonov-Bohm Effect
2.4 Tunnel Effect
2.5 Bose-Einstein Distribution
2.6 Random Motion and the Light Cone
2.7 Origin of the Universe
2.8 Classification of Boundary Points
2.9 Particle Theory of Electron Holography
2.10 Escherichia coli and Meson models
2.11 High-Temperature Superconductivity
Chapter 3 Momentum, Kinetic Energy, Locality
3.1 Momentum and Kinetic Energy
3.2 Matrix Mechanics
3.3 Function Representations of Operators
3.4 Expectation and Variance
3.5 The Heisenberg Uncertainty Principle
3.6 Kinetic Energy and Variance of Position
3.7 Theory of Hidden Variables
3.8 Einstein's Locality
3.9 Bell's Inequality
3.10 Local Spin Correlation Model
3.11 Long-Lasting Controversy and Random Motion
Chapter 4 Markov Processes
4.1 Time-Homogeneous Markov Proces-ses
4.2 Transformations by M-Functionals
4.3 Change of Time Scale
4.4 Duality and Time Reversal
4.5 Time Reversal, Last Occurrence Time
4.6 Time Reversal, Equations of Motion
4.7 Conditional Expectation
4.8 Paths of Brownian Motion
Chapter 5 Applications of Relative Entropy
5.1 Relative Entropy
5.2 Variational Principle
5.3 Exponential Family of Distributions
5.4 Existence of Entrance and Exit Functions
5.5 Cloud of Paths
5.6 Kac's Phenomenon of Propagation of Chaos
Chapter 6 Extinction and Creation
6.1 Extinction of Particles
6.2 Piecing-Together Markov Processes
6.3 Branching Markov Processes
6.4 Construction of Branching Markov Processes
6.5 Markov Processes with Age
6.6 Branching Markov Processes with Age
Bibliography
Index
