## Table of Contents

**Table of Contents**

Preface

Acknowledgements

Chapter 1. Understanding the Schrödinger Equation as a Kinematic Statement: A Probability-First Approach to Quantum

(James Daniel Whitfield, Department of Physics and Astronomy, Dartmouth College, Hanover, NH, USA)

Chapter 2. The Schrödinger Equation Written in the Second Quantization Formalism: Derivation from First Principles

(L. S. F. Olavo, S. S. João Augusto and Marcello Ferreira, Instituto de Física, Universidade de Brasília, Asa Norte, Brasília)

Chapter 3. Schrödinger Equation and Nonlinear Waves

(Nikolay K. Vitanov, Institute of Mechanics, Bulgarian Academy of Sciences, Sofia, Bulgaria)

Chapter 4. On the Wave Solutions of the Generalized Nonlinear Schrödinger-Like Equation of Formation of a Cosmogonical Body

(Alexander M. Krot, Laboratory of Self-Organization System Modeling, United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus)

Chapter 5. The Nonlinear Schrödinger Equation: A Mathematical Model with Its Wide Range of Applications

(Natanael Karjanto, Department of Mathematics, University College, Sungkyunkwan University, Natural Science Campus, Suwon, Gyeonggi Province, Republic of Korea)

Chapter 6. Self-Similar and Traveling-Wave Analysis of the Madelung Equations Obtained from the Schrödinger Equation

(Imre F. Barna and László Mátyás, Plasma Physics Department, Wigner Research Center for Physics, Budapest, Hungary, and Department of Bioengineering, Faculty of Economics, Socio-Human Sciences and Engineering, Sapientia Hungarian University of Transylvania, Miercurea Ciuc, Romania)

Chapter 7. Paradigm of Infinite Dimensional Phase Space

(E. E. Perepelkin, B. I. Sadovnikov and N. G. Inozemtseva, Department of Quantum Statistics and Field Theory, Lomonosov Moscow State University, Moscow, Russia, and Moscow Technical University of Communications and Informatics, Department of Physics, Moscow, Russia)

Chapter 8. From Classical to Quantum Physics: The Metatron

(Maurice A. de Gosson and Valentino A. Simpao, University of Vienna, Institute of Mathematics, Vienna, Austria, and Mathematical Consultant Services, Greenville, KY, USA, Research Adjunct Professor, Physics Department, Western Kentucky University, Bowling Green, KY, USA)

About the Editors

Index

**Reviews**

“The book contains various approaches to the Schrödinger equation (SE) as a fundamental equation of quantum mechanics. In Chapter 1, a new pedagogical paradigm is proposed which allows one to understand quantum mechanics as an extension of probability theory; its purpose is providing alternative methods to understand the Schrödinger equation. Chapter 2 is devoted to the derivation of SE from the classical Hamiltonian by some procedure of second quantization. In Chapters 3–5, the authors consider the nonlinear SE with many applications: from nonlinear waves in deep water to formation of a cosmogonical body, surface gravity waves, superconductivity and nonlinear optics. The goal of Chapter 6 is to establish the connection of Schrödinger, Madelung and Gross-Pitaevskii equations. Chapter 7, “Paradigm of infinite dimensional phase space,” describes the deep connection between SE and the infinite chain of equations for distribution functions of high-order kinematical values (Vlasov chain). The authors formulate the principles which allow one to combine and treat in unified form the physics of classical, statistical and quantum mechanical phenomena. And in final Chapter 8 it is shown that SE can be mathematically derived from Hamilton’s equation if one uses the metaplectic representation of canonical transformations. All that makes the book interesting for a wide community of physicists.” – <strong>E.E. Perepelkin, B.I. Sadovnikov (Lomonosov Moscow State University, Department of Quantum Statistics and Field Theory, Moscow, Russia), and N.G. Inozemtseva (Moscow Technical University of Communications and Informatics, Department of Physics, Moscow, Russia)