Understanding the Schrödinger Equation: Some [Non]Linear Perspectives


Valentino A. Simpao (Editor)
Mathematical Consultant Services, Greenville, KY, USA
Research Adjunct Professor, Physics Department, Western Kentucky University, Bowling Green, KY, USA

Hunter C. Little (Editor)
Creative Writing M.F.A. Program, Western Kentucky University, Bowling Green, KY, USA

Series: Mathematics Research Developments
BISAC: MAT007020

The current offering from Nova Science Publishers titled Understanding the Schrödinger Equation: Some [Non]Linear Perspectives is a collection of selectively invited manuscripts from some of the world’s leading workers in quantum dynamics; particularly as concerning Schrödinger’s wavefunction formalism. The work is dedicated to providing an “illustrative sketch” of a few of the numerous and sundry aspects of the Schrödinger equation; ranging from a new pedagogical teaching approach, to technical applications and foundational considerations. Towards this end, the work is generally of a theoretical nature; expounding various physical aspects of both linear and nonlinear Schrödinger systems and their attendant mathematical developments.

Expressly, the book contains

A chapter meant to give a new pedagogical paradigm for teaching an understanding of quantum mechanics, via the Schrödinger equation as an extension of probability theory….

A chapter addressing the Schrödinger equation written in the second quantization formalism, derived from first principles; towards a deeper understanding of classical-quantum correspondence….

A chapter discussing the connection between the Schrödinger equation and one of the most intuitive research fields in classical mechanics: the theory of nonlinear water waves….

A chapter which investigates wave solutions of the generalized nonlinear time-dependent Schrödinger-like equation describing a cosmogonical body formation….

A chapter addressing the nonlinear Schrödinger equation: a mathematical model with its wide-ranging applications and analytical results….

A chapter investigating analytical self-similar and traveling-wave solutions of the Madelung equations obtained from the Schrödinger equation….

A chapter which puts forth a novel paradigm of infinite dimensional quantum phase space extension of the Schrödinger equation….

A chapter which discusses a metaplectic Bohmian formalism from classical (Hamilton’s equations) to quantum physics (Schrödinger’s equation): the Metatron….

The book is written in a lucid style, nicely marrying physical intuition with mathematical insight. As such, it should be of interest to workers in Schrodinger theory and related areas, and generally, to those who seek a deeper understanding of some of the linear and nonlinear perspectives of the Schrödinger equation.

Table of Contents

Table of Contents



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



“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)

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