Overview

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 worlds leading workers in quantum dynamics; particularly as concerning Schrödingers 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 (Hamiltons equations) to quantum physics (Schrödingers 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.

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9781536176629

Table of Contents

Reviews

The book contains various approaches to the Schroedinger 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 Schroedinger equation. Chapter 2 is devoted to the derivation of SE from the classical Hamiltonian by some procedure of second quantization. In Chapters 35, 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 Schroedinger, 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 Hamiltons equation if one uses the metaplectic representation of canonical transformations. All that makes the book interesting for a wide community of physicists. - 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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