Quantum Mechanics in Nonlinear Systems

In the history of physics and science, quantum mechanics has served as the foundation of modern science. This book discusses the properties of microscopic particles in nonlinear systems, principles of the nonlinear quantum mechanical theory, and its applications in condensed matter, polymers and biological systems.The book is essentially composed of three parts. The first part presents a review of linear quantum mechanics, as well as theoretical and experimental fundamentals that establish the nonlinear quantum mechanical theory. The theory itself and its essential features are covered in the second part. In the final part, extensive applications of this theory in physics, biology and polymer are introduced. The whole volume forms a complete system of nonlinear quantum mechanics.The book is intended for researchers, graduate students as well as upper-level undergraduates. Preface Contents Chapter 1 Linear Quantum Mechanics: Its Successes and Problems 1.1 The Fundamental Hypotheses of the Linear Quantum Mechanics 1.2 Successes and Problems of the Linear Quantum Mechanics 1.3 Dispute between Bohr and Einstein 1.4 Analysis on the Roots of Problems of Linear Quantum Mechanics and Review on Recent Developments Bibliography Chapter 2 Macroscopic Quantum Effects and Motions of Quasi-Particles 2.1 Macroscopic Quantum Effects 2.1.1 Macroscopic quantum effect in superconductors 2.1.2 Macroscopic quantum effect in liquid helium 2.1.3 Other macroscopic quantum effects 2.2 Analysis on the Nature of Macroscopic Quantum Effect 2.3 Motion of Superconducting Electrons 2.3.1 Motion of electrons in the absence of external fields 2.3.2 Motion of electrons in the presence of an electromagnetic field 2.4 Analysis of Macroscopic Quantum Effects in Inhomogeneous Superconductive Systems 2.4.1 Proximity effect 2.4.2 Josephson current in S-I-S and S-N-S junctions 2.4.3 Josephson effect in SNIS junction 2.5 Josephson Effect and Transmission of Vortex Lines Along the Superconductive Junctions 2.6 Motion of Electrons in Non-Equilibrium Superconductive Systems 2.7 Motion of Helium Atoms in Quantum Superfluid Bibliography Chapter 3 The Fundamental Principles and Theories of Nonlinear Quantum Mechanics 3.1 Lessons Learnt from the Macroscopic Quantum Effects 3.2 Fundamental Principles of Nonlinear Quantum Mechanics 3.3 The Fundamental Theory of Nonlinear Quantum Mechanics 3.3.1 Principle of nonlinear superposition and Bäcklund transformation 3.3.2 Nonlinear Fourier transformation 3.3.3 Method of quantization 3.3.4 Nonlinear perturbation theory 3.4 Properties of Nonlinear Quantum-Mechanical Systems Bibliography Chapter 4 Wave-Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics 4.1 Invariance and Conservation Laws, Mass, Momentum and Energy of Microscopic Particles in the Nonlinear Quantum Mechanics 4.2 Position of Microscopic Particles and Law of Motion 4.3 Collision between Microscopic Particles 4.3.1 Attractive interaction (b > 0) 4.3.2 Repulsive interaction (b < 0) 4.3.3 Numerical simulation 4.4 Properties of Elastic Interaction between Microscopic Particles 4.5 Mechanism and Rules of Collision between Microscopic Particles 4.6 Collisions of Quantum Microscopic Particles 4.7 Stability of Microscopic Particles in Nonlinear Quantum Mechanics 4.7.1 “Initial” stability 4.7.2 Structural stability 4.8 Demonstration on Stability of Microscopic Particles 4.9 Multi-Particle Collision and Stability in Nonlinear Quantum Mechanics 4.10 Transport Properties and Diffusion of Microscopic Particles in Viscous Environment 4.11 Microscopic Particles in Nonlinear Quantum Mechanics versus Macroscopic Point Particles 4.12 Reflection and Transmission of Microscopic Particles at Interfaces 4.13 Scattering of Microscopic Particles by Impurities 4.14 Tunneling and Fraunhofer Diffraction 4.15 Squeezing Effects of Microscopic Particles Propagating in Nonlinear Media 4.16 Wave-corpuscle Duality of Microscopic Particles in a Quasiperiodic Perturbation Potential Bibliography Chapter 5 Nonlinear Interaction and Localization of Particles 5.1 Dispersion Effect and Nonlinear Interaction 5.2 Effects of Nonlinear Interactions on Behaviors of Microscopic Particles 5.3 Self-Interaction and Intrinsic Nonlinearity 5.4 Self-localization of Microscopic Particle by Inertialess Self-interaction 5.5 Nonlinear Effect of Media and Self-focusing Mechanism 5.6 Localization of Exciton and Self-trapping Mechanism 5.7 Initial Condition for Localization of Microscopic Particle 5.8 Experimental Verification of Localization of Microscopic Particle 5.8.1 Observation of nonpropagating surface water soliton in water troughs 5.8.2 Experiment on optical solitons in fibers Bibliography Chapter 6 Nonlinear versus Linear Quantum Mechanics 6.1 Nonlinear Quantum Mechanics: An Inevitable Result of Development of Quantum Mechanics 6.2 Relativistic Theory and Self-consistency of Nonlinear Quantum Mechanics 6.2.1 Bound state and Lorentz relations 6.2.2 Interaction between microscopic particles in relativistic theory 6.2.3 Relativistic dynamic equations in the nonrelativistic limit 6.2.4 Nonlinear Dirac equation 6.3 The Uncertainty Relation in Linear and Nonlinear Quantum Mechanics 6.3.1 The uncertainty relation in linear quantum mechanics 6.3.2 The uncertainty relation in nonlinear quantum mechanics 6.4 Energy Spectrum of Hamiltonian and Vector Form of the Non-linear Schrödinger Equation 6.4.1 General approach 6.4.2 System with two degrees of freedom 6.4.3 Perturbative method 6.4.4 Vector nonlinear Schrödinger equation 6.5 Eigenvalue Problem of the Nonlinear Schrödinger Equation 6.6 Microscopic Causality in Linear and Nonlinear Quantum Mechanics Bibliography Chapter 7 Problem Solving in Nonlinear Quantum Mechanics 7.1 Overview of Methods for Solving Nonlinear Quantum Mechanics Problems 7.1.1 Inverse scattering method 7.1.2 Bäcklund transformation 7.1.3 Hirota method 7.1.4 Function and variable transformations 7.2 Traveling-Wave Methods 7.2.1 Nonlinear Schrödinger equation 7.2.2 Sine-Gordon equation 7.3 Inverse Scattering Method 7.4 Perturbation Theory Based on the Inverse Scattering Transformation for the Nonlinear Schrödinger Equation 7.5 Direct Perturbation Theory in Nonlinear Quantum Mechanics 7.5.1 Method of Gorshkov and Ostrovsky 7.5.2 Perturbation technique of Bishop 7.6 Linear Perturbation Theory in Nonlinear Quantum Mechanics 7.6.1 Nonlinear Schrödinger equation 7.6.2 Sine-Gordon equation 7.7 Nonlinearly Variational Method for the Nonlinear Schrödinger Equation 7.8 D Operator and Hirota Method 7.9 Bäcklund Transformation Method 7.9.1 Auto-Bäcklund transformation method 7.9.2 Bäcklund transform of Hirota 7.10 Method of Separation of Variables 7.11 Solving Higher-Dimensional Equations by Reduction Bibliography Chapter 8 Microscopic Particles in Different Nonlinear Systems 8.1 Charged Microscopic Particles in an Electromagnetic Field 8.2 Microscopic Particles Interacting with the Field of an External Traveling Wave 8.3 Microscopic Particle in Time-dependent Quadratic Potential 8.4 2D Time-dependent Parabolic Potential-field 8.5 Microscopic Particle Subject to a Monochromatic Acoustic Wave 8.6 Effect of Energy Dissipation on Microscopic Particles 8.7 Motion of Microscopic Particles in Disordered Systems 8.8 Dynamics of Microscopic Particles in Inhomogeneous Systems 8.9 Dynamic Properties of Microscopic Particles in a Random Inhomogeneous Media 8.9.1 Mean field method 8.9.2 Statistical adiabatic approximation 8.9.3 Inverse-scattering transformation based statistical perturbation theory 8.10 Microscopic Particles in Interacting Many-particle Systems 8.11 Effects of High-order Dispersion on Microscopic Particles 8.12 Interaction of Microscopic Particles and Its Radiation Effect in Perturbed Systems with Different Dispersions 8.13 Microscopic Particles in Three and Two Dimensional Nonlinear Media with Impurities Bibliography Chapter 9 Nonlinear Quantum-Mechanical Properties of Excitons and Phonons 9.1 Excitons in Molecular Crystals 9.2 Raman Scattering from Nonlinear Motion of Excitons 9.3 Infrared Absorption of Exciton-Solitons in Molecular Crystals 9.4 Finite Temperature Excitonic Mössbauer Effect 9.5 Nonlinear Excitation of Excitons in Protein 9.6 Thermal Stability and Lifetime of Exciton-Soliton at Biological Temperature 9.7 Effects of Structural Disorder and Heart Bath on Exciton Localization 9.7.1 Effects of structural disorder 9.7.2 Influence of heat bath 9.8 Eigenenergy Spectra of Nonlinear Excitations of Excitons 9.9 Experimental Evidences of Exciton-Soliton State in Molecular Crystals and Protein Molecules 9.9.1 Experimental data in acetanilide 9.9.2 Infrared and Raman spectra of collagen, E. coli. and human tissue 9.9.3 Infrared radiation spectrum of human tissue and Raman spectrum of E. col. 9.9.4 Specific heat of ACN and protein 9.10 Properties of Nonlinear Excitations of Phonons Bibliography Chapter 10 Properties of Nonlinear Excitations and Motions of Protons, Polarons and Magnons in Different Systems 10.1 Model of Excitation and Proton Transfer in Hydrogen-bonded Systems 10.2 Theory of Proton Transferring in Hydrogen Bonded Systems 10.3 Thermodynamic Properties and Conductivity of Proton Transfer 10.4 Properties of Proton Collective Excitation in Liquid Water 10.4.1 States and properties of molecules in liquid water 10.4.2 Properties of hydrogen-bonded closed chains in liquid water 10.4.3 Ring electric current and mechanism of magnetization of water 10.5 Nonlinear Excitation of Polarons and its Properties 10.6 Nonlinear Localization of Small Polarons 10.7 Nonlinear Excitation of Electrons in Coupled Electron-Electron and Electron-Phonon Systems 10.8 Nonlinear Excitation of Magnon in Ferromagnetic Systems 10.9 Collective Excitations of Magnons in Antiferromagnetic Systems Bibliography Index This Book Discusses The Properties Of Microscopic Particles In Nonlinear Systems, Principles Of The Nonlinear Quantum Mechanical Theory, And Its Applications In Condensed Matter, Polymers And Biological Systems. The Book Is Essentially Composed Of Three Parts. The First Part Presents A Review Of Linear Quantum Mechanics, As Well As Theoretical And Experimental Fundamentals That Establish The Nonlinear Quantum Mechanical Theory. The Theory Itself And Its Essential Features Are Covered In The Second Part. In The Final Part, Extensive Applications Of This Theory In Physics, Biology And Polymer Are Introduced. The Whole Volume Forms A Complete System Of Nonlinear Quantum Mechanics. The Book Is Intended For Researchers And Graduate Students As Well As Upper-level Undergraduates.--book Jacket. Linear Quantum Mechanics : Its Success And Problems -- Macroscopic Quantum Effects And Motions Of Quasi-particles -- The Fundamental Principles And Theories Of Nonlinear Quantum Mechanics -- Wave-corpuscle Duality Of Microscopic Particles In Nonlinear Quantum Mechanics -- Nonlinear Interaction And Localization Of Particles -- Nonlinear Versus Linear Quantum Mechanics -- Problem Solving In Nonlinear Quantum Mechanics -- Microscopic Particles In Different Nonlinear Systems -- Nonlinear Quantum-mechanical Properties Of Excitons And Phonons -- Properties Of Nonlinear Excitations And Motions Of Protons, Polarons, And Magnons In Different Systems. Pang Xiao-feng, Feng Yuan-ping. Includes Bibliographical References And Index. In the history of physics and science, quantum mechanics has served as the foundation of modern science. This book discusses the properties of microscopic particles in nonlinear systems, principles of the nonlinear quantum mechanical theory, and its applications in condensed matter, polymers and biological systems. The book is essentially composed of three parts. The first part presents a review of linear quantum mechanics, as well as theoretical and experimental fundamentals that establish the nonlinear quantum mechanical theory. The theory itself and its essential features are covered in the