Quantum Mechanics In Nonlinear Systems

by Xiao-Feng, Pang; Yuan-ping, Feng; Pang, Xiao-feng; Feng, Yuan-ping

• ISBN13: 9789812562999
• ISBN10: 9812562990
• Format: Paperback
• Copyright: 2005-04-18
• Publisher: World Scientific Pub Co Inc

Summary

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.

Table of Contents

Preface	v
1. Linear Quantum Mechanics: Its Successes and Problems	1	(22)
1.1 The Fundamental Hypotheses of the Linear Quantum Mechanics	1	(4)
1.2 Successes and Problems of the Linear Quantum Mechanics	5	(5)
1.3 Dispute between Bohr and Einstein	10	(5)
1.4 Analysis on the Roots of Problems of Linear Quantum Mechanics and Review on Recent Developments	15	(6)
Bibliography	21	(2)
2. Macroscopic Quantum Effects and Motions of Quasi-Particles	23	(58)
2.1 Macroscopic Quantum Effects	23	(11)
2.1.1 Macroscopic quantum effect in superconductors	23	(5)
2.1.1.1 Quantization of magnetic flux	24	(1)
2.1.1.2 Structure of vortex lines in type-II superconductors	25	(1)
2.1.1.3 Josephson effect	26	(2)
2.1.2 Macroscopic quantum effect in liquid helium	28	(3)
2.1.3 Other macroscopic quantum effects	31	(3)
2.1.3.1 Quantum Hall effect	31	(2)
2.1.3.2 Spin polarized atomic hydrogen system	33	(1)
2.1.3.3 Bose-Einstein condensation of excitons	33	(1)
2.2 Analysis on the Nature of Macroscopic Quantum Effect	34	(43)
2.3 Motion of Superconducting Electrons	47	(7)
2.3.1 Motion of electrons in the absence of external fields	49	(1)
2.3.2 Motion of electrons in the presence of an electromagnetic field	50	(4)
2.4 Analysis of Macroscopic Quantum Effects in Inhomogeneous Superconductive Systems	54	(6)
2.4.1 Proximity effect	54	(2)
2.4.2 Josephson current in S-I-S and S-N-S junctions	56	(3)
2.4.3 Josephson effect in SNIS junction	59	(1)
2.5 Josephson Effect and Transmission of Vortex Lines Along the Superconductive Junctions	60	(6)
2.6 Motion of Electrons in Non-Equilibrium Superconductive Systems	66	(6)
2.7 Motion of Helium Atoms in Quantum Superfluid	72	(5)
Bibliography	77	(4)
3. The Fundamental Principles and Theories of Nonlinear Quantum Mechanics	81	(28)
3.1 Lessons Learnt from the Macroscopic Quantum Effects	81	(3)
3.2 Fundamental Principles of Nonlinear Quantum Mechanics	84	(5)
3.3 The Fundamental Theory of Nonlinear Quantum Mechanics	89	(12)
3.3.1 Principle of nonlinear superposition and Backlund transformation	89	(5)
3.3.2 Nonlinear Fourier transformation	94	(1)
3.3.3 Method of quantization	95	(5)
3.3.4 Nonlinear perturbation theory	100	(1)
3.4 Properties of Nonlinear Quantum-Mechanical Systems	101	(5)
Bibliography	106	(3)
4. Wave-Corpuscle Duality of Microscopic Particles in Nonlinear Quantum Mechanics	109	(124)
4.1 Invariance and Conservation Laws, Mass, Momentum and Energy of Microscopic Particles in the Nonlinear Quantum Mechanics	110	(7)
4.2 Position of Microscopic Particles and Law of Motion	117	(9)
4.3 Collision between Microscopic Particles	126	(17)
4.3.1 Attractive interaction (b > 0)	126	(13)
4.3.2 Repulsive interaction (b less than 0) 136
4.3.3 Numerical simulation	139	(4)
4.4 Properties of Elastic Interaction between Microscopic Particles	143	(6)
4.5 Mechanism and Rules of Collision between Microscopic Particles	149	(5)
4.6 Collisions of Quantum Microscopic Particles	154	(7)
4.7 Stability of Microscopic Particles in Nonlinear Quantum Mechanics	161	(8)
4.7.1 "Initial" stability	162	(2)
4.7.2 Structural stability	164	(5)
4.8 Demonstration on Stability of Microscopic Particles	169	(4)
4.9 Multi-Particle Collision and Stability in Nonlinear Quantum Mechanics	173	(5)
4.10 Transport Properties and Diffusion of Microscopic Particles in Viscous Environment	178	(10)
4.11 Microscopic Particles in Nonlinear Quantum Mechanics versus Macroscopic Point Particles	188	(5)
4.12 Reflection and Transmission of Microscopic Particles at Interfaces	193	(7)
4.13 Scattering of Microscopic Particles by Impurities	200	(9)
4.14 Tunneling and Fraunhofer Diffraction	209	(9)
4.15 Squeezing Effects of Microscopic Particles Propagating in Nonlinear Media	218	(3)
4.16 Wave-corpuscle Duality of Microscopic Particles in a Quasiperiodic Perturbation Potential	221	(7)
Bibliography	228	(5)
5. Nonlinear Interaction and Localization of Particles	233	(44)
5.1 Dispersion Effect and Nonlinear Interaction	233	(5)
5.2 Effects of Nonlinear Interactions on Behaviors of Microscopic Particles	238	(5)
5.3 Self-Interaction and Intrinsic Nonlinearity	243	(7)
5.4 Self-localization of Microscopic Particle by Inertialess Self-interaction	250	(2)
5.5 Nonlinear Effect of Media and Self-focusing Mechanism	252	(6)
5.6 Localization of Exciton and Self-trapping Mechanism	258	(5)
5.7 Initial Condition for Localization of Microscopic Particle	263	(4)
5.8 Experimental Verification of Localization of Microscopic Particle	267	(7)
5.8.1 Observation of nonpropagating surface water soliton in water troughs	269	(3)
5.8.2 Experiment on optical solitons in fibers	272	(2)
Bibliography	274	(3)
6. Nonlinear versus Linear Quantum Mechanics	277	(52)
6.1 Nonlinear Quantum Mechanics: An Inevitable Result of Development of Quantum Mechanics	277	(4)
6.2 Relativistic Theory and Self-consistency of Nonlinear Quantum Mechanics	281	(11)
6.2.1 Bound state and Lorentz relations	283	(3)
6.2.2 Interaction between microscopic particles in relativistic theory	286	(2)
6.2.3 Relativistic dynamic equations in the nonrelativistic limit	288	(3)
6.2.4 Nonlinear Dirac equation	291	(1)
6.3 The Uncertainty Relation in Linear and Nonlinear Quantum Mechanics	292	(11)
6.3.1 The uncertainty relation in linear quantum mechanics	292	(1)
6.3.2 The uncertainty relation in nonlinear quantum mechanics	293	(10)
6.4 Energy Spectrum of Hamiltonian and Vector Form of the Nonlinear Schrödinger Equation	303	(12)
6.4.1 General approach	304	(2)
6.4.2 System with two degrees of freedom	306	(3)
6.4.3 Perturbative method	309	(4)
6.4.4 Vector nonlinear Schrödinger equation	313	(2)
6.5 Eigenvalue Problem of the Nonlinear Schrödinger Equation	315	(6)
6.6 Microscopic Causality in Linear and Nonlinear Quantum Mechanics	321	(5)
Bibliography	326	(3)
7. Problem Solving in Nonlinear Quantum Mechanics	329	(68)
7.1 Overview of Methods for Solving Nonlinear Quantum Mechanics Problems	329	(7)
7.1.1 Inverse scattering method	330	(1)
7.1.2 Backlund transformation	330	(1)
7.1.3 Hirota method	331	(1)
7.1.4 Function and variable transformations	331	(5)
7.1.4.1 Function transformation	331	(1)
7.1.4.2 Variable transformation and characteristic line	332	(1)
7.1.4.3 Other variable transformations	332	(1)
7.1.4.4 Self-similarity transformation	333	(1)
7.1.4.5 Galilei transformation	334	(1)
7.1.4.6 Traveling-wave method	335	(1)
7.1.4.7 Perturbation method	335	(1)
7.1.4.8 Variational method	335	(1)
7.1.4.9 Numerical method	335	(1)
7.1.4.10 Experimental simulation	335	(1)
7.2 Traveling-Wave Methods	336	(4)
7.2.1 Nonlinear Schrödinger equation	336	(1)
7.2.2 Sine-Gordon equation	337	(3)
7.3 Inverse Scattering Method	340	(5)
7.4 Perturbation Theory Based on the Inverse Scattering Transformation for the Nonlinear Schrödinger Equation	345	(7)
7.5 Direct Perturbation Theory in Nonlinear Quantum Mechanics	352	(6)
7.5.1 Method of Gorshkov and Ostrovsky	352	(4)
7.5.2 Perturbation technique of Bishop	356	(2)
7.6 Linear Perturbation Theory in Nonlinear Quantum Mechanics	358	(8)
7.6.1 Nonlinear Schrödinger equation	359	(5)
7.6.2 Sine-Gordon equation	364	(2)
7.7 Nonlinearly Variational Method for the Nonlinear Schrödinger Equation	366	(9)
7.8 D Operator and Hirota Method	375	(4)
7.9 Backlund Transformation Method	379	(5)
7.9.1 Auto-Bäcklund transformation method	379	(3)
7.9.2 Bäcklund transform of Hirota	382	(2)
7.10 Method of Separation of Variables	384	(3)
7.11 Solving Higher-Dimensional Equations by Reduction	387	(7)
Bibliography	394	(3)
8. Microscopic Particles in Different Nonlinear Systems	397	(74)
8.1 Charged Microscopic Particles in an Electromagnetic Field	397	(4)
8.2 Microscopic Particles Interacting with the Field of an External Traveling Wave	401	(3)
8.3 Microscopic Particle in Time-dependent Quadratic Potential	404	(7)
8.4 2D Time-dependent Parabolic Potential-field	411	(4)
8.5 Microscopic Particle Subject to a Monochromatic Acoustic Wave	415	(4)
8.6 Effect of Energy Dissipation on Microscopic Particles	419	(4)
8.7 Motion of Microscopic Particles in Disordered Systems	423	(3)
8.8 Dynamics of Microscopic Particles in Inhomogeneous Systems	426	(5)
8.9 Dynamic Properties of Microscopic Particles in a Random Inhomogeneous Media	431	(7)
8.9.1 Mean field method	431	(2)
8.9.2 Statistical adiabatic approximation	433	(3)
8.9.3 Inverse-scattering transformation based statistical perturbation theory	436	(2)
8.10 Microscopic Particles in Interacting Many-particle Systems	438	(6)
8.11 Effects of High-order Dispersion on Microscopic Particles	444	(9)
8.12 Interaction of Microscopic Particles and Its Radiation Effect in Perturbed Systems with Different Dispersions	453	(6)
8.13 Microscopic Particles in Three and Two Dimensional Nonlinear Media with Impurities	459	(8)
Bibliography	467	(4)
9. Nonlinear Quantum-Mechanical Properties of Excitons and Phonons	471	(86)
9.1 Excitons in Molecular Crystals	471	(9)
9.2 Raman Scattering from Nonlinear Motion of Excitons	480	(7)
9.3 Infrared Absorption of Exciton-Solitons in Molecular Crystals	487	(6)
9.4 Finite Temperature Excitonic Mössbauer Effect	493	(8)
9.5 Nonlinear Excitation of Excitons in Protein	501	(9)
9.6 Thermal Stability and Lifetime of Exciton-Soliton at Biological Temperature	510	(10)
9.7 Effects of Structural Disorder and Heart Bath on Exciton Localization	520	(9)
9.7.1 Effects of structural disorder	521	(5)
9.7.2 Influence of heat bath	526	(3)
9.8 Eigenenergy Spectra of Nonlinear Excitations of Excitons	529	(7)
9.9 Experimental Evidences of Exciton-Soliton State in Molecular Crystals and Protein Molecules	536	(13)
9.9.1 Experimental data in acetanilide	536	(5)
9.9.1.1 Infrared absorption and Raman spectra	537	(1)
9.9.1.2 Dynamic test of soliton excitation in acetanilide	538	(3)
9.9.2 Infrared and Raman spectra of collagen, E. coli. and human tissue	541	(4)
9.9.2.1 Infrared spectra of collagen proteins	541	(3)
9.9.2.2 Raman spectrum of collagen	544	(1)
9.9.3 Infrared radiation spectrum of human tissue and Raman spectrum of E. col	545	(2)
9.9.4 Specific heat of ACN and protein	547	(2)
9.10 Properties of Nonlinear Excitations of Phonons	549	(2)
Bibliography	551	(6)
10. Properties of Nonlinear Excitations and Motions of Protons, Polarons and Magnons in Different Systems	557	(62)
10.1 Model of Excitation and Proton Transfer in Hydrogen-bonded Systems	557	(7)
10.2 Theory of Proton Transferring in Hydrogen Bonded Systems	564	(8)
10.3 Thermodynamic Properties and Conductivity of Proton Transfer	572	(5)
10.4 Properties of Proton Collective Excitation in Liquid Water	577	(9)
10.4.1 States and properties of molecules in liquid water	578	(1)
10.4.2 Properties of hydrogen-bonded closed chains in liquid water	579	(2)
10.4.3 Ring electric current and mechanism of magnetization of water	581	(5)
10.5 Nonlinear Excitation of Polarons and its Properties	586	(7)
10.6 Nonlinear Localization of Small Polarons	593	(3)
10.7 Nonlinear Excitation of Electrons in Coupled Electron-Electron and Electron-Phonon Systems	596	(5)
10.8 Nonlinear Excitation of Magnon in Ferromagnetic Systems	601	(6)
10.9 Collective Excitations of Magnons in Antiferromagnetic Systems	607	(6)
Bibliography	613	(6)
Index	619