QUANTUM
MECHANICS IN
NONLINEAR SYSTEMS

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QUANTUM
MECHANICS IN
NONLINEAR SYSTEMS

Pang Xiao-Feng
University of Electronic Science and Technology of China, China

Feng Yuan-Ping
National University of Singapore, Singapore

\[p World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Co vent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data
Pang, Xiao-Feng, 1945Quantum mechanics in nonlinear systems / Pang Xiao-Feng, Feng Yuan-Ping,
p. cm.
Includes bibliographical references and index.
ISBN 9812561161 (alk. paper) ISBN 9812562990 (pbk)
1. Nonlinear theories. 2. Quantum theory. I. Feng, Yuang-Ping. II. Title.
QC20.7.N6P36 2005
530.15'5252-dc22

2004060119

British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.

Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Printed in Singapore by World Scientific Printers (S) Pte Ltd

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Preface

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. It is intended for researchers,
graduate students, and upper level undergraduate students.
About the Book
Some materials in the book are based on the lecture notes for a graduate course
"Problems in nonlinear quantum theory" given by one of the authors (X. F. Pang)
in his university in the 1980s, and a book entitled "Theory of Nonlinear Quantum
Mechanics" (in Chinese) by the same author in 1994. However, the contents were
completely rewritten in this English edition, and in the process, we incorporated
recent results related to the nonlinear Schrodinger equations and the nonlinear
Klein-Gordon equations based on research of the authors as well as other scientists
in the field.
The following topics are covered in 10 chapters in this book, the necessity for
constructing a nonlinear quantum mechanical theory; the theoretical and experimental foundations on which the nonlinear quantum mechanical theory is based; the
elementary principles and the theory of nonlinear quantum mechanics; the wavecorpuscle duality of particles in the theory; nonlinear interaction and localization of
particles; the relations between nonlinear and linear quantum theories; the properties of nonlinear quantum mechanics, including simultaneous determination of position and momentum of particles, self-consistence and completeness of the theory;
methods of solving nonlinear quantum mechanical problems; properties of particles
in various nonlinear systems and applications to exciton, phonon, polaron, electron,
magnon and proton in physical, biological and polymeric systems. In particular,
an in-depth discussion on the wave-corpuscle duality of microscopic particles in
nonlinear systems is given in this book.
The book is organized as follows. We start with a brief review on the postulates of linear quantum mechanics, its successes and problems encountered by the
linear quantum mechanics in Chapter 1. In Chapter 2, we discuss some macroV

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Quantum Mechanics in Nonlinear Systems

scopic quantum effects which form the experimental foundation for a new nonlinear
quantum theory, and the properties of microscopic particles in the macroscopic
quantum systems which provide a theoretical base for the establishment of the nonlinear quantum theory. The fundamental principles on which the new theory is
based and the theory of nonlinear quantum mechanics as proposed by Pang et al.
are given in Chapter 3. The close relations among the properties of macroscopic
quantum effects; nonlinear interactions and soliton motions of microscopic particles
in macroscopic quantum systems play an essential role in the establishment of this
theory. In Chapter 4, we examine in details the wave-corpuscle duality of particles in nonlinear systems. In Chapter 5, we look into the mechanisms of nonlinear
interactions and their relations to localization of particles. In the next chapter,
features of the nonlinear and linear quantum mechanical theories are compared; the
self-consistence and completeness of the theory were examined; and finally solutions
and properties of the time-independent nonlinear quantum mechanical equations,
and their relations to the original quantum mechanics are discussed. We will show
that problems existed in the original quantum mechanics can be explained by the
new nonlinear quantum mechanical theory. Chapter 7 shows the methods of solving
various kinds of nonlinear quantum mechanical problems. The dynamic properties
of microscopic particles in different nonlinear systems are discussed in Chapter 8.
Finally in Chapters 9 and 10, applications of the theory to exciton, phonon, electron, polaron, proton and magnon in various physical systems, such as condensed
matter, polymers, molecules and living systems, are explored.
The book is essentially composed of three parts. The first part consists of Chapters 1 and 2, gives a review on the linear quantum mechanics, and the important
experimental and theoretical studies that lead to the establishment of the nonlinear
quantum-mechanical theory. The nonlinear theory of quantum mechanics itself as
well as its essential features are described in second part (Chapters 3-8). In the
third part (Chapters 9 and 10), we look into applications of this theory in physics,
biology and polymer, etc.
An Overview
Nonlinear quantum mechanics (NLQM) is a theory for studying properties and

motion of microscopic particles (MIPs) in nonlinear systems which exhibit quantum
features. It was named so in relation to the quantum mechanics established by Bohr,
Heisenberg, Schrodinger, and many others. The latter deals with only properties
and motion of microscopic particles in linear systems, and will be referred to as the
linear quantum mechanics (LQM) in this book.
The concept of nonlinearity in quantum mechanics was first proposed by de
Broglie in the 1950s in his book, "Nonlinear wave mechanics". LQM had difficulties
explaining certain problems right from the start, de Broglie attempted to clarify
and solve these problems of LQM using the concept of nonlinearity. Even though
a great idea, de Broglie did not succeed because his approach was confined to the

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Preface

vii

framework of the original LQM.
Looking back to the modern history of physics and science, we know that quantum mechanics is really the foundation of modern science. It had great successes
in solving many important physical problems, such as the light spectra of hydrogen and hydrogen-like atoms, the Lamb shift in these atoms, and so on. Jargons
such as "quantum jump" have their scientific origins and become ever fashionable in our normal life. In this particular case, the phrase "quantum jump" gives
a vivid description for major qualitative changes and is almost universally used.
However, it was also known that LQM has its problems and difficulties related
to the fundamental postulates of the theory, for example, the implications of the
uncertainty principle between conjugate dynamical variables, such as position and
momentum. Different opinions on how to resolve such issues and further develop
quantum mechanics lead to intense arguments and debates which lasted almost a
century. The long-time controversy showed that these problems cannot be solved
within the framework of LQM. It was also through such debates that the direction

to take for improving and further developing quantum mechanics became clear,
which was to extend the theory from the linear to the nonlinear regime. Certain
fundamental assumptions such as the principle of linear superposition, linearity of
the dynamical equation and the independence of the Hamiltonian of a system on
its wave function must be abandoned because they are the roots of the problems of
LQM. In other words, a new nonlinear quantum theory should be developed.
A series of nonlinear quantum phenomena including the macroscopic quantum
effects and motion of soli tons or solitary waves have, in recent decades, been discovered one after another from experiments in superconductors, superfluid, ferromagnetic, antiferromagnetic, organic molecular crystals, optical fiber materials
and polymer and biological systems, etc. These phenomena did underlie nonlinear quantum mechanics because they could not be explained by LQM. Meanwhile,
the theories of nonlinear partial differential equations and of solitary wave have
been very well established which build the mathematical foundation of nonlinear
quantum mechanics. Due to these developments of nonlinear science, a lot of new
branches of science, for example, nonlinear vibrational theory, nonlinear Newton
mechanics, nonlinear fluid mechanics, nonlinear optics, chaos, synergetics and fractals, have been established or being developed. In such a case, it is necessary to
build the nonlinear quantum mechanics described the law of motion of microscopic
particles in nonlinear systems.
However, how do we establish such a theory? Experiences in the study of quantum mechanics for several decades tell us that it is impossible to establish such a
theory if we followed the direction of de Broglie et al. A completely new way of
thinking, a new idea and method must be adopted and developed.
According to this idea we will, first of all, study the properties of macroscopic
quantum effects, which is a nonlinear quantum effect on macroscopic scale occurred
in some matters, for example, superconductors and superfluid. To be more precise,

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Quantum Mechanics in Nonlinear Systems


these effects occur in systems with ordered states over a long-range, or, coherent
states, or, Bose-like condensed states, which are formed through phase transitions
after a spontaneous symmetry breakdown in the systems by means of nonlinear
interactions. These results show that the properties of microscopic particles in the
macroscopic quantum systems cannot be well represented by LQM. In these systems
the microscopic particles are self-localized to become soliton with wave-corpuscle
duality. The observed macroscopic quantum effects are just a result produced by
soliton motions of the particles in these systems. Therefore, the macroscopic quantum effect is closely related to the nonlinear interaction and to solitary motion of
the particles. The close relations among them prompt us to propose and establish
the fundamental principles and the theory of NLQM which describes the properties
of microscopic particles in the nonlinear systems. We then demonstrate that the
NLQM is truely a self-consistent and complete theory. It has so far enjoyed great
successes in a wide range of applications in condensed matter, polymers and biological systems. In exploring these applications, we also obtain many important results
which are consistent with experimental data. These results confirm the correctness
of the NLQM on one hand, and provide further theoretical understanding to many
phenomena occurred in these systems on the other hand.
Therefore, we can say that the experimental foundation of the nonlinear quantum mechanics established is the macroscopic quantum effects, and the coherent
phenomena. Its theoretical basis is superconducting and superfluidic theories. Its
mathematical framework is the theories of nonlinear partial differential equations
and of solitary waves. The elementary principles and theory of the NLQM proposed
here are established on the basis of results of research on properties of microscopic
particles in nonlinear systems and the close relations among the macroscopic quantum effects, nonlinear interactions and soliton motions. The linearity in the LQM
is removed and dependence of Hamiltonian of systems on the state wave function
of particles is assumed in this theory. Through careful investigations and extensive
applications, we demonstrate that this new theory is correct, self-consistent and
complete. The new theory solves the problems and difficulties in the LQM.
One of the authors (X. F. Pang) has been studying the NLQM for about 25 years
and has published about 100 papers related to this topic. The newly established
nonlinear quantum theory has been reported and discussed in many international
conferences, for example, International Conference of Nonlinear Physics (ICNP), International Conference of Material Physics (ICMP), Asia Pacific Physics Conference

(APPC), International Workshop of Nonlinear Problems in Science and Engineering
(IWNPSE), National Quantum Mechanical Conference of China (NQMCC), etc..
Pang also published a monograph entitled "The problems for nonlinear quantum
theory" in 1985 and a book entitled "The theory of nonlinear quantum mechanics"
in 1994 in Chinese. Pang has also lectured in many Universities and Institutes on
this subject. Certain materials in this book are based on the above lecture materials
and book. It also incorporates many recent results published by Pang and other

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Preface

ix

scientists related to nonlinear Schrodinger equation and nonlinear Klein-Gordon
equations.
Finally, we should point out that the NLQM presented here is completely different from the LQM. It is intended for studying properties and motion of microscopic particles in nonlinear systems, in which the microscopic particles become
self-localized particles, or solitons, under the nonlinear interaction. Sources of
such nonlinear interation can be intrinsic nonlinearity or persistent self-interactions
through mechanisms such as self-trapping, self-condensation, self-focusing and selfcoherence by means of phase transitions, sudden changes and spontaneous breakdown of symmetry of the systems, and so on. In such cases, the particles have
exactly wave-corpuscle duality, and obey simultaneously the classical and quantum
laws of motion, i. e., the nature and properties of the microscopic particle are essentially changed from that in LQM. For example, the position and momentum of
a particle can be determined to a certain degree. Thus, the linear feature of theory
and the principles for independences of the Hamiltonian of the systems on the statewave function of particle are completely removed. However, this is not to deny the
validity of LQM. Rather we believe that it is an approximate theory which is only
suitable for systems with linear interactions and the nonlinear interaction is small
and can be neglected. In other words, LQM is a special case of the NLQM. This
relation between the LQM and the NLQM is similar to that between the relativity
and Newtonian mechanics. The NLQM established here is a necessary result of

development of quantum mechanics in nonlinear systems.
The establishment of the NLQM can certainly advance and facilitate further
developments of natural sciences including physics, biology and astronomy. Meanwhile, it is also useful in understanding the properties and limitations of the LQM,
and in solving problems and difficulties encountered by the LQM. Therefore, we
hope that by publishing this book on quantum mechanics in the nonlinear systems
would add some value to science and would contribute to our understanding of the
wonderful nature.
X. F. Pang and Y. P. Feng
2004

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Contents

Preface

v

1.

1

Linear Quantum Mechanics: Its Successes and Problems
1.1
1.2
1.3

1.4

2.

The Fundamental Hypotheses of the Linear Quantum Mechanics .
Successes and Problems of the Linear Quantum Mechanics
Dispute between Bohr and Einstein
Analysis on the Roots of Problems of Linear Quantum Mechanics
and Review on Recent Developments
Bibliography

15
21

Macroscopic Quantum Effects and Motions of Quasi-Particles

23

2.1

23
23
24
25
26
28
31
31
33
33

34
47
49
50

2.2
2.3

2.4

Macroscopic Quantum Effects
2.1.1 Macroscopic quantum effect in superconductors
2.1.1.1 Quantization of magnetic flux
2.1.1.2 Structure of vortex lines in type-II superconductors .
2.1.1.3 Josephson effect
2.1.2 Macroscopic quantum effect in liquid helium
2.1.3 Other macroscopic quantum effects
2.1.3.1 Quantum Hall effect
2.1.3.2 Spin polarized atomic hydrogen system
2.1.3.3 Bose-Einstein condensation of excitons
Analysis on the Nature of Macroscopic Quantum Effect
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
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
xi

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5
10

54
54
56


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Quantum Mechanics in Nonlinear Systems

2.4.3 Josephson effect in SNIS junction
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

59
60
66
72
77

The Fundamental Principles and Theories of Nonlinear Quantum
Mechanics

81


3.1
3.2
3.3

81
84
89

2.5

3.

4.

Lessons Learnt from the Macroscopic Quantum Effects
Fundamental Principles of Nonlinear Quantum Mechanics
The Fundamental Theory of Nonlinear Quantum Mechanics . . . .
3.3.1 Principle of nonlinear superposition and Backlund 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

89
94
95
100
101

106

Wave-Corpuscle Duality of Microscopic Particles in Nonlinear
Quantum Mechanics

109

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

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117
126
126
136
139
143
149
154
161
162
164
169
173
178
188


Contents

4.12
4.13
4.14
4.15

Reflection and Transmission of Microscopic Particles at Interfaces .
Scattering of Microscopic Particles by Impurities
Tunneling and Praunhofer Diffraction
Squeezing Effects of Microscopic Particles Propagating in Nonlinear
Media

4.16 Wave-corpuscle Duality of Microscopic Particles in a Quasiperiodic
Perturbation Potential
Bibliography
5.

193
200
209
218
221
228

Nonlinear Interaction and Localization of Particles

233

5.1
5.2

233

Dispersion Effect and Nonlinear Interaction
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
6.

xiii

Nonlinear versus Linear Quantum Mechanics
6.1
6.2

6.3

6.4

Nonlinear Quantum Mechanics: An Inevitable Result of Development of Quantum Mechanics
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
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 .
Energy Spectrum of Hamiltonian and Vector Form of the Nonlinear

Schrodinger Equation
6.4.1 General approach

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243
250
252
258
263
267
269
272
274
277
277
281
283
286
288
291
292
292
293
303
304


xiv


7.

Quantum Mechanics in Nonlinear Systems

6.4.2 System with two degrees of freedom
6.4.3 Perturbative method
6.4.4 Vector nonlinear Schrodinger equation
6.5 Eigenvalue Problem of the Nonlinear Schrodinger Equation . . . .
6.6 Microscopic Causality in Linear and Nonlinear Quantum Mechanics
Bibliography

306
309
313
315
321
326

Problem Solving in Nonlinear Quantum Mechanics

329

7.1

7.2

7.3
7.4
7.5


7.6

7.7
7.8
7.9

Overview of Methods for Solving Nonlinear Quantum Mechanics
Problems
7.1.1 Inverse scattering method
7.1.2 Backlund transformation
7.1.3 Hirota method
7.1.4 Function and variable transformations
7.1.4.1 Function transformation
7.1.4.2 Variable transformation and characteristic line . . .
7.1.4.3 Other variable transformations
7.1 A A Self-similarity transformation
7.1.4.5 Galilei transformation
7.1.4.6 Traveling-wave method
7.1.4.7 Perturbation method
7.1.4.8 Variational method
7.1.4.9 Numerical method
7.1.4.10 Experimental simulation
Traveling-Wave Methods
7.2.1 Nonlinear Schrodinger equation
7.2.2 Sine-Gordon equation
Inverse Scattering Method
Perturbation Theory Based on the Inverse Scattering Transformation for the Nonlinear Schrodinger Equation
Direct Perturbation Theory in Nonlinear Quantum Mechanics . . .
7.5.1 Method of Gorshkov and Ostrovsky

7.5.2 Perturbation technique of Bishop
Linear Perturbation Theory in Nonlinear Quantum Mechanics . . .
7.6.1 Nonlinear Schrodinger equation
7.6.2 Sine-Gordon equation
Nonlinearly Variational Method for the Nonlinear Schrodinger
Equation
D Operator and Hirota Method
Backlund Transformation Method
7.9.1 Auto-Backlund transformation method
7.9.2 Backlund transform of Hirota

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330
330
331
331
331
332
332
333
334
335
335
335
335
335
336
336

337
340
345
352
352
356
358
359
364
366
375
379
379
382


Contents

7.10 Method of Separation of Variables
7.11 Solving Higher-Dimensional Equations by Reduction
Bibliography
8. Microscopic Particles in Different Nonlinear Systems
8.1
8.2

Charged Microscopic Particles in an Electromagnetic Field
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
9. Nonlinear Quantum-Mechanical Properties of Excitons and Phonons
9.1
9.2
9.3
9.4
9.5
9.6

Excitons in Molecular Crystals
Raman Scattering from Nonlinear Motion of Excitons
Infrared Absorption of Exciton-Solitons in Molecular Crystals . . .
Finite Temperature Excitonic Mossbauer Effect
Nonlinear Excitation of Excitons in Protein
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

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384
387
394
397
397
401
404
411
415
419
423
426
431
431
433
436
438
444
453
459
467
471

471
480
487
493
501
510
520
521
526
529


xvi

Contents

9.9

Experimental Evidences of Exciton-Soliton State in Molecular
Crystals and Protein Molecules
9.9.1 Experimental data in acetanilide
9.9.1.1 Infrared absorption and Raman spectra
9.9.1.2 Dynamic test of soliton excitation in acetanilide . . .
9.9.2 Infrared and Raman spectra of collagen, E. coli. and human
tissue
9.9.2.1 Infrared spectra of collagen proteins
9.9.2.2 Raman spectrum of collagen
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

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

536
536
537
538
541
541
544
545

547
549
551

557
557
564
572
577
578
579
581
586
593
596
601
607
613
619

Index

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Chapter 1

Linear Quantum Mechanics: Its Successes
and Problems
The quantum mechanics established by Bohr, de Broglie, Schrodinger, Heisenberg

and Bohn in 1920s is often referred to as the linear quantum mechanics (LQM). In
this chapter, the hypotheses of linear quantum mechanics, the successes of and problems encountered by the linear quantum mechanics are reviewed. The directions
for further development of the quantum theory are also discussed.

1.1

The Fundamental Hypotheses of the Linear Quantum Mechanics

At the end of the 19th century, classical mechanics encountered major difficulties
in describing motions of microscopic particles (MIPs) with extremely light masses
(~ 10~23 - 10~26 g) and extremely high velocities, and the physical phenomena
related to such motions. This forced scientists to rethink the applicability of classical
mechanics and lead to fundamental changes in their traditional understanding of the
nature of motions of microscopic objects. The wave-corpuscle duality of microscopic
particles was boldly proposed by Bohr, de Broglie and others. On the basis of this
revolutionary idea and some fundamental hypotheses, Schrodinger, Heisenberg, etc.
established the linear quantum mechanics which provided a unique way of describing
quantum systems. In this theory, the states of microscopic particles are described by
a wave function which is interpreted based on statistics, and physical quantities are
represented by operators and are given in terms of the possible expectation values
(or eigenvalues) of these operators in the states (or eigenstates). The time evolution
of quantum states are governed by the Schrodinger equation. The hypotheses of
the linear quantum mechanics are summarized in the following.
(1) A state of a microscopic particle is represented by a vector in the Hilbert
space, \ip), or a wave function ip{r,t) in coordinate space. The wave function
uniquely describes the motion of the microscopic particle and reflects the wave
nature of microscopic particles. Furthermore, if /? is a constant, then both \ip) and
/3\ip) describe the same state. Thus, the normalized wave function, which satisfies
the condition (ipl'tp) = 1, is often used to describe the state of the particle.
l


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2

Quantum Mechanics in Nonlinear Systems

(2) A physical quantity, such as the coordinate X, the momentum P and the
energy E of a particle, is represented by a linear operator in the Hilbert space, and
the eigenvectors of the operator form a basis of the Hilbert space. An observable
mechanical quantity is represented by a Hermitian operator whose eigenvalues are
real. Therefore, the values a physical quantity can have are the eigenvalues of the
corresponding linear operator. The eigenvectors corresponding to different eigenvalues are orthogonal to each other. All eigenstates of a Hermitian operator span an
orthogonal and complete set, {IPL}- Any vector of state, ip(f,t), can be expanded
in terms of the eigenvectors:

^(r,t) = ^2cL^L(r,t),

or

\ij)(r,t)) = J^^M^PL)

L

(1.1)

L

where Ci = (tpL\ip) is the wave function in representation L. If the spectrum of

L is continuous, then the summation in (1.1) should be replaced by an integral:
JdL---.
Equation (1.1) can be regarded as a projection of the wave function
ip(f, t) of a microscopic particle system on to those of its subsystems and it is
the foundation of transformation between different representations in the linear
quantum mechanics. In the quantum state described by tjj(f,t), the probability of
getting the value L' in a measurement of L is \CL'\2 = KV'L'IV')!2 m t n e c a s e °f
discrete spectrum, or \(ipLi\ip)\2dL if the spectrum of the system is continuous. In
a single measurement of any mechanical quantity, only one of the eigenvalues of the
corresponding linear operator can be obtained, and the system is then said to be
in the eigenstate belonging to this eigenvalue. This is a fundamental assumption of
linear quantum mechanics concerning measurements of physical quantities.
(3) The average (A) of a physical quantity A in an arbitrary state \ip) is given

(1.2)

or

(A) = (v|i|V>),
if tp is normalized. Possible values of A can be obtained through the determination
of the above average. In order to obtain these possible values, we must find a wave
function in which A has a precise value. In other words, we must find a state such
that (AA)2 = 0, where (AA)2 = (A2) - (A)2. This leads to the following eigenvalue
problem for the operator A,
AtpL = AipL.

(1.3)

From the above equation we can determine the spectrum of eigenvalues of the operator A and the corresponding eigenfunctions ipL- The eigenvalues of A are possible
values observed from a measurement of the physical quantity. All possible values

of A in any other state are nothing but its eigenvalues in its own eigenstates. This

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Linear Quantum Mechanics: Successes and Problems

3

hypothesis reflects the statistical nature in the description of motion of microscopic
particles in the linear quantum mechanics.
(4) The Hilbert space in which the linear quantum mechanics is defined is a linear
space. The operator of a mechanical quantity is a linear operator in this space. The
eigenvectors of a linear operator satisfy the linear superposition principle. That is,
if two states, |T/>I) and l ^ ) are both eigenfunctions of a given linear operator, then
their linear combination
\1>) = Cl\ih) + C2\rh),

(1.4)

where C\ and C2 are constants, also describes a state of the same particle. The linear
superposition principle of quantum states is determined by the linear characteristics
of the operators and this is why the quantum theory is referred to as linear quantum
mechanics. It is noteworthy to point out that such a superposition is different from
that of classical waves, it does not result in changes in probability and intensity.
(5) The correspondence principle: If two classical mechanical quantities, A and
B, satisfy the Poisson brackets,
{

'


!

^[dqndpn

dPndqn)

where qn and pn are generalized coordinate and momentum in the classical system,
respectively, then the corresponding operators A and B in quantum mechanics
satisfy the following commutation relation:
[A, B] = (AB - BA) = -ih{A, B)

(1.5)

where i = y/—T and h is the Planck's constant. If A and B are substituted by qn
and pn respectively, we have:
\Pn,qm] = -ihSnm,

\pn,Pm] = 0,

This reflects the fact that values allowed for a physical quantity in a microscopic
system are quantized, and thus the name "quantum mechanics". Based on this fundamental principle, the Heisenberg uncertainty relation can be obtained as follows,
\ri\2

(A4)2 (AB)2 > J^L

(1.6)

where iC = [A,B] and AA = {A - {A}). For the coordinate and momentum
operators, the Heisenberg uncertainty relation takes the usual form

|Az||Ap>!
(6) The time dependence of a quantum state \ip) of a microscopic particle is
determined by the following Schrodinger equation:

-~W

= *W-

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(1.7)


4

Quantum Mechanics in Nonlinear Systems

This is a fundamental dynamic equation for microscopic particle in space-time. H
is the Hamiltonian operator of the system and is given by,
H = f + V = -^—W2 + V,
where T is the kinetic energy operator and V the potential energy operator. Thus,
the state of a quantum system at any time is determined by the Hamiltonian of the
system. As a fundamental equation of linear quantum mechanics, equation (1.7) is
a linear equation of the wave function ip which is another reason why the theory is
referred as a linear quantum mechanics.
If the quantum state of a system at time io is \ip(t0)), then the wave function
and mechanical quantities at time t are associated with those at time to by a unitary
operator U(t,to), i.e.
\m)


(1.8)

= U(t,to)\iP(to)),

where U(to,to) = 1 and U+U = UU+ = I. If we let U(t,0) = U(t), then the
equation of motion becomes
-~U(t)

(1.9)

= HU(t)

when H does not depend explicitly on time t and U(t) = e-l(H/h)t_
explicit function of time t, we then have
U(t)

= 1 + i / dhH{h)
™ Jo

+ — ^ f dhHih)
\lh) Jo

f ' dt2H(t2)
Jo

+ •••.

jf jj j g a n

(1.10)


Obviously, there is an important assumption here: the Hamiltonian operator of
the system is independent of its state, or its wave function. This is a fundamental
assumption in the linear quantum mechanics.
(7) Identical particles: No new physical state should occur when a pair of identical particles is exchanged in a system. In other words, the wave function satisfies
Pkj\ip) = A|"0)> where Pkj is an exchange operator and A = ± 1 . Therefore, the wave
function of a system consisting of identical particles must be either symmetric, ips,
(A = +1), or antisymmetric, ipa, (A = —1), and this property remains invariant
with time and is determined only by the nature of the particle. The wave function
of a boson particle is symmetric and that of a fermion is antisymmetric.
(8) Measurements of physical quantities: There was no assumption made about
measurements of physical quantities at the beginning of the linear quantum mechanics. It was introduced later to make the linear quantum mechanics complete.
However, this is a nontrivial and contraversal topic which has been a focus of scientific debate. This problem will not be discussed here. Interested reader can refer
to texts and references given at the end of this chapter.

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Linear Quantum Mechanics: Successes and Problems

1.2

5

Successes and Problems of the Linear Quantum Mechanics

On the basis of the fundamental hypotheses mentioned above, Heisenberg,
Schrodinger, Bohn, Dirac, and others established the theory of linear quantum mechanics which describes the properties and motions of microscopic particle systems.
This theory states that once the externally applied potential fields and initial states
of the particles are given, the states of the particles at any time later and any position can be determined by the linear Schrodinger equation, equations (1.7) and (1.8)

in the case of nonrelativistic motion, or equivalently, the Dirac equation and the
Klein-Gordon equation in the case of relativistic motion. The quantum states and
their occupations of electronic systems, atoms, molecules, and the band structure of
solid state matter, and any given atomic configuration are completely determined
by the above equations. Macroscopic behaviors of systems such as mechanical,
electrical and optical properties may also be determined by these equations. This
theory also describes the properties of microscopic particle systems in the presence
of external electromagnetic field, optical and acoustic waves, and thermal radiation.
Therefore, to a certain degree, the linear quantum mechanics describes the law of
motion of microscopic particles of which all physical systems are composed. It is
the foundation and pillar of modern physics.
The linear quantum mechanics had great successes in descriptions of motions of
microscopic particles, such as electron, phonon, photon, exciton, atom, molecule,
atomic nucleus and elementary particles, and in predictions of properties of matter
based on the motions of these quasi-particles. For example, energy spectra of atoms
(such as hydrogen atom, helium atom), molecules (such as hydrogen molecule) and
compounds, electrical, optical and magnetic properties of atoms and condensed
matters can be calculated based on linear quantum mechanics and the calculated
results are in good agreement with experimental measurements. Being the foundation of modern science, the establishment of the theory of quantum mechanics has
revolutionized not only physics, but many other science branches such as chemistry,
astronomy, biology, etc., and at the same time created many new branches of science, for example, quantum statistics, quantum field theory, quantum electronics,
quantum chemistry, quantum biology, quantum optics, etc. One of the great successes of the linear quantum mechanics is the explanation of the fine energy spectra
of hydrogen atom, helium atom and hydrogen molecule. The energy spectra predicted by linear quantum mechanics for these atoms and molecules are completely in
agreement with experimental data. Furthermore, modern experiments have demonstrated that the results of the Lamb shift and superfine structure of hydrogen atom
and the anomalous magnetic moment of the electron predicted by the theory of
quantum electrodynamics are in agreement with experimental data within an order
of magnitude of 10~5. It is therefore believed that the quantum electrodynamics is
one of most successful theories in modern physics.
Despite the great successes of linear quantum mechanics, it nevertheless en-


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6

Quantum Mechanics in Nonlinear Systems

countered some problems and difficulties. In order to overcome these difficulties,
Einstein had disputed with Bohr and others for the whole of his life and the difficulties still remained up to now. Some of the difficulties will be discussed in the next
section. These difficulties of the linear quantum mechanics are well known and have
been reviewed by many scientists. When one of the founders of the linear quantum
mechanics, Dirac, visited Australia in 1975, he gave a speech on the development
of quantum mechanics in New South Wales University. During his talk, Dirac mentioned that at the time, great difficulties existed in the quantum mechanical theory.
One of the difficulties referred to by Dirac was about an accurate theory for interaction between charged particles and an electromagnetic field. If the charge of a
particle is considered as concentrated at one point, we shall find that the energy
of the point charge is infinite. This problem had puzzled physicists for more than
40 years. Even after the establishment of the renormalization theory, no actual
progress had been made. Such a situation was similar to the unified field theory
for which Einstein had struggled for his whole life. Therefore, Dirac concluded his
talk by making the following statements: It is because of these difficulties, I believe
that the foundation for the quantum mechanics has not been correctly laid down.
As part of the current research based on the existing theory, a great deal of work
has been done in the applications of the theory. In this respect, some rules for getting around the infinity were established. Even though results obtained based on
such rules agree with experimental measurements, they are artificial rules after all.
Therefore, I cannot accept that the present foundation of the quantum mechanics
is completely correct.
However, what are the roots of the difficulties of the linear quantum mechanics
that evoked these contentions and raised doubts about the theory among physicists?
Actually, if we take a closer look at the history of physics, one would know that
not so many fundamental assumptions were required for all physical theories but

the linear quantum mechanics. Obviously, these assumptions of linear quantum
mechanics caused its incompleteness and limited its applicability.
It was generally accepted that the fundamentals of the linear quantum mechanics consist of the Heisenberg matrix mechanics, the Schrodinger wave mechanics,
Born's statistical interpretation of the wave function and the Heisenberg uncertainty principle, etc. These were also the focal points of debate and controversy. In
other words, the debate was about how to interpret quantum mechanics. Some of
the questions being debated concern the interpretation of the wave-particle duality,
probability explanation of the wave function, the difficulty in controlling interaction
between measuring instruments and objects being measured, the Heisenberg uncertainty principle, Bohr's complementary (corresponding) principle, single particle
versus many particle systems, the problems of microscopic causality and probability,
process of measuring quantum states, etc. Meanwhile, the linear quantum mechanics in principle can describe physical systems with many particles, but it is not easy
to solve such a system and approximations must be used to obtain approximate

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Linear Quantum Mechanics: Successes and Problems

7

solutions. In doing this, certain features of the system which could be important
have to be neglected. Therefore, while many enjoyed the successes of the linear
quantum mechanics, others were wondering whether the linear quantum mechanics
is the right theory of the real microscopic physical world, because of the problems
and difficulties it encountered. Modern quantum mechanics was born in 1920s, but
these problems were always the topics of heated debates among different views till
now. It was quite exceptional in the history of physics that so many prominent
physicists from different institutions were involved and the scope of the debate was
so wide. The group in Copenhagen School headed by Bohr represented the view of
the main stream in these discussions. In as early as 1920s, heated disputes on the
statistical explanation and completeness of wave function arose between Bohr and

other physicists, including Einstein, de Broglie, Schrodinger, Lorentz, etc.
The following is a brief summary of issues being debated and problems encountered by the linear quantum mechanics.
(1) First, the correctness and completeness of the linear quantum mechanics were
challenged. Is linear quantum mechanics correct? Is it complete and self-consistent?
Can the properties of microscopic particle systems be completely described by the
linear quantum mechanics? Do the fundamental hypotheses contradict each other?
(2) Is the linear quantum mechanics a dynamic or a statistical theory? Does
it describe the motion of a single particle or a system of particles? The dynamic
equation seems an equation for a single particle, but its mechanical quantities are
determined based on the concepts of probability and statistical average. This caused
confusion about the nature of the theory itself.
(3) How to describe the wave-particle duality of microscopic particles? What
is the nature of a particle defined based on the hypotheses of the linear quantum
mechanics? The wave-particle duality is established by the de Broglie relations. Can
the statistical interpretation of wave function correctly describe such a property?
There are also difficulties in using wave package to represent the particle nature
of microscopic particles. Thus describing the wave-corpuscle duality was a major
challenge to the linear quantum mechanics.
(4) Was the uncertainty principle due to the intrinsic properties of microscopic
particles or a result of uncontrollable interaction between the measuring instruments
and the system being measured?
(5) A particle appears in space in the form of a wave, and it has certain probability to be at a certain location. However, it is always a whole particle, rather than a
fraction of it, being detected in a measurement. How can this be interpreted? Is the
explanation of this problem based on wave package contraction in the measurement
correct?
Since these are important issues concerning the fundamental hypotheses of the
linear quantum mechanics, many scientists were involved in the debate. Unfortunately, after being debated for almost a century, there are still no definite answers
to most of these questions. We will introduce and survey some main views of this

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8

Quantum Mechanics in Nonlinear Systems

debate in the following.
As far as the completeness of the linear quantum mechanics was concerned, Von
Neumann provided a proof in 1932. According to Von Neumann, if O is a set of
observable quantities in the Hilbert space Q of dimension greater than one, then
the self-adjoint of any operator in this set represents an observable quantity in the
same set, and its state can be determined by the average (A) for the operator A.
If this average value satisfies (1) = 1, we have (rA) = r(A) for any real constant
r. If A is non-negative, then {A) > 0. If A,B,C,--- are arbitrary observable
quantities, then, there always exists an observable A + B + C + • • • such that
(A + B + C H ) = (A) + (B) + (C) H . Von Neumann proved that there exists a
self-adjoint operator A in Q such that {^4°) ^ {A)a. This implies that there always
exists an observable quantity A which is indefinite or does not have an accurate
value. In other words, the states as defined by the average value are dispersive
and cannot be determined accurately, which further implies that states in which all
observable quantities have accurate values simultaneously do not exist. To be more
concrete, not all properties of a physical system can possess accurate values. At
this stage, this was the best the theory can do. Whether it can be accepted as a
complete theory is subjective. It seemed that any further discussion would lead to
nowhere.
It was realized later that Von Neumann's theorem was mathematically flawless
but ambiguous and vague in physics. In 1957, Gleason made two modifications
to Von Neumann's assumptions: Q should be the Hilbert space of more than two
dimensions rather than one; and A, B,C, ••• should be limited to commutable selfadjoint operators in Q. He verified that Von Neumann's theorem is still valid with
these assumptions. Because the operators are commutable, the linear superposition

property of average values is, in general, independent of the order in which experiments are performed. Hence, these assumptions seem to be physically acceptable.
Furthermore, Von Neumann's conclusion ruled out some nontrivial hidden variable
theories in the Hilbert space with dimensions of more than two.
However, in 1966, Bell indicated that Gleason's theorem can essentially only
remove the hidden variable theories which are independent of environment and
arrangements before and after a measurement. It would be possible to establish
hidden variable theories which are dependent on environment and arrangements
before and after a measurement. At the same time, Bell argued that since there
are more input hidden variables in the hidden variable theory than in quantum
mechanics, there should be new results that may be compared with experiments,
thus to verify whether the quantum mechanics is complete.
Starting from an ideal experiment based on the localized hidden variables theory
and the average value q(a, b) = J Aa(X)Bb(X)d\, Bohm believed that some features
of a particle could be obtained once those of another particle which is remotely
separated from the first are measured. This indicates that correlation between
particles exists which could be described in terms of "hidden parameters". Based

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