ISSN 2077-8139
PROGRESS IN
NONLINEAR SCIENCE
Complex
Mechan ics
Unified Approach to Classical and Quantum Mechan ics
Ciann-Dong Yang
Asian Academic Publisher
Limited, Hong Kong, 2010
complete book download address
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complete book download address
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Preface
Complex mechan ics is a new mechan ics
treating the Nature as a complex-valued world wherein all physical quantities
are assumed to complex variables. In the complex space, probabilistic viewpoint
and deterministic viewpoint become identical so that classical mechan ics and quantum mechan ics
can be unified into the same framework and both can be learned according to the
same disciplines.
Consistency between probabilism and
determinism
The lack of dynamic
equations of motion with respect to time, which is one of the sources of
controversies of quantum mechan ics,
makes it impossible to analyze stability, chaos, bifurcation, and many other
nonlinear features existing in quantum systems. This impossibility has long
been taken for granted due to a common belief that the probabilistic nature of
quantum phenomena is in no way described or represented by deterministic
nonlinear models. However, probabilistic viewpoint and deterministic viewpoint
may not be as conflicting as we commonly think. Consider a scenario that a
dynamic motion occurs in the complex space but only its real-part motion can be
measured. Due to the influence of the unmeasurable imaginary-part motion and
its interaction with the real-part motion, the measured real-part motion is unpredictable
and can only be described probabilistically. On the other han d, the same motion, if viewed from the complex
space, is governed uniquely by a set of complex-valued nonlinear equations,
which are entirely deterministic. In such a situation, probabilistic interpretation
and deterministic interpretation can be equally applied to the same motion,
depending on which space we deal with.
Fig. A. When viewed from the
complex space, the motion of a particle is deterministic by following a
continuous path 1→2→3→...→ 8, while as viewed from the real space, the particle’s motion
is discontinuous and causality relation is broken. Because the data points 3,
4, 6 and 7 in the
imaginary space are unmeasurable, the causal connection between points 2 and 5
is missing as viewed from the real space, and it appears that the particle
jumps randomly from point 2 to point 5 and to point 8.
Figure A. is a schematic
explanation of why deterministic interpretation and probabilistic interpretation
can be applied simultaneously to the same physical motion. Consider a particle
moves deterministically in the complex space by following a continuous path 1→2→3→...→ 8. However, the same motion, as viewed from the real space, is
discontinuous and does not have any causal connection. Because the data points
3, 4, 6 and 7 appearing in the imaginary space are unmeasurable, the causal
connection between points 2 and 5 is missing as viewed from the real space, and
it appears that the particle jumps randomly from point 2 to point 5 and then to
point 8. In the absence of causal connections between points 2, 5 and 8, it is
very natural to treat the particle’s position as random variable and use
probability to describe its appearance.
In quantum mechan ics, we encounter the same scenario that physical
motions occur in the complex space, but what we sense and measure are merely
the real parts of the motions, which give rise to what we call quantum
phenomena. Therefore, understanding the how and the why of quantum mechan ics requires a viewpoint from the complex space.
The complex space is where a quantum motion takes place, while the real space
is where we take measurements of the motion. Quantum mechan ics
lays out the distribution and the evolution of the measurement data, while the
complex mechan ics describes the
quantum motion in the complex space before it is measured. In the complex
space, quantum motions are deterministic so that all the methods of classical
mechan ics can be applied. The
projection of the complex-valued motion into the real space recovers and
confirms the quantum phenomena observed from the measurement data. In recent
years, an excellent consistency of the projected solutions with the various
quantum effects has been justified under the framework of complex mechan ics and most of them will be discussed in this
book. By the publication of this book, the author wishes to share the delight of
teaching and learning complex mechan ics
with the readers.
Unification of classical and Quantum
Mechan ics
Complex mechan ics,
a unified approach to classical and quantum mechan ics,
provides a bridge between the probabilistic interpretation in the real space
and the deterministic interpretation in the complex space. Through this bridge,
researchers in classical mechan ics
can employ methods familiar to them to analyze quantum systems in the complex
space, and to predict and verify various quantum phenomena by projecting the
results of complex analysis into the real space. Complex mechan ics employs complex-extended Hamilton Hamilton Newton han ical problems can be solved by the
well-established methods in the classical mechan ics.
Figure B. highlights the features of complex mechan ics,
showing that by inputting classical motions to the complex space, we automatically
obtain the quantum motions as the output. In other words, the education of
quantum mechan ics is an integration
of engineering mechan ics (classical
mechan ics) and complex variable
theory. Therefore, undergraduate students completing the courses of engineering
mechan ics and complex variable
theory (engineering mathematics) have already gathered all the required ability
and tools to solve quantum mechan ical
problems.
Fig. B. Under the framework of complex mechan ics,
quantum mechan ics is equivalent to classical
mechan ics defined in the complex
domain so that the learning of the former becomes the learning of the latter
incorporated with complex variable theory.
Teaching Quantum Mechan ics Fully by Engineering Mechan ics
With advances in material
synthesis and device processing capabilities, the importance of quantum mechan ics in material science, electrical engineering
and applied physics, has dramatically increased over the last couple of
decades. The engineers can no longer just work with simplistic phenomenological
equations, but must understand a more fundamental origin of the phenomena.
Devices such as Josephson junctions, semiconductor lasers, transistors, and all
of the nanostructures cannot be fully understood in terms of simple classical
mechan ics. The big challenge does not stem from our ignorance of the importance of
nourishing engineering education with quantum elements, but from the seemingly
insurmountable task of teaching quantum mechan ical
concept in a class of engineering mechan ics.
The fundamental ideas of quantum mechan ics
and engineering mechan ics are so conflicting
that all the deterministic rules and causal relations found in engineering mechan ics are no longer valid in quantum mechan ics, which treats physical quantities as random
variables having only probabilistic nature. Some lecturers of quantum mechan ics even suggest students to forget temporarily
the impressions of determinism and causality gained from classical mechan ics, when they study quantum mechan ics. It appears that the knowledge and experience
learned from studying engineering mechan ics
cannot be conveyed directly to the study of quantum mechan ics.
These remarkable gaps between quantum mechan ics
and engineering mechan ics have
forbidden the possibility to teach quantum mechan ics
in a course of engineering mechan ics
or classical mechan ics.
To surmount the above-mentioned obstacles, a
teaching program based on complex mechan ics
was launched by the author at National Cheng Kung University (NCKU). This
teaching program has been tested in a course named “Engineering Quantum Mechan ics” in the Department of Aeronautics and Astronautics
at the semester years from 2006 to 2009. The preliminary validation of teaching
quantum mechan ics fully by the
concepts of classical mechan ics is
very successful in this course. Students enrolled in this course are benefited
greatly from the bridge provided by complex mechan ics
that allows them to accelerate and deepen the learning of quantum mechan ics by their previous knowledge and experience
gained from the course of engineering mechan ics.
The present book is the outgrowth of the lecture notes of this course. The success of the NCKU teaching program shows
that complex mechan ics shortens the
gap between quantum mechan ics and classical
mechan ics and makes the teaching and
learning of quantum mechan ics much easier.
Organization
All the materials
developed in this book are based on the unique assumption that the real world is a place we take
measurements of a physical motion, while the complex space is a place the
physical motion actually takes place. Unfortunately, this assumption is not
widely accepted in the contemporary physics and few references can be referred
to during the development of complex mechan ics.
El Naschie’s E-infinity theory is one
of the very few publications that hold the same view as complex mechan ics. Because of its independence from the mainstream
physics, the book is written and organized to be self-contained and
self-consistent. The book is
constituted by five parts: (a) the basic assumptions and definitions of complex
motions, (2) the fundamental principles derived from the assumptions and definitions,
(3) the predictions and verifications of various quantum phenomena by the
derived principles, (4) 3D complex motion in hydrogen atom, and (5) complex
motion in 4D spacetime. The topics covered by the five parts and their
interrelations are summarized in Fig. B.
(1)
Define physical motions in complex space: Chapter 1
Chapter 1 surveys the methods and concepts needed in the book to describe
physical motions in complex space, and introduces the necessary remedies for
the conventional complex variable theory to cope with complex variables with
internal dynamics.
(2)
Develop methods of
analysis and mathematical tools: Chapter 2 ~ Chapter 5
˜ Internal Dynamics of quantum states: In complex mechan ics, physical
quantities are complex variables with memory, i.e., they have internal dynamics,
which can be used to trace their past motions and predict their future motions.
Chapter 2 shows that internal dynamics is governed by deterministic Hamilton han ics can
be applied to investigate the internal dynamics of a quantum state.
˜ Complex representation of quantum operators: Chapter 3 points out that every quantum operator A' has a complex
representation A(x,p) in complex mechan ics. Knowing the expression for A(x,p) with generalized coordinate q and generalized momentum p satisfying Hamilton han ics now can be replaced by the role of A(x,p) in complex mechan ics.
˜
Quantization and time-averaged mean value: According to the time history of a physical
quantity f(x(t),p(t)) solved from its
internal dynamics, the time-averaged mean value f_ave will be computed and
compared to the probability-based mean value in Chapter 4. It is found that
quantization of f(x,p) is just the phenomenon
that its time-averaged value f_ave is independent of the
trajectory x(t) along which the time
average is taken. Both local and global trajectory independences are discussed in
Chapter 4. The former leads to the quantization of f_ave in a given state and the latter leads
to the conservation of f_ave.
Fig. C. The
topics covered by the book and their interrelations.
˜ Visualization of complex motions: Complex motion can be
visualized by analogy with potential flows. Chapter 5 aims to reveal a novel
analogy between probability flows and potential flows on the complex plane. For
a given complex-valued wavefunction Psi(z,t), x=x+iy, we first define a complex potential function Omega(z,t)=h/(im)ln(Psi(z,t))=phi(x,y,t)+i psi,y,t), and then prove that the streamline lines psi(x,y,t)=constant and the potential
lines phi(x,y,t)=constant in the potential flow
defined by Omega are equivalent to the
constant-probability lines and the constant-phase
lines in the quantum probability
flow defined by Psi. The discovered analogy is very useful in visualizing the
unobservable probability flow on the complex x+i y plane by analogy with
the 2D potential flow on the real x-y plane, which otherwise
can be visualized by using dye streaks in a fluid laboratory.
(2)
Predict various quantum
phenomena by complex mechan ics: Chapter 6 ~ Chapter 9
˜ Tunneling dynamics: Using the dynamic representation of a quantum state derived in Chapter 2,
Chapter 6 models tunneling dynamics exactly by quantum Hamilton Hamilton han ics
is twofold. It makes the tunneling time as simple as the usual time without the
necessity of defining any time operator and secondly, it provides the tunneling
trajectory in an unambiguous way such that trajectories in classical regions
and non-classical regions can be connected smoothly.
˜ Wave-particle duality: Wave
motion associated with a material particle is produced by projecting its
complex motion into real space. The aim of Chapter 7 is to verify this new
interpretation of matter wave. The equations of motion for a free particle is
solved therein to reveal how the interaction between real and imaginary motions
can produce the particle’s wave motion observed in real space.
˜ Feynman’ path integral: Chapter 8 explains Feynman’s
path integral in terms of the multiple paths derived in chapter 7. It is
revealed that under the framework of complex mechan ics
path integral trajectories can be parameterized continuously in terms of a free
parameter so that an infinite dimensional path integral can be transformed into
a one-dimensional normal integral over this free parameter.
˜ Quantum chaos: Chapter 9 shows that the
phenomena of quantum chaos is another kind of projection effects from the
complex space to the real space. A new chaotic behavior, called strong chaos,
is introduced in Chapter 9. Unlike classical chaos caused by the divergence of
two trajectories emerging from two nearby initial positions, strong chaos is
unique to quantum systems and is caused by the multi-path effect so that
infinitely many trajectories may emerge spontaneously from the same initial
position and diverge as time evolves.
(3)
Solve 3D quantum motion
in hydrogen atom by complex mechan ics: Chapter 10, 11
˜ Complex motion in hydrogen atom: Chapter 10 studies electronic quantum dynamics in hydrogen atom by complex
mechan ics. We will see there that
the quantizations of total energy and angular momentum are a natural
consequence of the electron’s complex motion. Meanwhile, the shell structure
observed in the hydrogen atom is shown to be a manifestation of the
quantum-potential structure, from which the quantum forces acting upon the
electron can be uniquely determined, the stability of atomic configuration can
be justified, and the electron’s continuous transition from the quantum world
to the classical world can be monitored as the quantum number n approaches infinity.
˜ Complex representation of spin
dynamics: The spin dynamics inherent in
the Schrödinger equation has long been overlooked since the inception of
quantum mechan ics. Chapter 11 aims
to report the discovery that the hydrogen ground-state wavefunctions solved
from the Schrödinger equation without any correction from Pauli or Dirac theory
clearly demonstrates the existence of an angular momentum h/2 when the orbital angular
momentum L^2 is zero. The
well-known spinless mode in the usual interpretation of the Schrödinger
equation is only one of the three spin modes contained in the hydrogen ground states,
while the remaining two modes inherent in the Schrödinger equation, namely, the
spin-down mode with angular momentum -h/2 and the spin-up mode
with angular momentum +h/2 have never been reported
before in the literature.
(4) Extend
complex motions to 4D spacetime: Chapter 12, 13
˜ Relativistic quantum mechan ics: Chapter 12 points out that extending special relativity to the
complex spacetime automatically leads to the relativistic quantum mechan ics. The complex spacetime is a bridge connecting
the causality in special relativity to the non-locality in quantum mechan ics. In the presence of quantum interactions, the
correct mass-energy relation is found to be E=mc^2sqrt(1-2Q/mc^2). When the quantum potential Q is zero, it reduces to
the famous result E=mc^2 in special relativity.
˜ Relativistic complex mechan ics: In the final chapter, we will learn how to describe particle’s complex
motion in general curvilinear coordinates, subjected to electromagnetic field and
gravitational field. This generalization is helpful for us to extend complex
mechan ics to relativistic domain,
wherein the space-time is curved and described by four-dimensional curvilinear
coordinates. At the end of this chapter, we will see how classical mechan ics, quantum mechan ics
and relativistic mechan ics are unified
under the framework of relativistic complex mechan ics.
Fig. D. A
Chinese couplet contrasts the yin-yang duality with the wave-particle duality
via the connection of complex mechan ics.
Tai Chi considers our universe as an entanglement of Yin and Yang. Yin denoted
by the black is the imaginary part of the universe and Yang denoted by the
white is the real part of the universe. Motivated by Tai Chi, complex mechan ics assumes that all physical quantities are
complex variables having real parts as well as imaginary parts.
Philosophical Motivation from Tai Chi
The assumptions made in complex
mechan ics are based on the
philosophy of Tai Chi, which advocates that the Nature
(the Tao) contains two parts: yang is the observable (real) part and yin is the unobservable (imaginary) part.
In the symbol of Tai Chi illustrated in Fig. C, yang is denoted by white
and yin denoted by black so that within the black, there is always some white,
and within the white, there is always some black. The figure of Tai Chi
symbolizes the totality of the combined real and imaginary worlds and the
entanglement between them. According to the
philosophy of Tai Chi, the Nature is a complex-valued world, while what
we sense and measure are only the real part of the world, which constitutes the
physical world we experience in daily lives.
Complex mechan ics is a scientific realization of Yin-Yang philosophy
via the language of complex variables, providing a bridge between the Yin-Yang
duality in Tai Chi and the wave-particle duality in quantum mechan ics. The recent progress in complex mechan ics has strengthened the fact that although the imaginary world cannot be directly sensed or
measured in the real world, its influences on the Nature can be definitely
detected via the measurement of its interaction with the real world. As will be
demonstrated throughout this book, the coupling connection between the real and
imaginary worlds gives rise to the various quantum phenomena that we have
observed in the real world. The Chinese couplet shown in Fig. C highlights the
role of the complex mechan ics as a
bridge between the Yin-Yang duality in Tai Chi and the wave-particle duality in
quantum mechan ics.
Acknowledgements
First of all, I would like to than k National
Cheng Kung
University Taiwan han ics (under the
grant of Top-University Projects). Being not on the mainstream of contemporary
physics, complex mechan ics has undergone
a painful experience in its publication. The publication of complex mechan ics would have been impossible without the decisive
supports from Prof. El Naschie (the founding editor of Chaos, Solitons and
Fractals) and Prof. Ji-Huan He
(Editor-in-chief, Nonlinear Science Letters A, International Journal of Nonlinear Sciences and Numerical Simulation). Their
appreciation and understanding of Chinese culture in Western science give me a
chan ce to introduce complex mechan ics to the public literature. The initiation of
the research of complex mechan ics is
inspired by the philosophy of Taoism, and I would like to dedicate this book to
the memory of all the ancient Chinese wisdoms.
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