| Prelude. || Can the System of Classical Physics Concepts Be Considered Logically Perfect?|
|1 ||Physics of the Microparticles|
| ||1.||Certain Characteristics and Properties of Microparticles|
| ||2.||Two Fundamental Ideas of Quantum Mechanics|
| ||3.||Uncertainty Relations|
| ||4.||Some Results Ensuing from the Uncertainty Relations|
| ||5.||Impossibility of Classical Representation of a Microparticle|
| ||6.||Rejection of Ideas of Classical Physics|
|Interlude. Is a "Physically Intuitive" Model of a Microparticle Possible?|
|2 ||Physical Foundations of Quantum Mechanics|
| ||7.||Some Basic Experiments|
| ||8.||Amplitudes of Transition Probabilities (Formulation of Basic Principles)|
| ||9.||Amplitudes of Transition Probabilities (Demonstration of Basic Principles)|
| ||10.||Superposition of States|
| ||11.||Measurement in Quantum Mechanics|
|Interlude. Are These the Same Waves? Or, Again on Waves in Quantum Mechanics|
| ||12.||Causality in Quantum Mechanics|
| ||13.||Microparticles with Two Basic States|
| ||14.||The Electron in a Magnetic Field|
| ||15.||The Wave Function|
| ||16.||Quantum Mechanics as a Qualitative Leap in Man's Knowledge of the Laws of Nature|
|Interlude. Do Quantum-Mechanical Concepts Contradict Our Common Sense?|
|3 ||Linear Operators in Quantum Mechanics|
| ||17.||A Brief Look at the Theory of Linear Operators|
| ||18.||From Hamiltonian Matrix to Energy Operator|
| ||19.||Linear Operators in Quantum Mechanics|
| ||20.||The Quantum-Mechanical Apparatus in Coordinate Representation|
| ||21.||Applications of the Schredinger Equation|
| ||22.||The Hamiltonian in Some Specific Problems|
| ||23.||Transition to the Momentum Representation|
| ||24.||An Electron in a Periodic Field|
| ||25.||The Probability of Quantum Transitions|
| ||26.||Ways of Describing Evolution of Microsystems with Time|
|On the History of Origin and Growth of Quantum Mechanics (A Brief Historical Survey)|
|Appendix A. Eigenvalues and Eigenfunctions of a Hermitian Operator|
|Appendix B. Transition from Quantum Mechanics to Classical Mechanics|
|Appendix C. Commutation Relations|
|Appendix D. Commutation of Operators M2 and Mi|
|Appendix E. Some Special Functions|
|Appendix F. Linear Harmonic Oscillators|
Some Preliminary Remarks
Research in physics, conducted at the end of the 19th
century and in the first half of the 20th century, revealed exceptionally
peculiar nature of the laws governing the behaviour of microparticles --
atoms, electrons, and so on. On the basis of this research a new physical
theory called quantum mechanics was founded.
The growth of quantum mechanics turned out to be quite complicated and
prolonged. The mathematical part of the theory, and the rules linking the
theory with experiment, were constructed relatively quickly (by the beginning
of the thirties). However, the understanding of the physical and philosophical
substance of the mathematical symbols used in the theory was unresolved for
decades. In Fock's words, The mathematical apparatus of
nonrelativistic quantum mechanics worked well and was free of contradictions;
but in spite of many successful applications to different problems of atomic
physics the physical representation of the mathematical scheme still remained
a problem to be solved.
Many difficulties are involved in a mathematical interpretation of the
quantum-mechanical apparatus. These are associated with the dialectics of the
new laws, the radical revision of the very nature of the questions which
a physicist "is entitled to put to nature", the reinterpretation of the role of
the observer vis a vis his surroundings, the new approach to the question
of the relation between chance and necessity in physical phenomena, and the
rejection of many accepted notions and concepts. Quantum mechanics was born in
an atmosphere of discussions and heated clashes between contradictory
arguments. The names of many leading scientists are linked with its
development, including N.Bohr, A.Einstein, M.Planck, E.Schr\"odinger,
M.Born, W.Pauli, A.Sommerfeld, L.deBroglie, P.Ehrenfest, E.Fermi,
W.Heisenberg, P.Dirac, R.Feynman, and others.
It is also not surprising
that even today anyone who starts studying quantum mechanics encounters some
sort of psychological barrier. This is not because of the mathematical
complexity. The difficulty arises from the fact that it is difficult to break
away from accepted concepts and to reorganize one's pattern of thinking which
is based on everyday experience.
Before starting a study of quantum mechanics, it is worthwhile getting
an idea about its place and role in physics. We shall consider (naturally in
the most general terms) the following three questions: What is quantum
mechanics? What is the relation between classical physics and quantum mechanics? What specialists need quantum mechanics? So, what is quantum
The question can be answered in different ways. First and foremost,
quantum mechanics is a theory describing the properties of matter at the
level of microphenomena -- it considers the laws of motion
of microparticles. Microparticles (molecules, atoms,
elementary particles) are the main "characters" in the drama of quantum
From a broader point of view quantum mechanics should be treated
as the theoretical foundation of the modern theory of the structure and
properties of matter. In comparison with classical physics, quantum
mechanics considers the properties of matter on a deeper and more
fundamental level. It provides answers to many questions which
remained unsolved in classical physics. For example, why is diamond hard?
Why does the electrical conductivity of a semiconductor increase with
temperature? Why does a magnet lose its properties upon heating? Unable to
get answers from classical physics to these questions, we turn to quantum
mechanics. Finally, it must be emphasized that quantum mechanics allows one
to calculate many physical parameters of substances.
Answering the question "What is quantum mechanics?", Lamb remarked:
The only easy one (answer) is that quantum mechanics is a discipline
that provides a wonderful set of rules for calculating physical properties
What is the relation of quantum mechanics to classical
physics? First of all quantum mechanics includes classical mechanics as
a limiting (extreme) case. Upon a transition from microparticles to
macroscopic bodies, quantum-mechanical laws are converted into the laws of
classical mechanics. Because of this it is often stated, though not very
accurately, that quantum mechanics "works" in the microworld and the
classical mechanics, in the macroworld. This statement assumes the existence
of an isolated "microworld" and an isolated "macroworld". In actual practice
we can only speak of microparticles (microphenomena) and macroscopic bodies
(macrophenomena). It is also significant that microphenomena form the basis
of macrophenomena and that macroscopic bodies are made up of microparticles.
Consequently, the transition from classical physics to quantum mechanics is
a transition not from one "world" to another, but from a shallower to
a deeper level of studying matter. This means that in studying the behaviour
of microparticles, quantum mechanics considers in fact the same
macro-particles, but on a more fundamental level. Besides, it must be
remembered that the boundary between micro- and macrophenomena in general is
quite conditional and flexible. Classical concepts are frequently found
useful when considering microphenomena, while quantum-mechanical ideas help
in the understanding of macrophenomena. There is even a special term
"quantum macrophysics" which is applied, in particular, to quantum
electronics, to the phenomena of superfluidity and superconductivity and to
a number of other cases.
In answering the question as to what specialists need quantum mechanics, we
mention beforehand that we have in mind specialists training in engineering
colleges. There are at least three branches of engineering for which a study of
quantum mechanics is absolutely essential. Firstly, there is the field of nuclear power and the application of radioactive isotopes to industry.
Secondly, the field of materials sciences (improvement of properties of
materials, preparation of new materials with preassigned properties). Thirdly,
the field of electronics and first of all the field of semiconductors and laser technology. If we consider that almost any
branch of industry uses new materials as well as electronics on a large scale,
it will become clear that a comprehensive training in engineering is impossible
without studying quantum mechanics.
The Structure of the Book
The aim of this
book is to acquaint the reader with the concepts and ideas of quantum mechanics
and the physical properties of matter; to reveal the logic of its new ideas, to
show how these ideas are embodied in the mathematical apparatus of linear
operators and to demonstrate the working of this apparatus using a number of
examples and problems. The book consists of three chapters. By way of an
introduction to quantum mechanics, the first chapter includes a study of
the physics of microparticles. Special attention has been paid to the
fundamental ideas of quantization and duality as well as to the uncertainty
relations. The first chapter aims at "introducing" the micro-particle, and at
showing the necessity of rejecting a number of concepts of classical physics.
The second chapter deals with the physical concepts of quantum mechanics.
The chapter starts with an analysis of a set of basic experiments which form
a foundation for a system of quantum-mechanical ideas. This system is based on
the concept of the amplitude of transition probability. The rules for working
with amplitudes are demonstrated on the basis of a number of examples, the
interference of amplitudes being the most important. The principle of
superposition and the measurement process are considered. This concludes the
first stage in the discussion of the physical foundation of the theory. In the
second stage an analysis is given based on amplitude concepts of the problems
of causality. The Hamiltonian matrix is introduced while considering causality
and its role is illustrated using examples involving microparticles with two
basic states, with emphasis on the example of an electron in a magnetic field.
The chapter concludes with a section of a general physical and philosophical
The third chapter deals with the application of linear operators in the
apparatus of quantum mechanics. At the beginning of the chapter the required
mathematical concepts from the theory of Hermitian and unitary linear operators
are introduced. It is then shown how the physical ideas can be "knitted" to the
mathematical symbols, thus changing the apparatus of operator theory into the
apparatus of quantum theory. The main features of this apparatus are further
considered in a concrete form in the framework of the coordinate
representation. The transition from the coordinate to the momentum
representation is illustrated. Three ways of describing the evolution of
microsystems in time, corresponding to the Schr\"odinger, Heisenberg and Dirac
representation, have been discussed. A number of typical problems are
considered to demonstrate the working of the apparatus; particular attention is
paid to the problems of the motion of an electron in a periodic field and to
the calculation of the probability of a quantum transition. The book contains
a number of interludes. These are dialogues in which the author has allowed
himself free and easy style of considering certain questions. The author was
motivated to include interludes in the book by the view that one need not take
too serious an attitude when studying serious subjects. And yet the reader
should take the interludes fairly seriously. They are intended not so much for
mental relaxation, as for helping the reader with fairly delicate questions,
which can be understood best through a flexible dialogue treatment. Finally,
the book contains many quotations. The author is sure that they will offer the
reader useful additional information.
The author wishes to express his deep gratitude to Prof. I.I.Gurevich,
Corresponding Member of the USSR Academy of Sciences, for the stimulating
discussions which formed the basis of this book. Prof. Gurevich discussed the
plan of the book and its preliminary drafts, and was kind enough to go through
the manuscript. His advice not only helped mould the structure of the book, but
also helped in the nature of exposition of the material.
The subsection "The Essence of Quantum Mechanics" in Sec.16 is a direct
consequence of Prof. Gurevich's ideas. The author would like to record the deep
impression left on him by the works on quantum mechanics by the leading
American physicist R.Feynman. While reading the sections in
this book dealing with the applications of the idea of probability amplitude,
superposition principle, microparticles with two basic states, the reader can
easily detect a definite similarity in approach with the corresponding parts in
Feynman's "Lectures in Physics". The author was considerably influenced by
N.Bohr (in particular by his wonderful essays Atomic Physics and Human
W.Pauli, P.Dirac, and also by the comprehensive
works of L.D.Landau and E.M.Lifshitz,
The author is especially indebted to Prof. M.I.Podgoretsky, D.Sc, for
a thorough and extremely useful analysis of the manuscript. He is also grateful
to Prof. Yu.A.Vdovin, Prof. E.E.Lovetsky, Prof. G.F.Drukarev, Prof.
V.A.Dyakov, Prof. Yu.N.Pchelnikov, and Dr. A.M.Polyakov, all of whom
took the trouble of going through the manuscript and made a number of valuable
comments. Lastly, the author is indebted to his wife Aldina Tarasova for her
constant interest in the writing of the book and her help in the preparation of
the manuscript. But for her efforts, it would have been impossible to bring the
book to its present form.
Prelude: Can the System of Classical Physics Concepts Be Considered Logically Perfect?
Participants: the Author and the Classical Physicist (Physicist of the older
generation, whose views have been formed on the basis of classical physics alone).
He who would study organic existence,
First drives out the soul with rigid persistence,
Then the parts in his hands he may hold and class
But the spiritual link is lost, alas!
Author: It is well known that the basic contents of a physical theory
are formed by a system of concepts which reflect the objective laws
of nature within the framework of the given theory. Let us take the
system of concepts lying at the root of classical physics. Can this
system be considered logically perfect?
Classical Physicist: It is quite perfect. The concepts of classical
physics were formed on the basis of prolonged human experience;
they have stood the test of time.
Author: What are the main concepts of classical physics?
Classical Physicist: I would indicate three main points: (a)
continuous variation of physical quantities; (b) the principle of
classical determinism; (c) the analytical method of studying objects
While talking about continuity, let us remember that the state of an
object at every instant of time is completely determined by
describing its coordinates and velocities, which are continuous
functions of time. This is what forms the basis of the concept of
motion of objects along trajectories. The change in the state of an
object may in principle be made as small as possible by reducing the
time of observation.
Classical determinism assumes that if the state of an object as well as
all the forces applied to it are known at some instant of time, we can
precisely predict the state of the object at any subsequent instant.
Thus, if we know the position and velocity of a freely falling stone at
a certain instant, we can precisely tell its position and velocity at any
other instant, for example, at the instant when it hits the ground.
Author: In other words, classical physics assumes an unambiguous
and inflexible link between present and future, in the same way as between
past and present.
Classical Physicist: The possibility of such a link is in close agreement
with the continuous nature of the change of physical quantities: for every
instant of time we always have an answer to two questions: "What are the
coordinates of an object?" and, "How fast do they change?" Finally, let us
discuss the analytical method of studying objects and phenomena. Here we
come to a very important point in the system of concepts of classical physics.
The latter treats matter as made up of different parts which, although they
interact with one another, may be investigated individually. This means that
firstly, the object may be isolated from its environments and treated as
an independent entity, and secondly, the object may be broken up, if
necessary, into its constituents whose analysis could lead to an understanding
of the nature of the object.
Author: It means that classical physics reduces the
question "what is an object like?" to "what is it made of?"
Classical Physicist: Yes, indeed. In order to understand
any apparatus we must "dismantle" it, at least in one's imagination, into
its constituents. By the way, everyone tries to do this in his childhood.
The same is applicable to phenomena: in order to understand the idea behind
some phenomenon, we have to express it as a function of time, i.e. to find
out what follows what.
Author: But surely such a step will destroy the notion of the
object or phenomenon as a single unit.
Classical Physicist: To some extent. However, the extent of
this "destruction" can be evaluated each time by taking into account the
interactions between different parts and relation between the time stages of
a phenomenon. It may so happen that the initially isolated object (a part of
it) may considerably change with time as a result of its interaction with
the surroundings (or interaction between parts of the object). However,
since these changes are continuous, the individuality of the isolated object
can always be returned over any period of time. It is worthwhile to stress
here the internal logical connections among the three fundamental notions of
Author: I would like to add that one special consequence of
the "principle of analysis" is the notion, characteristic of classical
physics, of the mutual independence of the object of observation and the
measuring instrument (or observer). We have an instrument and an object of
measurement. They can and should be considered separately, independently
from one another.
Classical Physicist: Not quite independently. The inclusion
of an ammeter in an electric circuit naturally changes the magnitude of the
current to be measured. However, this change can always be calculated if we
know the resistance of the ammeter.
Author: When speaking of the independence of the instrument and the
object of measurement, I just meant that their interaction may be simply
Classical Physicist: In that case I fully agree with you.
Author: Born has considered this point in. Characterizing the
philosophy of science which influenced "people of older generation", he
referred to the tendency to consider that the object of investigation and the
investigator are completely isolated from each other, that one can study
physical phenomena without interfering with their passage. Born called such
style of thinking "Newtonian", since he felt that this was reflected in
"Newton's celestial mechanics."
Classical Physicist: Yes, these are the notions of classical physics in
general terms. They are based on everyday commonplace experience and it may be
confidently stated that they are acceptable to our common sense, i.e. are
taken as quite natural. I rather believe that the "principle of analysis" is
not only a natural but the only effective method of studying matter. It is
incomprehensible how one can gain a deeper insight into any object or
phenomenon without studying its components. As regards the principle of
classical determinism, it reflects the causality of phenomena in nature and is
in full accordance with the idea of physics as an exact science.
Author: And yet there are grounds to doubt the "flawlessness" of
classical concepts even from very general considerations. Let us try to extend
the principle of classical determinism to the universe as a whole. We must
conclude that the positions and velocities of all "atoms" in the universe at
any instant are precisely determined by the positions and velocities of these
"atoms" at the preceding instant. Thus everything that takes place in the world
is predetermined beforehand, all the events can be
fatalistically predicted. According to Laplace, we could imagine some
"superbeing" completely aware of the future and the past. In his Theorie analytique des
probabilites, published in 1820, Laplace wrote: An intelligence
knowing at a given instant of time all forces acting in nature as
well as the momentary positions of all things of which the universe consists,
would be able to comprehend the motions of the largest bodies of the world and
those of the lightest atoms in one single formula, provided his intellect were
sufficiently powerful to subject all data to analysis, to him nothing would be
uncertain, both past and future would be present to his eyes. It can be
seen that an imaginary attempt to extend the principle of classical determinism
to nature in its entity leads to the emergence of the idea of fatalism, which
obviously cannot be accepted by common sense.
Next, let us try to apply the "principle of analysis" to an investigation of
the structure of matter. We shall, in an imaginary way, break the object
into smaller and smaller fractions, thus arriving finally at the molecules
constituting the object. A further "breaking-up" leads us to the conclusion
that molecules are made up of atoms. We then find out that atoms are made up
of a nucleus and electrons. Accustomed to the tendency of splitting, we
would like to know what an electron is made of. Even if we were able to get
an answer to this question, we would have obviously asked next: What are the
constituents, which form an electron, made of? And so on. We tend to accept
the fact that such a "chain" of questions is endless. The same common sense
will revolt against such a chain even though it is a direct consequence of
Attempts were made at different times to solve the
problem of this chain. We shall give two examples here. The first one is
based on Plato's views on the structure of matter. He assumed that matter is
made up of four "elements" -- earth, water, air and fire. Each of these
elements is in turn made of atoms having definite geometrical forms. The
atoms of earth are cubic, those of water are icosahedral, while the atoms of
air and fire are octahedral and tetrahedral, respectively. Finally, each
atom was reduced to triangles. To Plato, a triangle appeared as the simplest
and most perfect mathematical form, hence it cannot be made up of any
constituents. In this way, Plato reduced the chain to the purely
mathematical concept of a triangle and terminated it at this point.
The other example is characteristic for the beginning of the 20th century.
It makes use of the external similarity of form between the planetary model
of the atom and the solar system. It is assumed that
our solar system is nothing but an isolated atom of some other,
gigantic world, and an ordinary atom is a sort of "solar system" for some
third dwarfish world for which "our electron" is like a planet. In this case
we admit the existence of an infinite row of more and more dwarfish worlds,
just like more and more gigantic worlds. In such a system the structure of
matter is described in accordance with the primitive "chinese box"
principle. The "chinese box" principle of hollow tubes, according to which
nature has a more or less similar structure, was not accepted by all the
physicists of older generations. However, this principle is quite
characteristic of classical physics, it conforms to classical concepts, and
follows directly from the classical principle of analysis. In this
connection, criticizing Pascal's views that the smallest and the largest
objects have the same structure, Langevin pointed out that this would lead
to the same aspects of reality being revealed at all levels. The universe
should then be reflected in an absolutely identical fashion in all objects,
though on a much smaller scale. Fortunately, reality turns out to be much
more diverse and interesting. Thus, we are convinced that a successive
application of the principles of classical physics may, in some cases, lead
to results which appear doubtful. This indicates the existence of situations
for which classical principles are not applicable. Thus it is to be expected
that for a sufficiently strong "breaking-up" of matter, the principle of
analysis must become redundant (thus the idea of the independence of the
object of measurement from the measuring instrument must also become
obsolete). In this context the question "what is an electron made of?" would
simply appear to have lost its meaning.
If this is so, we must accept the relativity of the
classical concepts which are so convenient and dear to us, and replace them
with some qualitatively new ideas on the motion of matter. The classical
attempts to obtain an endless detalization of objects and phenomena mean
that the desire incalcated in us over centuries "to study organic existence"
leads at a certain stage to a "driving out of the soul" and a situation
arises, where, according to Goethe, "the spiritual link is lost".
Lev Vasilievich TARASOV
The author was born in 1934 and graduated from Moscow Engineering Physics
Institute in 1958 specializing in the field of theoretical nuclear physics.
He was awarded his PhD in Mathematics and Physics in 1968, was appointed
Associate Professor in 1969, and Professor in 1983. From 1989 to 1992 he was
Head of the Department of the Methodology for Natural Sciences Teaching at
the Moscow Institute for the Advanced Training for Teachers. Between 1992 and
1998 he was Head of the Department of Physics at Moscow State Pedagogical
In 1994 he was awarded the medal "For Excellence in Public Education" for
developing a new model of comprehensive school Ecology and Dialectics
together with its practical implementation at the level of an
intergovernmental pedagogical experiment.