Tarasov L.V. Basic Concepts of Quantum Mechanics Ed.3

Contents

Preface

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 mechanics?

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 mechanics.

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 of matter.

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 nature.

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.

Personal Remarks

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 Knowledge), V.A.Fock, W.Pauli, P.Dirac, and also by the comprehensive works of L.D.Landau and E.M.Lifshitz, D.I.Blokhintsev, E.Fermi, L.Schiff.

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).

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 and phenomena.

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 classical physics.

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 "ignored".

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 classical thinking.

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".

About author

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 Open University.

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.