Foundations of Classical and Quantum Electrodynamics E-Book

Igor N. Toptygin

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1054

Veröffentlichungsjahr: 2013

Beschreibung

This advanced textbook covers many fundamental, traditional and new branches of electrodynamics, as well as the related fields of special relativity, quantum mechanics and quantum electrodynamics. The book introduces the material at different levels, oriented towards 3rd-4th year bachelor, master, and PhD students. This is so as to describe the whole complexity of physical phenomena, instead of a mosaic of disconnected data. The required mathematical background is collated in Chapter 1, while the necessary physical background is included in the main text of the corresponding chapters and also given in appendices. The content is based on teaching material tested on students over many years, and their training to apply general theory for solving scientific and engineering problems. To this aim, the book contains approximately 800 examples and problems, many of which are described in detail. Some of these problems are designed for students to work on their own with only the answers and descriptions of results, and may be solved selectively. The examples are key ingredients to the theoretical course; the user should study all of them while reading the corresponding chapters. Equally suitable as a reference for researchers specialized in science and engineering.

Leseprobe

Contents

Preface

Fundamental Constants and Frequently Used Numbers

Basic Notation

1 The Mathematical Methods of Electrodynamics

1.1 Vector and Tensor Algebra

1.2 Vector and Tensor Calculus

1.3 The Special Functions of Mathematical Physics

1.4 Answers and Solutions

2 Basic Concepts of Electrodynamics: The Maxwell Equations

2.1 Electrostatics

2.2 Magnetostatics

2.3 Maxwell’s Equations. Free Electromagnetic Field

2.4 Answers and Solutions

3 The Special Theory of Relativity and Relativistic Kinematics

3.1 The Principle of Relativity and Lorentz Transformations

3.2 Kinematics of Relativistic Particles

3.3 Answers and Solutions

4 Fundamentals of Relativistic Mechanics and Field Theory

4.1 Four-Dimensional Vectors and Tensors

4.2 The Motion of Charged Particles in Electromagnetic Fields. Transformation of the Electric Field

4.3 The Four-Dimensional Formulation of Electrodynamics. Introduction to Field Theory

4.4 Answers and Solutions

5 Emission and Scattering of Electromagnetic Waves

5.1 Green’s Functions and Retarded Potentials

5.2 Emission in Nonrelativistic Systems of Charges and Currents

5.3 Emission by Relativistic Particles

5.4 Interaction of Charged Particles with Radiation

5.5 Answers and Solutions

6 Quantum Theory of Radiation Processes. Photon Emission and Scattering

6.1 Quantum Theory of the Free Electromagnetic Field

6.2 Quantum Theory of Photon Emission, Absorption, and Scattering by Atomic Systems

6.3 Interaction between Relativistic Particles

6.4 Answers and Solutions

7 Fundamentals of Quantum Theory of the Electron–Positron Field

7.1 Covariant Form of the Dirac Equation. Relativistic Bispinor Transformation

7.2 Covariant Quadratic Forms

7.3 Charge Conjugation and Wave Functions of Antiparticles

7.4 Secondary Quantization of the Dirac Field. Creation and Annihilation Operators for Field Quanta

7.5 Energy and Current Density Operators for Dirac Particles

7.6 Interaction between Electron–Positron and Electromagnetic Fields

7.7 Schrödinger Equation for Interacting Fields and the Evolution Operator

7.8 Scattering Matrix and Its Calculation

7.9 Calculations of Probabilities and Effective Differential Cross-Sections

7.10 Scattering of a Relativistic Particle with a Spin in the Coulomb Field

7.11 Green’s Functions of Electron–Positron and Electromagnetic Fields

7.12 Interaction between Electrons and Muons

7.13 Higher-Order Corrections

7.14 Answers and Solutions

Appendix A Conversion of Electric and Magnetic Quantities between the International System of Units and the Gaussian System

Appendix B Variation Principle for Continuous Systems

B.1 Vibrations of an Elastic Medium as the Vibration Limit of Discrete Point Masses

B.2 The Lagrangian Form of Equations of Motion for a Continuous Medium

Appendix C General Outline of Quantum Theory

C.1 Spectrum of Physical Values and the Wave Function

C.2 State Vector

C.3 Indistinguishability of Identical Particles

C.4 Operators and Their Properties

C.5 Some Useful Formulas of Operator Algebra

C.6 Wave Functions of the Hydrogen-Like Atom (the Lowest Levels)

References

Index

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Author

Dr. Igor N. ToptyginState Polytechnic UniversityDept. of Theoretical PhysicsSt. [email protected]

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Preface

This monograph presents the foundations and modern achievements of classical and quantum microscopic electrodynamics, combining classical and current results and classical and quantum mechanical formalisms. It is written for specialists of different levels – bachelors, masters, graduate students, and generally for researchers in various branches of physics, theoreticians, and experimentalists; it can also be useful for those who specialize in engineering. The book is self-sufficient and can used for self-education in various aspects of microscopic electrodynamics; it can serve as a collection of exercises, and it presents solutions to many specific problems described in a unified manner. The table of contents shows that the monograph contains a lot of material. This is achieved by the nontraditional composition of the monograph, which combines the style of a brief textbook and a collection of problems. The style is similar to that of the well-known monographs by Ryogo Kubo, that is, Statistical Mechanics (1965) and Thermodynamics (1968), but the material is different. Such a style is helpful for self-education and deeper study of the material.

The reader should not worry about the large number of pages. It is not obligatory to read all the material to be familiar with the basic laws of electromagnetic theory. Depending on the specific aims, some parts of the book can be skipped.

Chapter 1 contains the mathematical formalism which is widely used in theoretical physics. This material can be consulted whenever necessary. Physics is presented starting from Chapter 2. Each chapter contains material for theoretical studies and material for solving problems.

The material for theoretical studies takes up the smaller part of each chapter; it contains the formulations of the main principles and presents basic equations and other relations between physical quantities. Derivations of the majority of the mathematical relations are included in examples, and these are supplemented by detailed solutions. It is recommended to read these pages carefully and reproduce the derivations.

The material for solving problems takes up the larger part of each chapter. It contains formulations of problems, as well as answers with the analysis of the results and – in many cases – with solutions. Each chapter includes many problems (typically about 100). It is certainly not obligatory to solve all of them, but it is desirable to solve at least 10–15 problems from each chapter. This is recommended not only for theoreticians but for all physicists for better understanding of the basic relations. Especially useful problems are marked by a bold dot. Asterisk marks are used to give the reader an idea of the complexity of selected problems. High-complexity problems are marked with one asterisk and supercomplex problems with two asterisks.

Depending on the specific interests and level of study, some subjects can be skipped. Future bachelors can exclude from Section 2.3 the subsections on the partial polarization of waves and the analytical signal, and from Section 4.2 the subsections on the dynamics of orbital and spin magnetic moments and approximate methods. In addition, one can completely skip Sections 4.3 and 6.3 and Chapter 7, as well as some parts of Sections 5.3, 5.4, 6.1, and 6.2.

The master’s program includes additional sections, for instance, those which combine classical and quantum mechanical approaches. They are Section 4.3 (Lagrangian and Hamiltonian methods in field theory), Sections 6.1 and 6.2 (coherent quantum states and related problems), and Section 6.3 and Chapter 7 (relativistic Dirac equation and invariant perturbation theory). Depending on the direction of the master’s program, one may need other sections and problems. It would also be useful to look through additional literature, which is recommended at the end of each section. I have tried to cite the most valuable review articles and monographs.

The recommendations presented above can be useful for other potential readers, for instance, for engineers and young researchers who wish to improve their knowledge of electrodynamics. One should bear in mind that this monograph does not describe electromagnetic phenomena in matter. This important branch of electrodynamics is presented in another monograph: Electromagnetic Phenomena in Matter. Statistical and Quantum Approaches. Its publication by Wiley is currently in preparation. It is written in the same style and it also contains material of different levels.

While writing the present monograph, I tried to follow the best traditions of the Theoretical Physics Department of the St. Petersburg Polytechnical University in Russia. They have been formed since the 1920s by the founders of teaching theoretical physics at the Leningrad Polytechnic Institute, particularly, by A.A. Friedmann, V.R. Bursian, V.K. Frederiks, and Ya.I. Frenkel. In addition, I used my own experience of teaching theoretical physics at four physical faculties of the Saint Petersburg State Polytechnical University (the former Leningrad Polytechnic Institute) as well as my experience of writing earlier textbooks and monographs (Problems in Electrodynamics by V.V. Batygin and I.N. Toptygin, Fizmatgiz, 1962; Classical Electrodynamics by M.M. Bredov, V.V. Rumyantsev, and I.N. Toptygin, Nauka, 1985). Both books were published in several editions in Russian (with extensions and addenda). Moreover, they were translated and published in several languages.

The present monograph is essentially different from these earlier monographs. It is more general, contains a greater amount of material, and is updated with modern achievements, in the attempt to unify various branches of theoretical physics (sometimes expanding beyond electrodynamics itself) and make them most useful for scientists of various specializations. The author is very grateful to John Wiley & Sons for the decision to publish this monograph.

Initially, the present monograph was published in Russian by publishing house “Regular and Chaotic Dynamics” as volume 1 of a two-volume course entitled “Modern Electrodynamics”. There were two Russian editions, in 2003 and 2005. In comparison with the Russian editions, the present monograph is greatly updated. Chapter 6 was updated with material concerning quantum memory, atomic spectra and other matter. Chapter 7 is entirely new and describes relativistic invariant perturbation theory; it contains some basic problems of quantum electrodynamics of relativistic systems. Other chapters have also been updated and corrected. Volume 2 of the Russian edition will also be published by Wiley, but as a separate and updated monograph.

The overall design and outline of the book were developed jointly with Professor V.V. Batygin, my good friend and coauthor of many publications, who made an important and valuable contribution to this work. Unfortunately, he passed away in 1998 and could not participate in the practical realization of the project. D.V. Kupriyanov and I.M. Sokolov wrote a considerable part of the theoretical material in Sections 6.1 and 6.2, and suggested many problems. I greatly appreciate the assistance of my colleagues from the A.F. Ioffe Physical Technical Institute and from Saint Petersburg Polytechnical University. Publication of the English version of this book would have been impossible without the active help and financial support by means of grants of A.M. Bykov, A.M. Krasilshchikov, D.G. Yakovlev, V. Zelevinsky, A.I. Tsygan, V.S. Beskin, V.V. Dubov, D.V. Kupriyanov, N.V. Larionov, I.M. Sokolov, and K.Yu. Platonov. I am grateful to the translators of the book– A.B. Nemtsev (Chapters 1 and 2), N.V. Larionov (Chapter 6), and especially Yu.V. Morozov (Chapters 3, 4, 5, and 7). Professor D.V. Kupriyanov edited the translation of Chapters 6 and 7. Undergraduate students A. Egorov and F. Savenkov checked the solutions of the problems; A. Dubov and F. Savenkov improved the presentation of Chapter 7. Special thanks are due to A. Egorov for drawing a large number of the figures. I am grateful to the Russian Ministry of Education and Science (project 11.G34.31.0001) for partial financial support of the translation. Support from the External Fellowship Program of the Russian Quantum Center (reference number 86) is also greatly appreciated.

Igor N. Toptygin

Saint Petersburg, March 2013

Fundamental Constants and Frequently Used Numbers

Bohr magneton

Nuclear magneton

Classical electron radius

Compton wavelength of an electron

Compton wavelength of a proton

Basic Notation

1

The Mathematical Methods of Electrodynamics

1.1 Vector and Tensor Algebra

1.1.1 The Definition of a Tensor and Tensor Operations

In three-dimensional space, select a rectangular and rectilinear (Cartesian1)) system of coordinates x1, x2, x3. Regard the space as Euclidean. This means that all axioms of Euclidean geometry2) and their consequences considered in school courses on mathematics are valid in it. For instance, the square of the distance between two close points is given by the following expression:

Along with the original system of coordinates, consider some other systems of common origin yet rotated with respect to the original one (Figure 1.1).

Figure 1.1 The rotation of the Cartesian system of coordinates.

A scalar or invariant is a quantity that does not change when the system of coordinates is rotated, that is, it is the same in either the original or the rotated system of coordinates

(1.1)

(1.2)

(summing of elements over the repeated symbol β, from 1 to 3 is assumed). Here Vβ are the projections of the vector on an axis of the original system of coordinates, V′α are the projections of the vector on an axis of the rotated system, and aαβ are the coefficients of the transformation, which are the cosines of the angles between the β axis of the original system and the α axis of the rotated system. They may be written through the single vectors (orts) of the coordinate axes:

(1.3)

In three-dimensional space, a tensor of rank 2 is a nine-component quantity Tαβ (each index varies independently assuming three values: 1, 2, 3) which is defined in every system of coordinates and, when a coordinate system is rotated, is transformed as the products of the components of the two vectors AαVβ, in the following way:

(1.4)

In three-dimensional space, a tensor of rank s is a 3s-component quantity Tλ…v that is transformed as the product of s components of vectors:

(1.5)

Scalars and vectors may be regarded as tensors of rank 0 and 1, respectively.

Rotation matrix has the following properties:

1. Orthogonality

(1.6)

where

(1.7)

is Kronecker symbol3);

2. The determinant of a rotation matrix equals 1:

(1.8)

3. The product of two rotation matrices

(1.9)

describes the evolution of a system resulting from two consecutive rotations, first with matrix and then with matrix .4) In the general case, rotation matrices are noncommutative, that is,

(1.10)

As follows from property 1, a reverse matrix defined by the relation

(1.11)

results from the original matrix when the latter is transposed, that is, its columns are substituted for lines and vice versa:

(1.12)

The reverse transformation (1.2) looks like this:

(1.13)

All vectors are transformed identically according to rule (1.2) when a coordinate system is rotated. But they may behave in one of two ways if a system of coordinates is inverted, that is,

(1.14)

The sum of two tensors of the same rank produces a third tensor of the same rank with components

(1.15)

The direct products of the components of two tensors (without summing) constitute a tensor whose rank equals the sum of the ranks of the factors, for instance,

(1.16)

where Qαβγ is a tensor of rank 3.

(1.17)

is a scalar.

When an equality between tensors is written, the rule of the same tensor dimensionality must be observed: only tensors of the same rank may be equated. This means that the number of free symbols (over which no summation is done) must be the same in the first and second members of an equality. The number of pairs of “mute” symbols (those over which summing is done) may be any on the right and on the left.

Tensors may be symmetric (antisymmetric) with respect to a pair of indices α and β if their components satisfy the conditions

(1.18)

Tensor components may be either real or complex numbers. In the latter case, the concepts of Hermitian and anti-Hermitian tensors play an important role. The definition of a Hermitian tensor is as follows: