Compendium of Theoretical Physics

529 Pages·2006·2.562 MB·English

by  Wachter A., Hoeber H.

Compendium of Theoretical Physics
Armin Wachter Henning Hoeber
Compendium of Theoretical
Physics
Armin Wachter Henning Hoeber
Pallas Inc. de GeophysiqueCGG house
Brühl,Germany Compagnie Generale de Geophysique
[email protected] Feltham,Middlesex TW1401R
United Kingdom
hhoebercgg.com
Library ofCongress Control Number:2005927414
ISBN-10:0-387-25799-3 e-ISBN 0-387-29198-9
ISBN-13:978-0387-25799-0
Printed on acid-free paper.
© 2006 Springer Science+Business Media,Inc.
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Acknowledgements
We’d like to thank the Robert Gordon University, Aberdeen, where a large
partoftheelectrodynamicschapterwaswritten,foritskindsupport.(Where
better than in Aberdeen to write about electrodynamics?) Thanks to Klaus
Ka¨mpfforhisdonationoftextbooks;toourcolleagues,aswellasstudents,at
the Bergische Universita¨t Wuppertal and the Ho¨chstleistungsrechenzentrum
Ju¨lich (now John von Neumann Institute) for many discussions and sugges-
tions; to the Pallas GmbH and the Compagnie G´en´erale de G´eophysique for
their kind support in the final phases of our writing this book.
This translation is based on the 2nd edition of the German book, pub-
lished in 2004, which has undergone a thorough review in order to minimize
remaining errors. For the translation into English, we would like to thank
our friends John Mehegan and Andrew Ratcliffe for providing corrections
and suggestions to part of the book. The usual disclaimer holds: any errors
in this book are ours.
Thanks to Springer for letting us proceed at our own pace in providing
the final version of this translation.
Finally, we wish to thank our friends and families, without whom this
project would have simply been unthinkable.
Preface
Our book, Compendium of Theoretical Physics, contains the topics
• mechanics,
• electrodynamics,
• quantum mechanics, and
• statistical physics and thermodynamics,
which constitute the “canonical curriculum” of theoretical physics, taught
at the undergraduate level at most universities. It is oriented mainly toward
students of higher levels who are interested in a clearly arranged and co-
herent presentation of the curriculum, or who are preparing for their degree
examinations. Moreover, this book is also suited as an accompanying and
complementary textbook for students within the first semesters. For them,
it aims to provide a useful guideline to reclassify the content of the various
physics courses. Finally, physicists working in industry or research may also
benefit from this overview of theoretical physics.
There are, of course, many good textbooks for the above topics (some
suggestions can be found in our commented literature list). This book is
therefore not to be regarded as a substitute for these books; no student can
getbywithoutanextensivestudyofthecurriculumusingother,didactically
and historically well-edited presentations of theoretical physics. However, it
seemed necessary to us to write this book in order to offer a complementary
approachtotheoreticalphysics,inwhichthecomposition,thestructure,and,
lastbutnotleast,theeleganceofphysicaltheoriesareemphasizedandeasyto
recognize. We hope that this is achieved, among other things, by dispensing
with historical-phenomenological explanations.
Throughout,wepursueanaxiomatic-deductiveapproachinthatwestart
the discussion of each theory with its fundamental equations. Subsequently,
we derive the various physical relationships and laws in logical (rather than
chronological) order. Our aim is to emphasize the connections between the
individualtheoriesbyconsistentlyusingastandardizedpresentationandno-
tation.Think,forexample,oftheHamiltonformalism:itconstitutesafunda-
mentalconceptnotonlyinquantummechanicsbutalsoinstatisticalphysics.
VIII Preface
Inthefirstchapter,Mechanics,wepresenttheoftenoveremphasizedNew-
tonian approach to mechanics next to the Lagrange and Hamilton formula-
tions. Each of these equivalent representations distinguishes itself by spe-
cific advantages: whereas Newton’s approach is most easily accessible using
the concept of force and by finding the equations of motions, only the La-
grange and Hamilton formalisms provide a suitable platform for a deeper
understandingofmechanicsandothertheoreticalconcepts.Forexample,the
Lagrange formalism is better suited to seeing the relationship between sym-
metries and conservation laws. Accordingly, the first three sections of this
chapter deal with these three approaches and their connections in equitable
fashion.Furthermore,inthesectionRelativistic Mechanics,weintroducethe
correctLorentztensornotationinordertoeasethetransitiontotherelativis-
tic theory of electrodynamics, in which the disciplined use of this notation
turns out to be very useful and convenient.
The advantage of our deductive method may be particularly apparent in
the second chapter, Electrodynamics. In contrast to many other textbooks,
we start with Maxwell’s equations in their most general form. This allows us
immediately to see very clearly the structure of this theory. We quickly find
thegeneralsolutionstoMaxwell’sequationsusingtheveryimportantconcept
of gauge invariance. From this, the various laws of electrodynamics follow
naturallyinacleanandconcisemanner.Forexample,thesolutionsinempty
space,orthespecialcasesofelectro-andmagnetostatics,areeasilydeduced.
Basedonourresultsofrelativisticmechanics,weapplythecovariantnotation
to electrodynamics and discuss the Lagrange and Hamilton formalism with
respect to the field theoretical character of the theory.
In contrast to the other chapters we begin Quantum Mechanics with a
mathematical introduction in which some areas of linear algebra are recapit-
ulated using Dirac’s notation. In particular, the concepts of operators and
eigenvalue problems are discussed. These are of fundamental importance
in quantum mechanics. We then present the general structure of quantum
theory, where the fundamental concepts are established and discussed in a
representation-independent manner. Generally, throughout the whole chap-
ter, we try to avoid overemphasizing a particular representation.
Similarly to mechanics, there are different approaches in statistical me-
chanics/thermodynamics to describe many-particle systems. First, we have
thestatisticalansatzthatcombines(quantum)mechanicallawswithastatis-
tical principle. This results in a microscopic description in form of ensemble
theories. By contrast, thermodynamics is a purely phenomenological theory
based on purely macroscopic experiences. A third approach is given by in-
formation theory where a system is considered from the viewpoint of lack
of information. In order to highlight the inherent connections of these three
concepts, we discuss all of them in our chapter Statistical Physics and Ther-
modynamics and show their mutual equivalence.
Preface IX
Throughout this book important equations and relationships are sum-
marized in boxes containing definitions and theorems. We hope that this
facilitates structured learning and makes finding fundamental results easier.
Furthermore, we have arranged connected argumentations optically; in prin-
ciple, the reader should always be able to recognize the end of an argument.
At the end of each section we have placed a short summary as well as some
applications with solutions. These applications are intended to reaffirm, and
sometimes enhance, the understanding of the subject matter. Finally, in the
appendix,ashortcompilationofimportantandoften-usedmathematicalfor-
mulae are given.
It should be obvious that we make no claim of completeness. Instead,
the topics of the four chapters are chosen such that, on the one hand, they
containthefundamentalideasandconceptsand,ontheotherhand,coverthe
areas we have found, quite subjectively, most relevant for examinations and
day-to-day work. To complement this book, we make some further literature
suggestions in the appendix.
Overall, we hope that we have written a book that works as a broker
between textbooks, lecture notes, and formula compilations. It would make
usveryhappyifithelpedyoutobetterunderstandtheconceptsoftheoretical
physics.
Cologne and Newcastle Armin Wachter
October 2005 Henning Hoeber
Table of Contents
Acknowledgements ........................................... V
Preface ....................................................... VII
List of Applications..........................................X. VII
1. Mechanics ................................................ 1
1.1 Newtonian Mechanics ................................... 2
1.1.1 Coordinate Systems and Vectors ................... 3
1.1.2 Newton’s Axioms................................. 4
1.1.3 Physical Consequences, Conservation Laws .......... 8
1.1.4 Accelerated Coordinate Systems
and Inertial Systems, Galilei Invariance ............. 13
1.1.5 N-Particle Systems ............................... 19
Applications ........................................... 22
1.2 Lagrangian Mechanics................................... 26
1.2.1 Constraining Forces, d’Alembert’s Principle,
and Lagrange Equations........................... 27
1.2.2 Conservation Laws ............................... 33
1.2.3 Hamilton Principle and Action Functional ........... 35
Applications ........................................... 40
1.3 Hamiltonian Mechanics ................................. 47
1.3.1 Hamilton Equations .............................. 47
1.3.2 Conservation Laws ............................... 49
1.3.3 Poisson Bracket .................................. 50
1.3.4 Canonical Transformations ........................ 52
1.3.5 Hamilton-Jacobi Equation......................... 56
Applications ........................................... 59
1.4 Motion of Rigid Bodies.................................. 62
1.4.1 General Motion of Rigid Bodies .................... 63
1.4.2 Rotation of Rigid Bodies Around a Point............ 65
1.4.3 Euler Angles and Lagrange Equations............... 67
Applications ........................................... 69
XII Table of Contents
1.5 Central Forces ......................................... 72
1.5.1 Two-Particle Systems ............................. 73
1.5.2 Conservative Central Problems, 1/r-Potentials ....... 74
1.5.3 Kepler’s Laws and Gravitational Potential........... 78
1.5.4 Elastic One-Particle Scattering by a Fixed Target .... 82
1.5.5 Elastic Two-Particle Scattering .................... 86
Applications ........................................... 90
1.6 Relativistic Mechanics .................................. 95
1.6.1 Axioms, Minkowski Space, Lorentz Transformations .. 95
1.6.2 Relativistic Effects ............................... 99
1.6.3 Causality Principle,
Space-, Light- and Time-like Vectors................ 101
1.6.4 Lorentz-Covariant Formulation
of Relativistic Mechanics .......................... 102
1.6.5 Lagrange Formulation of Relativistic Mechanics ...... 106
Applications ........................................... 109
2. Electrodynamics.......................................... 113
2.1 Formalism of Electrodynamics ........................... 114
2.1.1 Maxwell’s Equations and Lorentz Force ............. 114
2.1.2 Interpretation of Maxwell’s Equations............... 116
2.1.3 Energy and Momentum Conservation ............... 120
2.1.4 Physical Units ................................... 123
Applications ........................................... 126
2.2 Solutions of Maxwell’s Equations in the Form of Potentials .. 128
2.2.1 Scalar and Vector Potential........................ 128
2.2.2 Gauge Transformations ........................... 129
2.2.3 General Solution of the Homogeneous
Wave Equations.................................. 132
2.2.4 Specific Solution of the Inhomogeneous
Wave Equation, Retarded Potentials ................ 133
Applications ........................................... 136
2.3 Lorentz-Covariant Formulation of Electrodynamics.......... 139
2.3.1 Lorentz Tensors .................................. 139
2.3.2 Lorentz-Covariant Formulation of Maxwell’s Equations 141
2.3.3 Transformational Behavior of Electromagnetic Fields . 143
2.3.4 Lorentz Force and Covariance...................... 144
2.3.5 Energy and Momentum Conservation ............... 146
Applications ........................................... 147
2.4 Radiation Theory ...................................... 150
2.4.1 Li´enard-Wiechert Potentials ....................... 150
2.4.2 Radiation Energy ................................ 153
2.4.3 Dipole Radiation ................................. 156
Applications ........................................... 159
Table of Contents XIII
2.5 Time-Independent Electrodynamics....................... 161
2.5.1 Electrostatics and Magnetostatics .................. 161
2.5.2 Multipole Expansion of Static Potentials and Fields... 165
2.5.3 Boundary Problems in Electrostatics I .............. 169
2.5.4 Boundary Problems in Electrostatics II ............. 175
2.5.5 Field Distributions in Magnetostatics ............... 179
Applications ........................................... 182
2.6 Electrodynamics in Matter .............................. 185
2.6.1 Macroscopic Maxwell Equations.................... 185
2.6.2 Material Equations ............................... 191
2.6.3 Continuity Conditions at Boundaries................ 193
Applications ........................................... 195
2.7 Electromagnetic Waves.................................. 198
2.7.1 Plane Waves in Nonconducting Media............... 198
2.7.2 Reflection and Refraction.......................... 201
2.7.3 Superposition of Waves, Wave Packets .............. 206
2.7.4 Plane Waves in Conducting Media.................. 210
2.7.5 Cylindrical Hollow Conductor...................... 211
Applications ........................................... 213
2.8 Lagrange Formalism in Electrodynamics................... 216
2.8.1 Lagrange and Hamilton Functions
of a Charged Particle ............................. 216
2.8.2 Lagrange Density of the Electromagnetic Field ....... 217
2.8.3 Conservation Laws and the Noether Theorem ........ 220
2.8.4 Internal Symmetries and Gauge Principle............ 222
Applications ........................................... 225
3. Quantum Mechanics...................................... 227
3.1 Mathematical Foundations of Quantum Mechanics.......... 229
3.1.1 Hilbert Space .................................... 229
3.1.2 Linear Operators................................. 233
3.1.3 Eigenvalue Problems.............................. 236
3.1.4 Representation of Vectors and Linear Operators...... 239
Applications ........................................... 242
3.2 Formulation of Quantum Theory ......................... 245
3.2.1 Limits of Classical Physics......................... 245
3.2.2 Postulates of Quantum Mechanics .................. 247
3.2.3 Quantum Mechanical Measurement................. 249
3.2.4 Schro¨dinger Picture and Schro¨dinger Equation ....... 253
3.2.5 Other Pictures of Quantum Theory................. 255
3.2.6 Representations in Quantum Mechanics ............. 259
Applications ........................................... 264
3.3 One-Dimensional Systems ............................... 266
3.3.1 Aspects of the Schro¨dinger Equation
in Coordinate Space .............................. 267