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Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen Uwe Krey · Anthony Owen Basic Theoretical Physics Uwe Krey · Anthony Owen Basic Theoretical Physics AConciseOverview With 31 Figures 123 Prof. Dr. Uwe Krey University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Dr. rer nat habil Anthony Owen University of Regensburg (retired) FB Physik Universitätsstraße 31 93053 Regensburg, Germany E-mail: [email protected] Library of Congress Control Number: 2007930646 ISBN 978-3-540-36804-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting and production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: eStudio Calamar S.L., F. Steinen-Broo, Pau/Girona, Spain Printed on acid-free paper SPIN 11492665 57/3180/YL - 5 4 3 2 1 0 Preface This textbook on theoretical physics (I-IV) is based on lectures held by one of the authors at the University of Regensburg in Germany. The four ‘canonical’ parts of the subject have been condensed here into a single volume with the following main sections : I = Mechanics and Basic Relativity; II = Electrodynamics and Aspects of Optics; III = Quantum Mechanics (non-relativistic theory), and IV = Thermodynamics and Statistical Physics. Our compendium is intended primarily for revision purposes and/or to aid in a deeper understanding of the subject. For an introduction to theoretical physics many standard series of textbooks, often containing seven or more volumes, are already available (see, for example, [1]). Exercises closely adapted to the book can be found on one of the authors websites [2], and these may be an additional help. We have laid emphasis on relativity and other contributions by Einstein, since the year 2005 commemorated the centenary of three of his ground- breaking theories. In Part II (Electrodynamics) we have also treated some aspects with which every physics student should be familiar, but which are usually neglected in textbooks, e.g., the principles behind cellular (or mobile) phone technology, synchrotron radiation and holography. Similarly, Part III (Quantum Mechan- ics) additionally covers aspects of quantum computing and quantum cryp- tography. We have been economical with figures and often stimulate the reader to sketch his or her own diagrams. The frequent use of italics and quotation marks throughout the text is to indicate to the reader where a term is used in a specialized way. The Index contains useful keywords for ease of reference. Finally we are indebted to the students and colleagues who have read parts of the manuscript and to our respective wives for their considerable support. Regensburg, Uwe Krey May 2007 Anthony Owen Contents Part I Mechanics and Basic Relativity 1 Space and Time .......................................... 3 1.1 PreliminariestoPartI .................................. 3 1.2 GeneralRemarksonSpaceandTime ..................... 3 1.3 SpaceandTimeinClassicalMechanics.................... 4 2 Force and Mass ........................................... 5 2.1 Galileo’s Principle (Newton’s First Axiom) . 5 2.2 Newton’s Second Axiom: Inertia; Newton’s Equation ofMotion ............................................. 5 2.3 Basic and Derived Quantities; Gravitational Force . 6 2.4 Newton’s Third Axiom (“Action and Reaction . ”) . 8 3 Basic Mechanics of Motion in One Dimension ............ 11 3.1 GeometricalRelationsforCurvesinSpace................. 11 3.2 One-dimensionalStandardProblems...................... 13 4 Mechanics of the Damped and Driven Harmonic Oscillator ................................................. 17 5 The Three Classical Conservation Laws; Two-particle Problems .................................... 23 5.1 Theorem for the Total Momentum (orfortheMotionoftheCenterofMass).................. 23 5.2 TheoremfortheTotalAngularMomentum................ 24 5.3 TheEnergyTheorem;ConservativeForces................. 26 5.4 TheTwo-particleProblem............................... 29 6 Motion in a Central Force Field; Kepler’s Problem ....... 31 6.1 Equations of Motion in Planar Polar Coordinates . 31 6.2 Kepler’sThreeLawsofPlanetaryMotion.................. 32 6.3 Newtonian Synthesis: From Newton’s Theory ofGravitationtoKepler................................. 33 6.4 PerihelionRotation..................................... 34 VIII Contents 6.5 Newtonian Analysis: From Kepler’s Laws toNewtonianGravitation ............................... 36 6.5.1 Newtonian Analysis I: Law of Force fromGivenOrbits............................... 36 6.5.2 Newtonian Analysis II: From the String Loop 2 Construction of an Ellipse to the Law Fr = −A/r ... 36 6.5.3 Hyperbolas;Comets ............................. 37 6.5.4 Newtonian Analysis III: Kepler’s Third Law andNewton’sThirdAxiom....................... 38 6.6 The Runge-Lenz Vector as an Additional Conserved Quantity 39 7 The Rutherford Scattering Cross-section ................. 41 8 Lagrange Formalism I: Lagrangian and Hamiltonian ...... 45 8.1 The Lagrangian Function; Lagrangian Equations oftheSecondKind..................................... 45 8.2 An Important Example: The Spherical Pendulum withVariableLength ................................... 46 8.3 The Lagrangian Equations of the 2nd Kind . 47 8.4 Cyclic Coordinates; Conservation of Generalized Momenta . 49 8.5 TheHamiltonian....................................... 50 8.6 The Canonical Equations; Energy Conservation II; PoissonBrackets....................................... 51 9 Relativity I: The Principle of Maximal Proper Time (Eigenzeit) ............................................... 55 9.1 Galilean versus Lorentz Transformations. 56 9.2 Minkowski Four-vectors and Their Pseudo-lengths; ProperTime........................................... 58 9.3 The Lorentz Force and its Lagrangian . 60 9.4 The Hamiltonian for the Lorentz Force; KineticversusCanonicalMomentum...................... 61 10 Coupled Small Oscillations ............................... 63 10.1 Definitions; Normal Frequencies (Eigenfrequencies) andNormalModes ..................................... 63 10.2 Diagonalization: Evaluation of the Eigenfrequencies andNormalModes ..................................... 65 10.3 A Typical Example: Three Coupled Pendulums withSymmetry ........................................ 65 10.4ParametricResonance:ChildonaSwing.................. 68 11 Rigid Bodies .............................................. 71 11.1 Translational and Rotational Parts of the Kinetic Energy . 71 Contents IX 11.2 Moment of Inertia and Inertia Tensor; Rotational Energy andAngularMomentum ................................ 72 11.3 Steiner’s Theorem; Heavy Roller; Physical Pendulum . 74 11.4 Inertia Ellipsoids; Poinsot Construction . 77 11.5TheSpinningTopI:Torque-freeTop...................... 78 11.6 Euler’s Equations of Motion and the Stability Problem . 79 11.7 The Three Euler Angles ϕ, ϑ and ψ; the Cardani Suspension . 81 11.8TheSpinningTopII:HeavySymmetricTop............... 83 12 Remarks on Non-integrable Systems: Chaos .............. 85 13 Lagrange Formalism II: Constraints ...................... 89 13.1D’Alembert’sPrinciple.................................. 89 13.2 Exercise: Forces of Constraint for Heavy Rollers onanInclinedPlane.................................... 91 14 Accelerated Reference Frames ............................ 95 14.1 Newton’s Equation in an Accelerated Reference Frame . 95 14.2CoriolisForceandWeatherPattern....................... 97 14.3 Newton’s “Bucket Experiment” and the Problem ofInertialFrames ...................................... 98 14.4 Application: Free Falling Bodies with Earth Rotation . 99 15 Relativity II: E=mc2 ..................................... 101 PartIIElectrodynamicsandAspectsofOptics 16 Introduction and Mathematical Preliminaries to Part II .. 109 16.1 Different Systems of Units in Electromagnetism . 109 16.2 Mathematical Preliminaries I: Point Charges and Dirac’s δ Function.................................. 112 16.3 Mathematical Preliminaries II: Vector Analysis . 114 17 Electrostatics and Magnetostatics ........................ 119 17.1ElectrostaticFieldsinVacuo............................. 119 17.1.1 Coulomb’s Law and the Principle of Superposition . 119 17.1.2 Integral for Calculating the Electric Field . 120 17.1.3 Gauss’sLaw.................................... 121 17.1.4 Applications of Gauss’s Law: Calculating the Electric Fields for Cases of Spherical or Cylindrical Symmetry . 123 17.1.5 The Curl of an Electrostatic Field; TheElectrostaticPotential....................... 124 XContents 17.1.6 General Curvilinear, Spherical andCylindricalCoordinates...................... 126 17.1.7
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