Mathematica for Theoretical Physics: Electrodynamics, Quantum Mechanics, General Relativity and Fractals

407 Pages·2005·2.619 MB·English

by  Gerd Baumann

®
Mathematica for Theoretical Physics
®
Mathematica
for Theoretical Physics
Electrodynamics,
Quantum Mechanics,
General Relativity,
and Fractals
Second Edition
Gerd Baumann
CD-ROMIncluded
GerdBaumann
DepartmentofMathematics
GermanUniversityinCairoGUC
NewCairoCity
MainEntranceofAlTagamoaAlKhames
Egypt
[email protected]
This is a translated, expanded, and updated version of the original German version of
the work “Mathematica® in der Theoretischen Physik,” published by Springer-Verlag
Heidelberg,1993©.
LibraryofCongressCataloging-in-PublicationData
Baumann,Gerd.
[MathematicaindertheoretischenPhysik.English]
Mathematicafortheoreticalphysics/byGerdBaumann.—2nded.
p.cm.
Includesbibliographicalreferencesandindex.
Contents:1. Classicalmechanicsandnonlineardynamics—2. Electrodynamics,quantum
mechanics,generalrelativity,andfractals.
ISBN0-387-21933-1
1. Mathematicalphysics—Dataprocessing. 2. Mathematica(Computerfile) I. Title.
QC20.7.E4B3813 2004
530′.285′53—dc22 2004046861
ISBN-10:0-387-21933-1 e-ISBN0-387-25113-8 Printedonacid-freepaper.
ISBN-13:978-0387-21933-2
©2005SpringerScience+BusinessMedia,Inc.
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The use in this publication of trade names, trademarks, service marks, and similar terms, even if
theyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornot
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Mathematica,MathLink,andMathSourceareregisteredtrademarksofWolframResearch,Inc.
PrintedintheUnitedStatesofAmerica. (HAM)
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To Carin,
for her love, support, and encuragement.
Preface
As physicists, mathematicians or engineers, we are all involved with
mathematical calculations in our everyday work. Most of the laborious,
complicated, and time-consuming calculations have to be done over and
over again if we want to check the validity of our assumptions and
derive new phenomena from changing models. Even in the age of
computers, we often use paper and pencil to do our calculations.
However, computer programs like Mathematica have revolutionized our
working methods. Mathematica not only supports popular numerical
calculations but also enables us to do exact analytical calculations by
computer. Once we know the analytical representations of physical
phenomena, we are able to use Mathematica to create graphical
representations of these relations. Days of calculations by hand have
shrunk to minutes by using Mathematica. Results can be verified within
a few seconds, a task that took hours if not days in the past.
The present text uses Mathematica as a tool to discuss and to solve
examples from physics. The intention of this book is to demonstrate the
usefulness of Mathematica in everyday applications. We will not give a
complete description of its syntax but demonstrate by examples the use
of its language. In particular, we show how this modern tool is used to
solve classical problems.
viii Preface
This second edition of Mathematica in Theoretical Physics seeks to
prevent the objectives and emphasis of the previous edition. It is
extended to include a full course in classical mechanics, new examples
in quantum mechanics, and measurement methods for fractals. In
addition, there is an extension of the fractal's chapter by a fractional
calculus. The additional material and examples enlarged the text so
much that we decided to divide the book in two volumes. The first
volume covers classical mechanics and nonlinear dynamics. The second
volume starts with electrodynamics, adds quantum mechanics and
general relativity, and ends with fractals. Because of the inclusion of
new materials, it was necessary to restructure the text. The main
differences are concerned with the chapter on nonlinear dynamics. This
chapter discusses mainly classical field theory and, thus, it was
appropriate to locate it in line with the classical mechanics chapter.
The text contains a large number of examples that are solvable using
Mathematica. The defined functions and packages are available on CD
accompanying each of the two volumes. The names of the files on the
CD carry the names of their respective chapters. Chapter 1 comments on
the basic properties of Mathematica using examples from different fields
of physics. Chapter 2 demonstrates the use of Mathematica in a
step-by-step procedure applied to mechanical problems. Chapter 2
contains a one-term lecture in mechanics. It starts with the basic
definitions, goes on with Newton's mechanics, discusses the Lagrange
and Hamilton representation of mechanics, and ends with the rigid body
motion. We show how Mathematica is used to simplify our work and to
support and derive solutions for specific problems. In Chapter 3, we
examine nonlinear phenomena of the Korteweg–de Vries equation. We
demonstrate that Mathematica is an appropriate tool to derive numerical
and analytical solutions even for nonlinear equations of motion. The
second volume starts with Chapter 4, discussing problems of
electrostatics and the motion of ions in an electromagnetic field. We
further introduce Mathematica functions that are closely related to the
theoretical considerations of the selected problems. In Chapter 5, we
discuss problems of quantum mechanics. We examine the dynamics of a
free particle by the example of the time-dependent Schrödinger equation
and study one-dimensional eigenvalue problems using the analytic and
Preface ix
numeric capabilities of Mathematica. Problems of general relativity are
discussed in Chapter 6. Most standard books on Einstein's theory discuss
the phenomena of general relativity by using approximations. With
Mathematica, general relativity effects like the shift of the perihelion
can be tracked with precision. Finally, the last chapter, Chapter 7, uses
computer algebra to represent fractals and gives an introduction to the
spatial renormalization theory. In addition, we present the basics of
fractional calculus approaching fractals from the analytic side. This
approach is supported by a package, FractionalCalculus, which is not
included in this project. The package is available by request from the
author. Exercises with which Mathematica can be used for modified
applications. Chapters 2–7 include at the end some exercises allowing
the reader to carry out his own experiments with the book.
Acknowledgments Since the first printing of this text, many people
made valuable contributions and gave excellent input. Because the
number of responses are so numerous, I give my thanks to all who
contributed by remarks and enhancements to the text. Concerning the
historical pictures used in the text, I acknowledge the support of the
http://www-gapdcs.st-and.ac.uk/~history/ webserver of the University of
St Andrews, Scotland. My special thanks go to Norbert Südland, who
made the package FractionalCalculus available for this text. I'm also
indebted to Hans Kölsch and Virginia Lipscy, Springer-Verlag New
York Physics editorial. Finally, the author deeply appreciates the
understanding and support of his wife, Carin, and daughter, Andrea,
during the preparation of the book.
Cairo, Spring 2005
Gerd Baumann
Contents
Volume I
Preface vii
1 Introduction 1
1.1 Basics 1
1.1.1 Structure of Mathematica 2
1.1.2 Interactive Use of Mathematica 4
1.1.3 Symbolic Calculations 6
1.1.4 Numerical Calculations 11
1.1.5 Graphics 13
1.1.6 Programming 23
2 Classical Mechanics 31
2.1 Introduction 31
2.2 Mathematical Tools 35
2.2.1 Introduction 35
2.2.2 Coordinates 36
2.2.3 Coordinate Transformations and Matrices 38
2.2.4 Scalars 54
2.2.5 Vectors 57
2.2.6 Tensors 59
2.2.7 Vector Products 64
2.2.8 Derivatives 69
2.2.9 Integrals 73
2.2.10 Exercises 74
xii Contents
2.3 Kinematics 76
2.3.1 Introduction 76
2.3.2 Velocity 77
2.3.3 Acceleration 81
2.3.4 Kinematic Examples 82
2.3.5 Exercises 94
2.4 Newtonian Mechanics 96
2.4.1 Introduction 96
2.4.2 Frame of Reference 98
2.4.3 Time 100
2.4.4 Mass 101
2.4.5 Newton's Laws 103
2.4.6 Forces in Nature 106
2.4.7 Conservation Laws 111
2.4.8 Application of Newton's Second Law 118
2.4.9 Exercises 188
2.4.10 Packages and Programs 188
2.5 Central Forces 201
2.5.1 Introduction 201
2.5.2 Kepler's Laws 202
2.5.3 Central Field Motion 208
2.5.4 Two-Particle Collisons and Scattering 240
2.5.5 Exercises 272
2.5.6 Packages and Programs 273
2.6 Calculus of Variations 274
2.6.1 Introduction 274
2.6.2 The Problem of Variations 276
2.6.3 Euler's Equation 281
2.6.4 Euler Operator 283
2.6.5 Algorithm Used in the Calculus of Variations 284
2.6.6 Euler Operator for qDependent Variables 293
2.6.7 Euler Operator for q + p Dimensions 296
2.6.8 Variations with Constraints 300
2.6.9 Exercises 303
2.6.10 Packages and Programs 303
2.7 Lagrange Dynamics 305
2.7.1 Introduction 305
2.7.2 Hamilton's Principle Hisorical Remarks 306