Mathematics for the Physical Sciences

249 Pages·2002·8.034 MB·English

by  James B. Seaborn (auth.)

Mathematics for
the Physical Sciences
Springer
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James B. Seaborn
Mathematics for
the Physical Sciences
With 96 Illustrations
, Springer
James B. Seaborn
Department of Physics
University of Richmond
VA 23173, USA
[email protected]
Library of Congress Cataloging-in-Publication Data
Seaborn, James B.
Mathematics for the physical sciences / James B. Seaborn.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4419-2959-4
1. Mathematical physics. I. Title.
QC20.M364 2001
530.l5--dc21 2001049271
Printed on acid-free paper.
© 2002 Springer-Verlag New York, Inc.
Sof'tcover reprint of the hardcover 1s t edition 2002
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ISBN 978-1-4419-2959-4 ISBN 978-1-4684-9279-8 (eBook)
DOI 10.1007/978-1-4684-9279-8
Springer-Verlag New York Berlin Heidelberg
A member of BertelsmannSpringer Science+Business Media GmbH
To my friend, Matthias Wagner
Preface
This book is intended to provide a mathematical bridge from a general physics
course to intermediate-level courses in classical mechanics, electricity and mag
netism, and quantum mechanics. The book begins with a short review of a few
topics that should be familiar to the student from a general physics course. These
examples will be used throughout the rest of the book to provide physical con
texts for introducing the mathematical applications. The next two chapters are
devoted to making the student familiar with vector operations in algebra and cal
culus. Students will have already become acquainted with vectors in the general
physics course. The notion of magnetic flux provides a physical connection with
the integral theorems of vector calculus. A very short chapter on complex num
bers is sufficient to supply the needed background for the minor role played by
complex numbers in the remainder of the text. Mathematical applications in in
termediate and advanced undergraduate courses in physics are often in the form
of ordinary or partial differential equations. Ordinary differential equations are
introduced in Chapter 5. The ubiquitous simple harmonic oscillator is used to il
lustrate the series method of solving an ordinary, linear, second-order differential
equation.
The one-dimensional, time-dependent SchrOdinger equation provides an illus
tration for solving a partial differential equation by the method of separation of
variables in Chapter 6. Two examples are discussed in detail-one from quantum
mechanics (the quantum harmonic oscillator) and the other from classical elec
trostatics (a conducting sphere in a uniform electric field). In both cases, physical
boundary conditions are used to constrain the parameters introduced in the equa
tions. A more general discussion of boundary value problems is given in Chapter
7. Two more examples are considered in some detail-again, one from classical
viii Preface
physics (a vibrating drumhead) and one from quantum physics (a particle in a
one-dimensional box).
Orthogonal functions are treated in Chapter 8. The chapter opens with a brief
discussion of the failure of classical physics beginning around 1900. The need for
a new physics provides the motive for discussing mathematical operators, eigen
value equations, and the virtues of orthogonal functions. The quantum harmonic
oscillator illustrates the formulation of an eigenvalue problem and the appearance
of the discrete nature of physical quantities in quantum physics. This treatment
of the quantum harmonic oscillator is then seen as a special case of orthogonal
functions arising in the larger context of Sturm-Liouville theory. The usefulness
of orthogonal functions is further illustrated in the remainder of this chapter in
connection with Fourier integrals, Fourier series, and periodic functions applied
to problems in both classical and quantum physics.
The matrix formulation of an eigenvalue problem is introduced in Chapter 9
with a view toward laying a mathematical foundation for Heisenberg's matrix
mechanics. Two applications from classical physics are also considered-a system
of coupled harmonic oscillators and the principal axes of a rotating rigid body. A
brief discussion of variational calculus and a derivation of the Euler-Lagrange
equation are given in Chapter 10.
A set of exercises is provided at the end of each chapter to give the student
experience in applying mathematics to problems in physics. Illustrative examples
are worked out in the text. A number of exercises call for graphical representations
of the solutions. Some are particularly amenable to solution by numerical methods.
Most science and mathematics students have some expertise in using software
systems such as Mathematica and Maple or plotting routines like Physica and will
be able to apply their computational skills to these exercises. However, to avoid
giving the impression that computers are necessary for the book to be useful, I
have not treated numerical and computational methods in the text.
The purpose of the book is to collect in one place some essential mathematical
background material commonly encountered by undergraduate students in upper
level physics and chemistry courses. There is nothing new here, of course. There
are a number of well-written, comprehensive texts on mathematical methods in
physics that offer much more extensive treatments of this material along with a
wide range of other mathematical topics. All of this material is also treated to some
extent in various upper-level undergraduate physics texts, but to my knowledge
not all of it in a single book. I have listed in the References those works that have
been most helpful to me in putting together this one. For the reader who wants
more detail or a different view, some additional sources are recommended in a
short Bibliography.
University of Richmond James B. Seaborn
July 2001
Contents
Preface vii
1 A Review 1
1.1 Electrostatics.
1.2 Electric Current 3
1.3 Magnetic Flux 4
1.4 Simple Harmonic Motion 5
1.5 A Rigid Rotator 7
1.6 Exercises ......... 8
2 Vectors 13
2.1 Representations of Vectors 13
2.2 The Scalar Product of Two Vectors 16
2.3 The Vector Product of Two Vectors 18
2.4 Exercises .............. 24
3 Vector Calculus 29
3.1 Partial Derivatives . . . . . . . 30
3.2 A Vector Differential Operator 32
3.3 Components of the Gradient . 34
3.4 Flux .. 40
3.5 Exercises ........... 59
x Contents
4 Complex Numbers 69
4.1 Why Study Complex Numbers? . 69
4.2 Roots of a Complex Number 72
4.3 Exercises ............ . 74
5 Differential Equations 77
5.1 Infinite Series ......... . 79
5.2 Analytic Functions ........ . 80
5.3 The Classical Harmonic Oscillator 82
5.4 Boundary Conditions 84
5.5 Polynomial Solutions 88
5.6 Elementary Functions 90
5.7 Singularities 91
5.8 Exercises ...... . 92
6 Partial Differential Equations 103
6.1 The Method of Separation of Variables 103
6.2 The Quantum Harmonic Oscillator . . 105
6.3 A Conducting Sphere in an Electric Field . 109
6.4 The Schr6dinger Equation for a Central Field. 115
6.5 Exercises.................... 119
7 Eigenvalue Problems 127
7.1 Boundary Value Problems . . . . . . . 127
7.2 A Vibrating Drumhead ....... . 128
7.3 A Particle in a One-Dimensional Box. 131
7.4 Exercises .... 132
8 Orthogonal Functions 137
8.1 The Failure of Classical Physics. 137
8.2 Observables and Their Measurement 138
8.3 Mathematical Operators . . . . . . 139
8.4 Eigenvalue Equations . . . . . . . 139
8.5 The Quantum Harmonic Oscillator 143
8.6 Sturm-Liouville Theory 145
8.7 The Dirac Delta Function 148
8.8 Fourier Integrals .. 153
8.9 Fourier Series ... 156
8.10 Periodic Functions. 161
8.11 Exercises ..... . 165
9 Matrix Formulation of the Eigenvalue Problem 179
9.1 Reformulating the Eigenvalue Problem 180
9.2 Systems of Linear Equations .. 180
9.3 Back to the Eigenvalue Problem .... 182
Contents xi
9.4 Coupled Hannonic Oscillators 193
9.5 A Rotating Rigid Body 195
9.6 Exercises ........... . 198
10 Variational Principles 207
10.1 Fennat's Principle . . . . . . . . 207
10.2 Another Variational Calculation . 209
10.3 The Euler-Lagrange Equation 211
10.4 Exercises ............ . 221
Appendix A Vector Relations 227
A.1 Vector Identities . . . . . . . . . 227
A.2 Integral Theorems . . . . . . . . 227
A.3 The Functions of Vector Calculus 228
Appendix B Fundamental Equations of Physics 229
B.1 Poisson's Equation .. 229
B.2 Laplace's Equation ........... . 229
B.3 Maxwell's Equations ......... . 229
B.4 Time-Dependent SchrOdinger Equation. 230
Appendix C Some Useful Integrals and Sums 231
C.1 Integrals 231
C.2 Sums ................. . 234
Appendix D Algebraic Equations 235
D.I Quadratic Equation 235
D.2 Cubic Equation ..... 235
References 237
Bibliography 239
Index 241