Mathematical Methods for Physicists

1036 Pages·1995·19.09 MB·English

MATHEMATICAL
METHODS
FOR PHYSICISTS
Fourth Edition
George B. Arfken
Miami University
Oxford, Ohio
Hans J. Weber
University of Virginia
Charlottesville, Virginia
Academic Press
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Library of Congress Cataloging-in-Publication Data
Arfken, George B. (George Brown), date.
Mathematical methods for physicists / by George B. Arfken,
Hans-Jurgen Weber. - 4th ed.
p. cm.
Includes bibliographical references (p. ) and index.
ISBN 0-12-059815-9
ISBN 0-12-059816-7 (International paper edition)
1. Mathematics. 2. Mathematical physics. I. Weber, Hans-Jurgen.
II. Title.
QA37.2.A74 1995
515'.l-dc20 94-24911
CIP
PRINTED IN THE UNITED STATES OF AMERICA
95 96 97 98 99 00 DO 9 8 7 6 5 4 3 2 1
To Carolyn and Edith Enzian
PREFACE
There are many additions, revisions and some deletions in this fourth edition of
Mathematical Methodsfor Physicists. In detail, the linear properties ofscalar and
vector products are emphasized to motivate their definitions in Chapter 1. A new
Section 1.15 on Dirac's delta function collects portions that were scattered over
several chapters.
Chapter 2 on vectors and tensors has been shortened by moving the section on
separation ofvariables to Chapter 8, where they are presented as a method ofre
ducing partial to ordinary differential equations, and by deleting the sections on
dyadics andelasticity, topics thatare rarely taught in physics today.
Chapter3 is now closely linkedto Chapter 1by using the geometricalaspects of
scalarand vector products systematically to motivate the conceptand definition of
determinants and to solve linearequations. Theconceptofmatrices and their mul
tiplication is based on rotations as special lineartransformations. The product the
oremlinking matricesanddeterminantsisincluded. Diracgammamatricesarenow
based on the metric and conventions of Bjorken and Drell, Relativistic Quantum
Mechanics, which are becoming standard in the literature. A briefnew subsection
oncommonly used functions ofmatrices is included.
The group theoretical sections have been collected and expanded into a separate
Chapter4.Generatorsaretreatedinmoredetail.Ladderoperatorshavebeenadapted
fromChapter12onsphericalharmonics. AngularmomentumcouplingandClebsch
Gordancoefficientsaredevelopedalongwithsphericaltensoroperators. Sectionson
theLorentzgroupandMaxwell'sequationsareadaptedtotheBjorken-Drellmetric,
inaccord withJackson'scorrespondingrevisionofClassical Electrodynamics.
Chapter 5 contains minor additions to Bernoulli numbers. The integral conver
gence test is extended to alternating and otherseries. An examplefrom Fourierse
ries in Chapter 14is included.
Multivalent functions and their branch cuts are given more emphasis in Chapter
6, and this is continued in Chapter7. The Mittag-Leffler pole expansion ofmero
morphic functions and product expansions ofentire functions are included, along
with applications such as Rouche's theorem.
xv
xvi Preface
In Chapter 8 characteristics are briefly introduced. A soliton solution of a nonlin-
ear partial differential equation is included. The separation of variables from Chap-
ter 1 has been moved here. Green's functions have been moved here from Chapter 16
and have been moved into Chapter 9 as well. The connections of Chapter 9 with the
linear algebra of Chapters 1-4 are emphasized early on.
The product expansion of the Gamma function in Chapter 10 is now tied to prod-
uct expansions of analytic functions discussed in Chapter 7.
In Chapter 12 vector spherical harmonics are adapted to the notation in some an-
gular momentum texts.
As an application of Fourier series, Chapter 14 now contains the functional equa-
tion of the Riemann zeta function. The discussion expands the connections of this
topic with Chapter 5 on Bernoulli numbers, and ties it in with Chapter 6 as an ex-
ample of analytic continuation. The connection to analytic number theory is men-
tioned in more detail.
A new Chapter 18 on nonlinear methods takes into account a few of the major as-
pects of this vast and rapidly expanding field.
The problem sets have been examined closely. Some problems have been deleted
and a number of new problems have been added.
The 4th edition is based on the advice and help of many people. Some of the ad-
ditions and many revisions are in response to the comments of reviewers. We are
grateful to them and to Senior Editor Robert Kaplan who organized the early stages
of the revision. Dr. Michael Bozoian, Dr. Nelson Max, and Professor Philip A.
Macklin have been most helpful with numerous suggestions and corrections in the
text and problem sets. The final form of the 4th edition owes much to the expertise
of Senior Editor Peter Renz and Production Editor Jacqueline Garrett.
INTRODUCTION
Many of the physical examples used to illustrate the applications of
mathematics are taken from electromagnetism and quantum mechanics. For
convenience the main equations are listed below and the symbols identified.
References are also given.
ELECTROMAGNETIC THEORY
Maxwell's Equations (MKS Units-Vacuum)
V D = ρ V XE =
VB = 0 V χ Η V J
Here Ε is the electric field defined in terms of force on a static charge and Β
the magnetic induction defined in terms of force on a moving charge. The
related fields D and Η are given (in vacuum) by
D = £0E and Β = μ0Η.
The quantity ρ represents free charge density while J is the corresponding
current. The electric field Ε and the magnetic induction Β are often expressed
in terms of the scalar potential ψ and the magnetic vector potential A:
V(p Β = V X A.
For additional details see J. M. Marion, Classical Electromagnetic Radiation.
New York: Academic Press (1965); J. D. Jackson, Classical Electrodynamics,
2nd ed. New York: Wiley (1975).
xvii
xviii INTRODUCTION
Note that Marion and Jackson prefer Gaussian units. A glance at the last
two texts and the great demands they make upon the student's mathematical
competence should provide considerable motivation for the study of this
book.
QUANTUM MECHANICS
Schrodinger Wave Equation (Time Independent)
ψ is the (unknown) wave function. The potential energy, often a function of
position, is denoted by V while Ε is the total energy of the system. The mass
of the particle being described by ψ is m. h is Planck's constant h divided by
In. Among the extremely large number of beginning or intermediate texts we
might note: A. Messiah, Quantum Mechanics, 2 vols. New York; Wiley
(1961); E. Merzbacher, Quantum Mechanics, 2nd ed. New York: Wiley
(1970); G. Baym, Lectures on Quantum Mechanics, 2nd printing. Reading,
MA: Benjamin (1973); J. J. Sakurai, Modern Quantum Mechanics, rev. ed.
Reading, MA: Addison-Wesley (1994).
1 VECTOR
ANALYSIS
•HJ 1.1 DEFINITIONS, ELEMENTARY APPROACH
In science and engineering we frequently encounter quantities that have
magnitude and magnitude only: mass, time, and temperature. These we label
scalar quantities. In contrast, many interesting physical quantities have
magnitude and, in addition, an associated direction. This second group
includes displacement, velocity, acceleration, force, momentum, and angular
momentum. Quantities with magnitude and direction are labeled vector quan-
tities. Usually, in elementary treatments, a vector is defined as a quantity
having magnitude and direction. To distinguish vectors from scalars, we
identify vector quantities with boldface type, that is, V.
Our vector may be conveniently represented by an arrow with length
proportional to the magnitude. The direction of the arrow gives the direction
of the vector, the positive sense of direction being indicated by the point. In
this representation vector addition
C = A + B (1.1)
consists in placing the rear end of vector Β at the point of vector A. Vector C
is then represented by an arrow drawn from the rear of A to the point of B.
This procedure, the triangle law of addition, assigns meaning to Eq. (1.1) and
is illustrated in Fig. 1.1. By completing the parallelogram, we see that
C = A + B = B + A, (1.2)
as shown in Fig. 1.2. In words, vector addition in commutative.
1
2 1 VECTOR ANALYSIS
Β
Figure 1.1 Triangle law of vector
addition.
Figure 1.2 Parallelogram law of vector
addition.
For the sum of three vectors
D = A + Β + C,
Fig. 1.3, we may first add A and Β
A + Β = Ε.
Then this sum is added to C
D = Ε + C.
Similarly, we may first add Β and C
Β + C = F.
Then
D = A + F.
In terms of the original expression,
(A + B) + C = A + (B + C).
Vector addition is associative.
Figure 1.3 Vector addition is associative.
1.1 Definitions, Elementary Approach 3
Figure 1.4 Equilibrium of forces. ¥ + F = -F .
l 2 3
A direct physical example of the parallelogram addition law is provided by
a weight suspended by two cords. If the junction point (O in Fig. 1.4) is in
equilibrium, the vector sum of the two forces Fj and F must just cancel the
2
downward force of gravity, F . Here the parallelogram addition law is subject
3
1
to immediate experimental verification.
Subtraction may be handled by defining the negative of a vector as a vector
of the same magnitude but with reversed direction. Then
A - Β = A + (-B).
In Fig. 1.3
A = Ε - B.
Note that the vectors are treated as geometrical objects that are independent
of any coordinate system. Indeed, we have not yet introduced a coordinate
system. This concept of independence of a preferred coordinate system is
developed in considerable detail in the next section.
The representation of vector A by an arrow suggests a second possibility.
2
Arrow A (Fig. 1.5), starting from the origin, terminates at the point
(A,A,A). Thus, if we agree that the vector is to start at the origin, the
x y z
1
Strictly speaking the parallologram addition was introduced as a definition. Experiments
show that if we assume that the forces are vector quantities and we combine them by parallelogram
addition the equilibrium condition of zero resultant force is satisfied.
2
The reader will see that we could start from any point in our cartesian reference frame; we
choose the origin for simplicity. This freedom of shifting the origin of the coordinate system
without affecting the geometry is called translation invariance.