Modern Physics
for Scientists and Engineers


Modern Physics

for Scientists and Engineers

Second Edition

John C. Morrison
Physics Department, University of Louisville, Louisville, KY, USA

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Dedication
This book is dedicated to the scientists and mathematicians
in the Holy Lands who are striving for peace

in a spiritually and culturally rich part of the world.

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Online applets are available to solve realistic problems in atomic and
condensed matter physics.
You can find the applets at:
/>
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Preface
Modern Physics for Scientists and Engineers presents the ideas that have shaped modern physics and provides an
introduction to current research in the different fields of physics. Intended as the text for a first course in modern physics
following an introductory course in physics with calculus, the book begins with a brief and focused account of historical
events leading to the formulation of modern quantum theory, while ensuing chapters go deeper into the underlying physics.
This book helps prepare engineering students for the upper division courses on devices they will later take and provides
engineering and physics majors an overview of contemporary physics. The course in modern physics is the last course in
physics most engineering students will ever take. For this reason, this book covers a few topics that are ordinarily taught at
the junior/senior level. I include these advanced topics because they are relevant and interesting to engineering students and
because these topics would ordinarily be unavailable to them. Topics such as Bloch’s theorem, heterostructures, quantum
wells and barriers, and a phenomenological description of semiconductor lasers help to give engineering students the physics
background they need for the courses they will later take on semiconductor devices, while subjects like the Hartree-Fock
theory, Bose-Einstein condensation, the relativistic Dirac equation, and particle physics help students appreciate the range
and scope of contemporary physics. This course helps physics majors by giving them a substantial introduction to quantum
theory and to the various fields of modern physics. The books I have used to prepare later chapters of this book are just the
books used in upper-division courses in the various fields of contemporary physics.

THIS NEW EDITION

The challenge in preparing this new edition has been to describe the developments that have occurred in physics since the
first edition of this book appeared in January 2010. I would like to thank Keith Ellis of the Theory Group at Fermilab for
discussing recent developments in particle physics with me and correcting the two new sections I have written on local
gauge invariance and the discovery of the Higgs Boson. Thanks are also due to Chris Quigg at Fermilab, Ken Hicks at Ohio
University, and Wafaa Khater at Birzeit University. My writing of the two new sections on graphene and carbon nanotubes
was also greatly helped by Fendinand Evers at Karlsruher Institute of Technology and by Gamini Sumanesekera and Shi-Yu
Wu at University of Louisville.

NEW FEATURES
In this new edition of Modern Physics for Scientists and Engineers, I have included a description of simulations from the
educational software package PhET developed at the University of Colorado. These simulations, which can be accessed
online, enable students to gain an intuitive understanding of how waves interfere with each other and how waves can be
combined to form wave packets. The new edition also contains many exercises using the software package MATLAB. A
new appendix on MATLAB has been added. Students are shown how to use MATLAB to plot functions, solve differential
equations, and evaluate integrals. To make these techniques available to as large a group of students as possible, I also
show how the free software package Octave can be used. The MATLAB programs in the first six chapters of this book run
unchanged in either MATLAB or Octave. As I shall show, however, the MATLAB programs in later chapters of the book
must be modified slightly to run in Octave.
Many of the electrical devices that have been developed within recent years are quantum devices. The finite potential well
provides a fairly realistic description of the active region of a semiconductor laser. This book includes MATLAB programs
that can be used to find the energy levels and wave functions for electrons confined to finite wells. Another MATLAB
program enables one to calculate the transmission and reflections coefficients for electrons incident upon a potential step
where the potential energy changes discontinuously. Potential steps of this kind occur naturally at the interface between
two different materials. By expressing the relation between the incoming and outgoing amplitudes of electrons incident
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xii


Preface

upon an interface in matrix form, one can calculate the transmission and reflection coefficients for complex systems by
multiplying the matrices for the individual parts. MATLAB and Octave programs described in Chapter 10 enable one to
calculate transmission coefficients for barriers where the potential energy assumes a different value for a short interval and
for more complex structures with two or three barriers. Interesting interference effects occur for more than a single interface.
This new edition also has new exercises using MATLAB and many more problems at the end of each chapter. In response
to the request of several teachers of modern physics, all of the figures in the book will be placed at the website of the book
and a digital copy of the book will be made available to teachers of modern physics upon request. Having the figures and a
digital copy available makes it easier for teachers to prepare PowerPoint lectures.

THE NATURE OF THE BOOK
As can be seen from the table of contents, Modern Physics for Scientists and Engineers covers atomic and solid-state
physics before covering relativity theory. When I was beginning to teach modern physics, I led off with the special theory of
relativity as do most books, but I found that this approach had a number of disadvantages. Following the short treatment of
relativity, there was invariably an uncertain juncture when I made the transition back to a nonrelativistic framework in order
to introduce the ideas of wave mechanics. The students were asked to make this transition when they were just getting started
in the course. Then, the important applications of relativity theory to particle and nuclear physics came at the end of the
course when we had not used the relativistic formalism for some time. I found it to be better to develop nonrelativistic
wave mechanics at the beginning of the course and “go relativistic” in the last 3 or 4 weeks. The course flows better
that way.
The first three chapters of this book give an introduction to quantum mechanics at an elementary level. Chapters 4-6 are
devoted to atomic physics and the development of lasers. Chapter 7 is devoted to statistical physics and Chapters 8-10 are
devoted entirely to condensed matter physics. Each of these chapters has special features that cannot be found in any other
book at this level. The new version of the Hartree-Fock applet described in Chapter 5 enables students to do Hartree-Fock
calculations on any atom in the periodic table using the Hartree-Fock applet at the website of the book. With the HartreeFock program of Charlottte Fischer in the background and a Java interface, the applet comes up showing the periodic table.
A student can initiate a Hartree-Fock calculation by choosing a particular atom in the periodic table and clicking on the red
arrow in the lower right-hand corner of the web page. The wave functions of the atom immediately appear on the screen
and tabs along the upper edge of the web page enable students to gain additional information about the properties of the

atom. One can find the average distance of each electron from the nucleus and evaluate the two-electron Slater integrals and
the spin-orbit constant of the outer electrons. When I cover the chapters on atomic physics in my course, I keep the focus
on the underlying physics. As one moves from one atom to the next along a row of the periodic table, the nuclear charge
increases. As a result, the electrons are drawn in toward the nucleus, and the distance between the electrons decreases. The
Coulomb interaction between the electrons increases and the “LS” term structure expands. All of this can be understood in
simple physical terms.
With the addition of MATLAB to Chapter 7, students can evaluate the probability that the values of the variables of
particles lie within a particular range. This enables one to calculate the probability that the velocities of molecules in the
upper atmosphere of a planet are greater than the escape velocity with the planet losing its atmosphere, and it enables one to
calculate the fractional number of electrons in a semiconductor with an energy above the Fermi energy. In this new edition,
Chapter 8 has a detailed description of graphene and carbon nanotubes. One of my surprises in preparing the new edition
was to find that the charge carriers of graphene are Fermions with zero mass that are accurately described by the Dirac
equation. Physics is a whole with all of the individual pieces fitting together.
Chapters 11 and 12, which are devoted to relativity theory, include a careful treatment of the Dirac equation and a
qualitative description of quantum electrodynamics. Chapter 13 on particle physics includes a description of the conservation
laws of lepton number, baryon number, and strangeness. Also included is a treatment of the parity and charge conjugation
symmetries, isospin, and the flavor and color SU(3) symmetries. The chapter on particle physics concludes with two new
sections on local gauge invariance and the recent discovery of the Higgs boson.
Most chapters of this book are fewer than 40 pages long, making it possible for an instructor to cover the main topics in
each chapter in 1 week. To give myself some flexibility in presenting the material, I usually choose two or three chapters
that I will not cover apart from a few qualitative remarks and then choose another three chapters that I will only expect my
students to know in a qualitative way. My selection of the subjects I cover more extensively depends upon the interests of
the particular class. Typically, the students might be expected to be able to work problems for the first three chapters and the
first section of Chapter 4, for Chapter 7 on Boltzman and Fermi-Dirac statistics, for Chapter 8 on condensed matter physics,
for Chapter 11 and the first two sections of Chapter 12 on relativity theory, and for the first two sections of Chapter 13 on

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Preface


xiii

particle physics. The students might then be asked qualitative quiz questions for Chapters 5, 6, 9, and 10, and the concluding
sections of Chapters 12 and 13. Suitable quiz questions and test problems can be found at the end of each chapter. In my
own classes, I typically give six quizzes and two tests. The practice of giving frequent quizzes keeps students up on the
reading and better prepared for discussion in class. Also, as a practical matter, our physics courses are always competing
with the engineering program for the study time of our students. Only by requiring in some concrete way that students keep
up with our courses can we expect a continuous investment of effort on their part.
I feel strongly that any class in physics should reach out to the broad majority of students, but that the class should
also allow students the opportunity to follow their interests beyond the level of the general course. Each chapter of this
book begins with a sound, rudimentary treatment of the fundamental subject matter, but then treats subjects such as the
Dirac theory that challenges the abilities of my better students. I always encourage my students to do extra-credit projects
in which they have a special interest and to work additional problems in areas that have been reserved for the quizzes. The
few physics majors I have had in my class often choose more advanced topics in which they have a special interest. For the
physics majors, my course gives them a valuable overview of the fields of contemporary physics that helps them with the
specialty course they later take as juniors and seniors.

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Acknowledgments
Many people have helped me to produce this book. I would like to thank Leslie Friesen who drew all of the figures for the
two editions of this book and responded to numerous suggestions that I have made of how the figures could be improved.
Special thanks is also due Ken Hicks at Ohio University who suggested that I use MATLAB to solve the problems that arise
in modern physics and provided many of the MATLAB exercises and problems in the text. Ken wrote the first draft of the
appendix on MATLAB. I would also like to thank Thomas Ericsson of the Mathematics Department of Göteborg University
for bringing our MATLAB exercises and problems up to the level of modern books on mathematics and Geoffrey Lentner
of the Department of Physics and Astronomy at University of Louisville for helping me with Octave. I appreciate the kind
help Charlotte Fischer provided me so that our applet could take advantage of all of the special features of her atomic

Hartree-Fock program and the work of Simon Rochester who wrote the current version of our Hatree-Fock applet.
I would like to express my appreciation for the help I have received from many physicists who are at the forefront of
their research areas and have helped me during the course of producing this book. In the area of condensed matter physics, I
would like to thank Jim Davenport, Dick Watson, and Vic Emory for their hospitality in the Condensed Matter Theory Group
of Brookhaven National Laboratory during the summer when I wrote my first draft of the solid-state chapter. I appreciate
the guidance of John Wilkins of Ohio State University, who has served as the Chair of the condensed matter section of the
American Physical Society. In the area of particle physics, I would like to thank Keith Ellis and Chris Quigg at Fermilab,
William Palmer at Ohio State University, and Howard Georgi, who allowed me to attend his class on group theory and
particle physics at Harvard University.
Several well-known physicists have distinguished themselves not only for their research but also for their teaching and
writing. I would like to thank Dirk Walecka at College of William and Mary, Dick Furnstahl at Ohio State University,
and I would like to thank Thomas Moore at Pomona College whose writings on elementary physics have been a source of
inspiration for me.
This book evolved over a number of years and several of the early reviewers of the manuscript played an important
role in its development. For their ideas and guidance, I would like to thank Massimilliano Galeazzi at University of Miami,
Amitabh Lath at SUNY Rutgers, and Mike Santos and Michael Morrison at University of Oklahoma.
Finally, I would like to thank the teachers of modern physics, who have sent me valuable suggestions and extended
to me their hospitality when I have visited their university. Thanks are due to Jay Tang at Brown University; W. Andreas
Schroeder at University of Illinois, Chicago; Roger Bengtson at University of Texas; Michael Jura at UCLA; Dmitry Budker
at University of California Berkeley; Paul Dixon at California State San Bernadino; Murtadha Khakoo at California State
Fullerton; Charlotte Elster at Ohio University; Ronald Reifenberger at Purdue University; Michael Schulz at University
of Missouri, Rolla; Bill Skocpol at Boston University; David Jasnow at University of Pittsburgh; Sabine Lammers, Lisa
Kaufman, and Jon Urheim at Indiana University; Connie Roth and Fereydoon Family at Emery University; David Maurer
at Auburn University; Xuan Gao and Peter Kerman at Case Western Reserve; and Cheng Cen and Earl Scime at University
of West Virginia. I would also like to thank Lee Larson, Dave Brown, Chris Davis, Humberto Gutierrez, Christian Tate,
Kyle Stephen, and Joseph Brock at University of Louisville; Matania Ben-Atrzi at Hebrew University in Jerusalem; Ramzi
Rihan, Aziz Shawabka, Henry Jagaman, Wael Qaran, and Wafaa Khater at Birzeit University in Ramallah; and Jacob Katriel
at Technion University in Haifa.
I welcome the suggestions and the questions of any teacher who takes to the phone or keyboard and wants to talk about
a particular topic.

Louisville, Kentucky
John C. Morrison
October 21, 2014


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Introduction
Every physical system can be characterized by its size and the length of time it takes for processes occurring within it to evolve.
This is as true of the distribution of electrons circulating about the nucleus of an atom as it is of a chain of mountains rising up
over the ages.

Modern physics is a rich field including decisive experiments conducted in the early part of the twentieth century and more
recent research that has given us a deeper understanding of fundamental processes in nature. In conjunction with our growing
understanding of the physical world, a burgeoning technology has led to the development of lasers, solid-state devices, and
many other innovations. This book provides an introduction to the fundamental ideas of modern physics and to the various
fields of contemporary physics in which discoveries and innovation are going on continuously.

I.1 THE CONCEPTS OF PARTICLES AND WAVES
While some of the ideas currently used to describe microscopic systems differ considerably from the ideas of classical
physics, other important ideas are classical in origin. We begin this chapter by discussing the important concepts of a
particle and a wave which have the same meaning in classical and modern physics. A particle is an object with a definite
mass concentrated at a single location in space, while a wave is a disturbance that propagates through space. The first
section of this chapter, which discusses the elementary properties of particles and waves, provides a review of some of the
fundamental ideas of classical physics. Other elements of classical physics will be reviewed later in the context for which
they are important. The second section of this chapter describes some of the central ideas of modern quantum physics and
also discusses the size and time scales of the physical systems considered in this book.


I.1.1

The Variables of a Moving Particle

The position and velocity vectors of a particle are illustrated in Fig. I.1. The position vector r extends from the origin to the
particle, while the velocity vector v points in the direction of the particle’s motion. Other variables, which are appropriate
for describing a moving particle, can be defined in terms of these elementary variables.
The momentum p of the particle is equal to the product of the mass and velocity v of the particle
p = mv.
We shall find that the momentum is useful for describing the motion of electrons in an extended system such as a crystal.
The motion of a particle moving about a center of force can be described using the angular momentum, which is defined
to be the cross product of the position and momentum vectors
= r × p.
The cross product of two vectors is a vector having a magnitude equal to the product of the magnitudes of the two
vectors times the sine of the angle between them. Denoting the angle between the momentum and position vectors by θ as
in Fig. I.1, the magnitude of the angular momentum vector momentum can be written
| | = |r| |p| sin θ.
This expression for the angular momentum may be written more simply in terms of the distance between the line of
motion of the particle and the origin, which is denoted by r0 in Fig. I.1. We have
| | = r0 |p|.
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xviii

Introduction


v

q

r

O

m
r0

FIGURE I.1 The position r and the velocity v of a moving particle of mass m. The point O denotes the origin, and r0 denotes the distance between the
line of motion and the origin.

The angular momentum is thus equal to the distance between the line of motion of the particle and the origin times the
momentum of the particle. The direction of the angular momentum vector is generally taken to be normal to the plane of the
particle’s motion. For a classical particle moving under the influence of a central force, the angular momentum is conserved.
The angular momentum will be used in later chapters to describe the motion of electrons about the nucleus of an atom.
The kinetic energy of a particle with mass m and velocity v is defined by the equation
KE =

1 2
mv ,
2

where v is the magnitude of the velocity or the speed of the particle. The concept of potential energy is useful for describing
the motion of particles under the influence of conservative forces. In order to define the potential energy of a particle, we
choose a point of reference denoted by R. The potential energy of a particle at a point P is defined as the negative of the work
carried out on the particle by the force field as the particle moves from R to P. For a one-dimensional problem described by
a variable x, the definition of the potential energy can be written

VP = −

P

F(x)dx.

(I.1)

R

As a first example of how the potential energy is defined we consider the harmonic oscillator illustrated in Fig. I.2(a).
The harmonic oscillator consists of a body of mass m moving under the influence of a linear restoring force
F = −kx,

(I.2)

where x denotes the distance of the body from its equilibrium position. The constant k, which occurs in Eq. (I.2), is called
the force constant. The restoring force is proportional to the displacement of the body and points in the direction opposite to
the displacement. If the body is displaced to the right, for instance, the restoring force points to the left. It is natural to take
the reference position R in the definition of the potential energy of the oscillator to be the equilibrium position for which
x = 0. The definition of the potential energy (I.1) then becomes
x

V(x) = −

(−kx )dx =

0

1 2

kx .
2

(I.3)

Here x is used within the integration in place of x to distinguish the variable of integration from the limit of integration.
If one were to pull the mass shown in Fig. I.2(a) from its equilibrium position and release it, the mass would oscillate
with a frequency independent of the initial displacement. The angular frequency of the oscillator is related to the force
constant of the oscillator and the mass of the particle by the equation
ω=

k/m.

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Introduction

xix

V(x) = ½m w 2 x 2
m
k

(a)

O

x


x

(b)

FIGURE I.2 (a) A simple harmonic oscillator in which a mass m is displaced a distance x from its equilibrium position. The mass is attracted toward its
equilibrium position by a linear restoring force with force constant k. (b) The potential energy function for a simple harmonic oscillator.

or
k = mω2 .
Substituting this expression for k into Eq. (I.3), we obtain the following expression for the potential energy of the
oscillator
1
V(x) = mω2 x2 .
(I.4)
2
The oscillator potential is illustrated in Fig. I.2(b). The harmonic oscillator provides a useful model for a number of
important problems in physics. It may be used, for instance, to describe the vibration of the atoms in a crystal about their
equilibrium positions.
As a further example of potential energy, we consider the potential energy of a particle with electric charge q moving
under the influence of a charge Q. According to Coulomb’s law, the electromagnetic force between the two charges is
equal to
F=

1
4π

Qq
,
2
0 r


where r is the distance between the two charges and 0 is the permittivity of free space. The reference point for the potential
energy for this problem can be conveniently chosen to be at infinity where r = ∞ and the force is equal to zero. Using
Eq. (I.1), the potential energy of the particle with charge q at a distance r from the charge Q can be written
V(r) = −

Qq
4π 0

r

1

dr .
r2
Evaluating the above integral, one finds that the potential energy of the particle is
V(r) =

∞

Qq 1
.
4π 0 r

An application of this last formula will arise when we consider the motion of electrons in an atom. For an electron with
charge −e moving in the field of an atomic nucleus having Z protons and hence a nuclear charge of Ze, the formula for the
potential energy becomes
V(r) = −

Ze2 1

.
4π 0 r

(I.5)

The energy of a body is defined to be the sum of its kinetic and potential energies
E = KE + V.
For an object moving under the influence of a conservative force, the energy is a constant of the motion.

I.1.2

Elementary Properties of Waves

We consider now some of the elementary properties of waves. Various kinds of waves arise in classical physics, and we
shall encounter other examples of wave motion when we apply the new quantum theory to microscopic systems.

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xx

Introduction

Traveling Waves
If one end of a stretched string is moved abruptly up and down, a pulse will move along the string as shown in Fig. I.3(a).
A typical element of the string will move up and then down as the pulse passes. If instead the end of the string moves up
and down with the time dependence,
y = sin ωt,
an extended sinusoidal wave will travel along the string as shown in Fig. I.3(b). A wave of this kind which moves up and
down with the dependence of a sine or cosine is called a harmonic wave.

The wavelength of a harmonic wave will be denoted by λ and the speed of the wave by v. The wavelength is the distance
from one wave crest to the next. As the wave moves, a particular element of the string which is at the top of a crest will move
down as the trough approaches and then move back up again with the next crest. Each element of the string oscillates up
and down with a period, T. The frequency of oscillation f is equal to 1/T. The period can also be thought of as the time for a
crest to move a distance of one wavelength. Thus, the wavelength, wave speed, and period are related in the following way:
λ = vT.
Using the relation, T = 1/f , this equation can be written
λf = v.

(I.6)

The dependence of a harmonic wave upon the space and time coordinates can be represented mathematically using
the trigonometric sine or cosine functions. We consider first a harmonic wave moving along the x-axis for which the
displacement is
y(x, t) = A sin[2π(x/λ − t/T)],

(I.7)

where A is the amplitude of the oscillation. One can see immediately that as the variable x in the sine function increases by
an amount λ or the time increases by an amount T, the argument of the sine will change by an amount 2π , and the function
y(x, t) will go through a full oscillation. It is convenient to describe the wave by the angular wave number,
k=

2π
,
λ

(I.8)

ω=


2π
.
T

(I.9)

and the angular frequency,

Using the relation, T = 1/f , the second of these two equations can also be written
ω = 2π f .

(I.10)

The angular wave number k, which is defined by Eq. (I.8), has SI units of radians per meter, while ω, which is defined
by Eq. (I.9), has SI units of radians per second. Using Eqs. (I.8) and (I.9), the wave function (I.7) can be written simply
y(x, t) = A sin(kx − ωt).

(I.11)

y

y
v

v
x

x


(a)

(b)

FIGURE I.3 (a) A pulse moving with velocity v along a stretched string. (b) An extended sinusoidal wave moving along a string.

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Introduction

xxi

Equation (I.11) describes a traveling wave. We can see this by considering the crest of the wave where the value of the
phase of the sine function in Eq. (I.11) is equal to π/2. The location of the crest is given by the equation
kxcrest − ωt =

π
.
2

Solving this last equation for xcrest , we get
xcrest − =

ωt
π
+ .
k
2k


An expression for the velocity of the wave crest can be obtained by taking the derivative of xcrest with respect to time
to obtain
v=

dxcrest
ω
= .
dt
k

(I.12)

Equation (I.11) thus describes a sinusoidal wave moving in the positive x-direction with a velocity of ω/k. Equation (I.12)
relating the velocity of the wave to the angular wave number k and angular frequency ω can also be obtained by solving
Eq. (I.8) for λ and solving Eq. (I.10) for f . Equation (I.12) is then obtained by substituting these expressions for λ and f
into Eq. (I.6).
Using the same approach as that used to understand the significance of Eq. (I.11), one can show that
y(x, t) = A sin(kx + ωt)

(I.13)

describes a sinusoidal wave moving in the negative x-direction with a velocity of ω/k.
Figure I.4(a) illustrates how the harmonic function (I.11) varies with position at a fixed time chosen to be t = 0. Setting
t equal to zero, Eq. (I.11) becomes
y(x, 0) = A sin kx.

(I.14)

The wave described by the function A sin x and illustrated in Fig. I.4(a) does not depend upon the time. Such a wave,
which is described by its dependence upon a spatial coordinate, is called a stationary wave. As for the traveling wave (I.11),

the angular wave number k is related to the wavelength by Eq. (I.8). Similarly, Fig. I.4(b) shows how the function (I.11
varies with time at a fixed position chosen to be x = 0. Setting x equal to zero in Eq. (I.11) and using the fact that the sine
is an odd function, we obtain
y(0, t) = −A sin ωt.

(I.15)

The wave function (I.15) oscillates as the time increases with an angular frequency ω given by Eq. (I.10).
y

l

x

(a)
y
T

t

(b)
FIGURE I.4 (a) The x-dependence of the sinusoidal function for a fixed time t = 0. (b) The time dependence of the sinusoidal function at the fixed
point x = 0.

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xxii

Introduction


Standing Waves
Suppose two waves travel simultaneously along the same stretched string. Let y1 (x, t) and y2 (x, t) be the displacements of
the string due to the two waves individually. The total displacement of the string is then
y(x, t) = y1 (x, t) + y2 (x, t).
This is called the principle of superposition. The displacement due to two waves is generally the algebraic sum of the
displacements due to the two waves separately. Waves that obey the superposition principle are called linear waves and
waves that do not are called nonlinear waves. It is found experimentally that most of the waves encountered in nature obey
the superposition principle. Shock waves produced by an explosion or a jet moving at supersonic speeds are uncommon
examples of waves that do not obey the superposition principle. In this text, only linear waves will be considered. Two
harmonic waves reinforce each other or cancel depending upon whether or not they are in phase (in step) with each other.
This phenomena of reinforcement or cancelation is called interference.
We consider now two harmonic waves with the same wavelength and frequency moving in opposite directions along the
string. The two waves having equal amplitudes are described by the wave functions
y1 (x) = A sin(kx − ωt)
and
y2 (x) = A sin(kx + ωt).
According to the principle of superposition, the combined wave is described by the wave function
y(x, t) = y1 (x) + y2 (x) = A [sin(kx − ωt) + sin(kx + ωt)] .

(I.16)

Using the trigonometric identity,
sin(A ± B) = sin A cos B ± cos A sin B,

(I.17)

u(x, t) = [2A cos ωt] sin kx.

(I.18)


Eq. (I.16) may be written

This function describes a standing wave.
At a particular time, the quantity within square brackets in Eq. (I.18) has a constant value and may be thought of as the
amplitude of the wave. The amplitude function 2A cos ωt varies with time having both positive and negative values. The
function sin kx has the spatial form illustrated in Fig. I.5 being zero at the points satisfying the equation
kx = nπ ,

for n = 0, 1, 2, . . . .

Substituting k = 2π/λ into this equation, we get
λ
x = n , for n = 0, 1, 2, . . . .
2
The function sin kx is thus equal to zero at points separated by half a wavelength. At these points, which are called nodes,
the lateral displacement is always equal to zero. An example of a standing wave is provided by the vibrating strings of a
guitar. The ends of the guitar strings are fixed and cannot move. In addition to the ends of the strings, other points along
the strings separated by half a wavelength have zero displacements. We shall find many examples of traveling and standing
waves later in the book when we consider microscopic systems.
One can gain an intuitive understanding of the properties of waves by using the PhET simulation package developed at
the University of Colorado. The simulations can be found at the Web site: phet.colorado.edu/en/simulations. Choosing the

x

FIGURE I.5 Function describing the spatial form of a standing wave. The nodes, which have zero displacement, are represented by dots.

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Introduction

xxiii

FIGURE I.6 A simulation of wave interference in the PhET simulation package developed at the University of Colorado.

categories “physics” and “sound and waves,” one can initiate the simulation called “wave interference.” Choosing the tab
“water” and the option “one drip,” one sees the waves spreading across a body of water when drips from a single faucet
strike the water surface. Choosing then the option “two drips,” one sees the waves, produced by the drips of two faucets
striking the water surface. This figure is shown in Fig. I.6. As we have just described the waves from the two disturbances
add together and destructively interfere to produce a complex disturbance on the surface of the water. One can observe
similar effects with sound and light waves by choosing the tabs “sound” and “light.”

The Fourier Theorem
We have thus far considered sinusoidal waves on a string and would now like to consider wave phenomenon when the shape
of the initial disturbance is not sinusoidal. In the decade of the 1920s, Jean Baptiste Fourier showed that any reasonably
continuous function f (x), which is defined in the interval 0 ≤ x < L, can be represented by a series of sinusoidal waves
f (x) =

Sn sin nkx,

for 0 ≤ x ≤ L,

(I.19)

n=1,2,...

where k = 2π/L and
Sn =


L

2
L

sin nkxf (x)dx.

(I.20)

0

A sketch of the derivation of Eq. (I.20) is given in Problem 4.
As an example, we consider a square wave
f (x) =

−A if 0 ≤ x < 12 L
+A if 12 L < x ≤ L

(I.21)

Using Eqs. (I.19) and (I.20), the square wave (I.21) can be shown to be equal to the following infinite sum of sinusoidal
waves
f (x) = −A

4
π

sin kx +

1

1
sin 3kx + sin 5kx + · · ·
3
5

where k is the angular wave number of the fundamental mode of vibration.

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,

(I.22)


xxiv

Introduction

FIGURE I.7 A representation of a square wave function formed by adding harmonic waves together using the PhET simulation package developed at the
University of Colorado.

We can gain some insight into how harmonic waves combine to form the square wave function (I.21) by using the simulation package “Fourier: Making Waves” at the Web site: />A reproduction of the window that comes up is shown in Fig. I.7. With “Preset Function” set to “sine/cosine” and “Graph
controls” set at “Function of: space (x)” and “sin,” one can begin by setting A1 = 1 and A3 = 0.33 and then gradually adding
A5 = 0.20, A7 = 0.14, A9 = 0.11, and A11 = 0.09. As one adds more and more sine functions of higher frequency, the sum
of the waves shown in the lower screen becomes more and more like a square wave. One can understand in qualitative terms
how the harmonic waves add up to produce the square wave. Using the window reproduced by Fig. I.7, one can view each
sinusoidal wave by setting the amplitude of the wave equal to one and all other amplitudes equal to zero. The amplitude A1
corresponds to the fundamental wave for which a single wavelength stretches over the whole region. This sinusoidal wave—
like the square wave—is zero at the center of the region and assumes negative values to the left of center and positive values
to the right of center. The sinusoidal waves with amplitudes A3 , A5 , A7 , A9 , and A11 all have these same properties but being

sinusoidal waves of higher frequencies they rise more rapidly from zero as one moves to the right from the center of the
region. By adding waves with higher frequencies to the fundamental wave, one produces a wave which rises more rapidly
as one moves to the right from center and declines more rapidly as one moves to the left from center; however, the sum of
the waves oscillate with a higher frequency than the fundamental frequency in the region to the right and left of center. As
one adds more and more waves, the oscillations due to the various waves of high frequency destructively interfere and one
obtains the square wave.
The above result can also be obtained using the MATLAB software package. A short introduction to MATLAB can
be found in Appendix C and a more extensive presentation in Appendix CC. MATLAB Program I.1 given below adds
sinusoidal waves up to the fifth harmonic. The first three lines of the program define the values of A, L, and k, and the next
line defines a vector x with elements between −L/2, and +L/2 with equal steps of L/100. The plot of x versus y produced by
this MATLAB program is shown in Fig. I.8. This figure is very similar to Fig. I.7 produced by the PhET simulation package.
MATLAB Program I.1
This program adds the Fourier components up to the fifth harmonic to produce a square wave of amplitude 1.0 and width 1.0.
A=1;
L=1;
k=2*pi/L;
x = -L/2 : L/100 : L/2;
y = (A*4/pi)*( sin(k*x)+(1/3)*sin(3*k*x)+(1/5)*sin(5*k*x) );
plot(x,y)

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Introduction

xxv

1.5
1.0
0.5

0
−0.5
−1.0

−0.5 −0.4 −0.3 −0.2 −0.1

0

0.1

0.2

0.3

0.4

0.5

FIGURE I.8 Plot produced by MATLAB Program I.1.

The Fourier theorem has wide-ranging consequences. No matter what the shape of a disturbance, one can think of the
disturbance as being a sum of harmonic waves.

Representation of Waves Using Exponentials
It is often convenient to represent waves using exponential functions. For instance, a stationary wave can be described by
the function
ψ(x) = Aeikx ,

(I.23)


where the constant A is a real number. The function ψ(x) can be resolved into its real and imaginary parts using Euler’s
equation,
eiθ = cos θ + i sin θ,

(I.24)

to obtain
ψ(x) = A cos(kx) + iA sin(kx).
Notice that the imaginary part of the function ψ(x) is equal to A sin(kx). This function corresponds to the stationary wave
shown in Fig. I.4(a). Similarly, the real part of the function ψ(x) is equal to the function A cos(kx) which can be obtained
by shifting the function shown in Fig. I.4(a) to the left by an amount π/2.
A traveling wave can be described by the exponential function
ψ(x, t) = Aei(kx−ωt) .

(I.25)

Using Euler’s equation (I.24), one may readily show that the imaginary part of the right-hand side of this last equation
is equal to the sinusoidal function appearing in Eq. (I.11).
The exponential function has mathematical properties which makes it more convenient to use than the trigonometric
functions. For instance, the product of an exponential function eA and a second exponential function eB can be evaluated by
simply adding up the exponents
eA · eB = eA+B .
We now consider stationary waves for which the direction in which the value of the function changes most rapidly does
not coincide with the x-direction, and we consider traveling waves moving in other directions than the positive and negative
x-directions. Imagine that in a particular region of space, we identify a point where the wave function has a local maximum
and we identify other points near our original point that are also local maxima. A surface passing through these points is
called a wave front. We denote by k a vector pointing in a direction perpendicular to the wave fronts with a magnitude
|k| =

2π

.
λ

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xxvi

Introduction

k

Wave front

x
r

q

FIGURE I.9 The wave vector k and the position vector r for a wave traveling in three dimensions is shown together with a particular wave front. The
angle θ appearing in Eq. (I.26) is also shown.

The magnitude of k will be denoted by k. The vector k, which is called the wave vector, is shown together with a position
vector r and a particular wave front in Fig. I.9. The scalar product k · r can be written
k · r = |k| |r| cos θ.

(I.26)

Notice that the quantity |r| cos θ shown in Fig. I.9 is the projection of the position vector r upon the direction of the
vector k. All of the points on a wave front correspond to the same value of |r| cos θ . The quantity k · r is the product of k

and the distance to a wave front measured along the vector k. Hence, k · r plays the same role as kx does for waves in one
dimension. The wave function for a stationary wave in three dimensions can be written
ψ(r) = Aeik·r .
Similarly, a traveling wave in three dimensions can be described by the function
ψ(r) = Aei(k·r−ωt) .
The wave vector k is perpendicular to the wave fronts pointing in the direction the wave propagates.

I.1.3

Interference and Diffraction Phenomena

The variation of amplitude and intensity that occur when waves encounter a physical barrier can be understood using
Huygens’ principle, which states that each point on a wave front may be considered as a source of secondary waves. The
position of the wave front at a later time can be found by superimposing these secondary waves. Waves emitted by the wave
front thus serve to regenerate the wave and enable us to analyze its propagation in space. This is illustrated in Fig. I.10.
Wave front formed
from secondary waves

Secondary
waves

Waves emitted
by source S

S

FIGURE I.10 Huygens’ principle states that every point on a wave front may be considered as a source of secondary waves.

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Introduction

xxvii

B
P
S

P0

h
H
A q
L1

h sin θ

L2

FIGURE I.11 Interference produced by two slits.

The word interference is used to describe the superposition of two waves, while diffraction is interference produced by
several waves. For both interference and diffraction phenomena, Huygens’ principle enables us to reconstruct subsequent
wave fronts and to calculate the resulting intensities.
A good example of interference effects is provided by the two slit interference experiment shown schematically in
Fig. I.11. In the experiment, the light source S, which lies in the focal plane of the lens L1 , produces a beam of parallel rays
falling perpendicularly upon the plane containing the double slit. The interference of secondary waves emitted by the two
apertures leads to a variation of the intensity of the transmitted light in the secondary focal plane of the lens L2 . Whether or
not constructive interference occurs at the point P depends upon whether the number of waves along the upper path (BP)

shown in Fig. I.11 differs from the number of waves along the lower segment (AP) by an integral number of wavelengths.
The difference in the length of the two paths is equal to the length of the segment AH. If we denote the distance between
the two slits by h, then the length of AH is equal to h sin θ, and the condition for constructive interference is
h sin θ = nλ.

(I.27)

Constructive interference occurs when the difference in path lengths is equal to an integral number of wave lengths.
The intensity distribution of the light incident upon the screen at the right is illustrated in Fig. I.12(a). A photograph of the
interference pattern produced by a double slit is shown in Fig. I.12(b).
The two slit interference experiment which we have discussed clearly illustrates the ideas of constructive and destructive
interference. The bright fringes produced in the experiment corresponds to angles at which light traveling through the two
slits arrive at the focal plane of the second lens in phase with each other, while the dark fringes correspond to angles for
which the distance traveled by light from the two slits differ by an odd number of half wavelengths and the light destructively
interferes.
An optical grating can be made by forming a large number of parallel equidistant slits. A grating of this kind is illustrated
in Fig. I.13. As in the case of a double slit, intensity maxima can be observed in the focal plane of the lens L2 . The brightest
maxima occur at points corresponding to the values of θ satisfying Eq. (I.27) where h here represents the distance between
the centers of neighboring slits. At such points, light from all of the different slits arrive with the same phase. Equation (I.27)
thus gives all of the angles for which constructive interference occurs for the double slit interference experiment and the
angles for which the principal maxima occur for a grating. For a grating, however, a large number of secondary maxima
occur separated by a corresponding number of secondary minima. The gratings used in modern spectroscopic experiments
consist typically of aluminum, silver-coated, or glass plates which have thin lines ruled on them by a fine diamond needle.
A grating having several hundred thousand lines produces a number of narrow bright lines on a dark background, each line
corresponding to a different value of n in Eq. (I.27). Using this equation and the measured angles of the maxima, one may
readily calculate the frequency of the incident light.

Electromagnetic Waves
The wave model may be used to describe the propagation of electromagnetic radiation. The frequencies and wavelengths of
the most important forms of electromagnetic radiation are shown in Fig. I.14. The human eye can perceive electromagnetic

radiation (light) with wavelengths between 400 and 700 nm (that is between 400 × 10−9 and 700 × 10−9 m). The
wavelength of light is also commonly given in angstrom. One angstrom (Å) is equal to 1 × 10−10 m or one-tenth of a
nanometer. When wavelengths are expressed in angstrom, the wavelength of visible light is between 4000 and 7000 Å.

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Introduction

I

sin q
– 2l
—
h

(a)

λ
–—
h

0

λ
+—
h


+ 2λ
—
h

(b)
FIGURE I.12 (a) Intensity distribution produced on a screen by light passing through a double slit. (b) A photograph of an interference pattern produced
by two slits.

P

S
q
L1
L2
FIGURE I.13 Diffraction by a grating.

Ultraviolet light, X-rays, and γ-rays have wavelengths which are shorter than the wavelength of visible light, while infrared
light, microwaves, and radio waves have wavelengths which are longer. We shall denote the speed of light in a vacuum by c.
Substituting c for v in Eq. (I.6), we have
λf = c.

(I.28)

Solving this equation for f , we get
c
.
(I.29)
λ
According to Eq. (I.29), the frequency and the wavelength of light are inversely related to each other. If the wavelength
increases, for instance, the frequency will decrease. The speed of light is given together with other physical constants in

f =

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Introduction

700

600

Red

500

Orange
Green
Yellow

xxix

400

Blue

Violet

Visible spectrum
Wavelength (m)
107


105

103

Long waves
102

104

10 −1

10

10 −3

Radio waves
106

108

10 −5

10 −7

Infrared
1010

1012


1014

10 −9

10 −11

Ultraviolet X-rays
1016

1018

10 −13

10 −15

Gamma rays
1020

1022

1024

Frequency (Hz)

FIGURE I.14 The frequencies and wavelengths of electromagnetic radiation.

TABLE I.1 Wavelength (λ)
and Frequency (f ) of Light
Color


λ (nm)

f (Hz)

Red

700

4.28 × 1014

Violet

400

7.49 × 1014

Appendix A. Using Eq. (I.29) and the value of c given in this appendix, one may easily obtain the values of the frequency
given in Table I.1.
The unit of one cycle per second (s−1 ) is referred to as a Hertz and abbreviated Hz. In the SI system of units, 103 is
denoted by kilo (k), 106 is denoted by mega (M), 109 is denoted by giga (G), and 1012 is denoted by tera (T). Red light thus
has a frequency of 428 THz and violet light has frequency of 749 THz.

Example I.1
Calculate the wavelength of electromagnetic radiation having a frequency of f = 100 MHz = 100 × 106 s−1 .
Solution
Solving Eq. (I.28) for λ, we obtain
λ=

c
.

f

We then substitute the value of c given in Appendix A and f = 100 × 106 s−1 into this last equation to obtain
λ=

2.998 × 108 m/s
= 2.998 m.
100 × 106 s−1

The radiation, which has a wavelength of about 3 m, corresponds to radio waves.

I.2 AN OVERVIEW OF QUANTUM PHYSICS
Microscopic systems differ in a number of ways from macroscopic systems for which the laws of classical physics apply.
One of the most striking new features of physical systems on a microscopic level is that they display a wave-particle duality.
Certain phenomena can be understood by considering radiation or matter as consisting of particles, while other phenomena
demand that we think of radiation or matter as consisting of waves.

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Introduction

At the beginning of the twentieth century, electromagnetic radiation was thought of as a continuous quantity described by
waves, while it was thought that matter could be resolved into constituent particles. The first evidence that electromagnetic
radiation had a discrete quality appeared in 1900 when Max Planck succeeded in explaining the radiation field within a
cavity. In his theory, Planck assumed that the electromagnetic field interchanged energy with the walls of the cavity in
integral multiples of hf where h is a physical constant now called Planck’s constant and f is the frequency of the radiation.
While Planck was careful to confine his assumption to the way the radiation field exchanges energy with its environment,

Albert Einstein broke entirely with the tenets of classical physics 5 years later when he proposed a theory of the photoelectric
effect. The photoelectric effect refers to the emission of electrons by a metal surface when light is incident upon the surface.
Einstein was able to explain the observed features of the photoelectric effect by supposing that the radiation field associated
with the incident light consisted of quanta of energy. In keeping with the earlier work of Planck, Einstein supposed that
these quanta have an energy
E = hf .

(I.30)

The theory of the radiation field developed by Planck will be described in Chapter 7, while the theory of the photoelectric
effect of Einstein will be described in Chapter 1.
The theories of Planck and Einstein, which have since been confirmed by experiment, were the first indication that
electromagnetic radiation has a dual wave-particle quality. While the interference and diffraction phenomena discussed in
the previous section require that we think of light as consisting of electromagnetic waves, the phenomena associated with
the absorption, and emission of radiation demand that we think of light as consisting of quanta of energy, which we call
photons. In 1923, Louis de Broglie suggested that just as light has both a wave and a particle character, the objects we
think of as particles should also display a wave-particle dualism. This remarkable suggestion, which placed the theories
of radiation and matter on the same footing, has since been confirmed by experiment. While a beam of electrons passing
through a magnetic field is deflected in the way charged particles would be deflected, a beam of electrons, which is reflected
by the planes of atoms within a crystal, displays the same interference patterns that we would associate with waves. The
electron and the particles that constitute the atomic nucleus all have this dual wave-particle character. The theory of de
Broglie and experiments that confirm his theory are described in Chapter 1.
The dual nature of waves and particles determines to a considerable extent the mathematical form of modern theories.
The distinctive feature of modern theories is that they are formulated in terms of probabilities. The equations of modern
quantum theory are not generally used to predict with certainty the outcome of an observation but rather the probability of
obtaining a particular possible result. To give some idea of how the concept of probability arises from the wave-particle
dualism, we consider again the interference experiment shown in Fig. I.11. In this experiment, light is incident upon the two
slits shown in the figure and an interference pattern is formed on the screen to the right. As we have seen, the intensity pattern
produced on the screen, which is shown in Fig. I.12(a), can be interpreted in terms of the interference of secondary waves
emitted by the two slits. This intensity pattern is predicted unequivocally by classical optics. The concept of probability

enters the picture when we consider the interference experiment from the particle point of view. The beam of light can be
thought of not only as a superposition of waves but also as a stream of photons. If the screen were made of a light-sensitive
material and the intensity of the light were sufficiently low, the impact of each photon could be recorded. The cumulative
effect of all of the photons passing through the slits and striking the screen would produce the effect illustrated in Fig. I.12(a).
Each photon has an equal probability of passing through either slit. The density of the image produced at a particular point
on the screen is proportional to the probability that a photon would strike the screen at that point. We are thus led to use
the concept of probability not due to any shortcoming of classical optics, but due to the fact that the incident light can be
described both as waves and as particles.
Ideas involving probability play an important role in our description of all microscopic systems. To show how this occurs,
we shall conduct a thought experiment on a collection of hydrogen atoms. Hydrogen is the lightest and simplest atom with a
single electron moving about a nucleus. Imagine we have a sensitive camera which can record the position of the electron of
a hydrogen atom on a photographic plate. If we were to take a large number of pictures of the electrons in different hydrogen
atoms superimposed on a single photographic plate, we would get a picture similar to that shown in Fig. I.15 in which the
hydrogen nucleus is surrounded by a cloud. The density of the cloud at each point in space is related to the probability
of finding an electron at that point. In modern quantum theory, the electron is described by a wave function ψ which is
a solution of an equation called the Schrödinger equation. The probability of finding the electron at any point in space is
proportional to the absolute value squared of the wave function |ψ|2 . The situation is entirely analogous to the two-slit
interference experiment we have just considered. The photon in the interference pattern and the electrons surrounding the

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Introduction

xxxi

FIGURE I.15 Cloud surrounding the hydrogen nucleus. The density of the cloud is related to the probability of finding an electron.

nucleus of an atom are both particles corresponding to waves which are accurately described by the theory. The concept of
probability is required to describe the position of a particle that corresponds to a wave and is thus due to the wave-particle

nature of photons and electrons.
The electron in the hydrogen atom moves about a nucleus which has a single proton with a positive charge. As we shall
see in Chapter 4, the radius of the cloud surrounding the hydrogen nucleus is equal to a0 = 0.529 Å or 0.529 × 10−10 m, and
the diameter of the cloud is thus approximately 1 Å or one-tenth of a nanometer. Atoms of helium and lithium, which follow
hydrogen in the Periodic Table, have two and three electrons, respectively, and their nuclei have corresponding numbers of
protons. Since electrons have a negative charge, the cloud surrounding the nucleus of an atom can be interpreted as a charge
cloud. As one moves from one atom to the next along a row of the Periodic Table, the nuclear charge increases by one
and an additional electron is added to the charge cloud. The positive electric charge of atomic nuclei attracts the negatively
charged electrons and draws the electron cloud in toward the nucleus. For this reason, the size of atoms increases only very
slowly as the number of electrons increases. Xenon which has 56 electron is only two to three times larger than the helium
atom which has two electrons.
Atoms and nuclei emit and absorb radiation in making transitions from one state to another. The basic principles of
radiative transitions were given by Niels Bohr in 1913. Bohr proposed that an atom has stationary states in which it has
well-defined values of the energy and that it emits or absorbs a photon of light when it makes a transition from one state to
another. In an emission process, the atom makes a transition to a state in which it has less energy and emits a single photon,
while in an absorption process, the atom absorbs a photon and makes a transition to a state in which it has more energy.
These ideas have been found to apply generally to molecules and nuclei as well.
The energies of atoms depend upon the nature of the electron charge cloud. These energies can conveniently be expressed
in electron volts (eV). One electron volt is the kinetic energy an electron would have after being accelerated through a
potential difference of 1 V. The energy of a photon that has been emitted or absorbed is given by Eq. (I.30), which is due
to Planck and Einstein. An expression for the energy of a photon in terms of the wavelength of light can be obtained by
substituting the expression for the frequency provided by Eq. (I.29) into Eq. (I.30) giving
hc
E= .
(I.31)
λ
Using the values of Planck’s constant and the speed of light given in Appendix A, one may easily show that the product
of the constants appearing in the above equation is
hc = 1240 eV nm.
This is a good number to remember for later reference.


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(I.32)


xxxii

Introduction

Example I.2
Calculate the energy of the photons for light having a wavelength λ = 400 nm.
Solution
Substituting the wavelength λ of the light and the product hc given by Eq. (I.32) into Eq. (I.31) gives
E=

1240 eV nm
= 3.1 eV.
400 nm

The light may thus be thought of as consisting of photons having an energy of 3.1 eV.

Equation (I.32) for the product hc is convenient for visible light and for atomic transitions. The transitions made by nuclei
typically involve millions of electron volts (MeV) of energy. For problems involving nuclear radiation, it is convenient to
write the product of the constants h and c as follows:
hc = 1240 MeV fm,

(I.33)

which expresses hc in terms of MeV and Fermi (fm). One MeV is equal to 106 eV, and one Fermi (fm), which is the

approximate size of an atomic nucleus, is equal to 10−15 m or 10−6 × 10−9 m. The unit MeV appearing in Eq. (I.33) is thus
one million times larger than the unit eV appearing in Eq. (I.32), while the unit fm appearing in Eq. (I.33) is one million
times smaller than the unit nm appearing in Eq. (I.32).
The following table, which gives a few typical photon energies, is arranged according to decreasing wavelength or
increasing photon energy.
The photons of visible light have energies of a few eV. Red light with a wavelength of 700 nm has the least energetic
photons and violet light with a wavelength of 400 nm has the most energetic photons in the visible region. We shall find
that the outer electrons of an atom are bound to the atom by a few electron volts of energy. The third row of Table I.2 gives
the photon energy for light with a wavelength of 1 Å, which is a typical distance separating the atoms of a crystal. One
angstrom is equal to 10−10 m which is the same as 0.1 nm. Light of this wavelength known as X-rays is commonly used to
study crystal structures. As can be seen from the third row of the table, X-ray photons have energies of tens of thousands of
electron volts. The gamma rays emitted by nuclei range in energy up to about 1 MeV. For this reason, we chose the photon
energy in the last row of the table to be 1.0 MeV and then used Eqs. (I.31) and (I.33) to obtain the wavelength corresponding
to this radiation.
As can be seen from the entries in Table I.2, the electromagnetic radiation emitted by atoms and nuclei has a much longer
wavelength than the size of the species that emits the radiation. Violet light has a wavelength of 4000 Å and is thus four
thousand times larger than an atom. Very energetic 1 MeV gamma rays have a wavelength 1240 times the size of the nucleus.
Less energetic gamma rays, which are commonly emitted by nuclei, have even longer wavelengths. Such considerations
will be important in later chapters when we study radiation processes.
We can think about the time scales appropriate for describing atomic processes in the same way as we think about time
in our own lives. Our life is a process which begins with our birth and extends on until the day we die. It takes us a certain
length of time to overcome different kinds of adversities and to respond to changes in our environment. The same could be
said of the motion of the electrons in an atom; however, the time scale is different. As we shall see in Chapter 4, the unit of
time that is appropriate for describing an electron in an atom is 2.4 × 10−17 s. We can think of this as the time required for
an electron in an atom to circulate once about the nuclear center. It is like the pulse rate of our own bodies. Atoms generally
decay from excited states to the ground state or readjust to changes in their environment in about a nanosecond which is

TABLE I.2 Typical Photon Energies
Wavelength (λ)


Photon Energy (E)

Type of Radiation

700 nm

1.77 eV

Red light

400 nm

3.10 eV

Violet light

1 Å = 0.1 nm

12.4 keV

X-rays

1240 fm

1.0 MeV

Gamma rays

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