Principles of Modern Physics doc

principles of modern physics
principles of
modelrn
physics
NEIL ASHBY
STANLEY C. MILLER
University of Colorado
HOLDEN-DAY, INC.
San Francisco
Cambridge
London
Amsterdam
o Copyright 1970 by
Holden-Day, Inc.,
500 Sansome Street
San Francisco, California
All rights reserved.
No part of this book
may be reproduced in any form,
by mimeograph or any
other means, without
permission in writing from
the publisher.
library of Congress Catalog
Card Number:
71-l
13182
Manufactured in
the United States of America
HOLDEN-DAY SERIES IN PHYSICS
McAllister Hull and David S. Saxon, Editors

preface
This book is intended as a general introduction to modern physics for science and
engineering students. It is written at a level which presurnes a prior
tull
year’s
course in classical physics, and a knowledge of elementary differential and
integral calculus.
The material discussed here includes probability, relativity, quantum me-
chanics, atomic physics, statistical mechanics, nuclear physics and elementary
particles. Some of these
top&,
such as statistical mechanics and probability, are
ordinarily not included in textbooks at this level. However, we have felt that for
proper understanding of many topics in modern physics such as quaIlturn me-
chanics and its applications this material is essential. It is our opilnion that
present-day science and engineering students should be able to worlk quanti-
tatively with the concepts of modern physics. Therefore, we have attempted to
present these ideas in a manner which is logical and fairly rigorous. A number of
topics, especially in
quantum1
mechanics, are presented in greater depth than is
customary. In many cases, unique ways of presentation are given which greatly
simplify the discussion of there topics. However, few of the developments require
more mathematics than elementary calculus and the algebra of complex nurn-
bers; in a few places, familiarity with partial differentiation will be necessary.
Unifying concepts which halve important applications throughout modern
physics, such as relativity, probability and the laws of conservation, have been
stressed. Almost all theoretical developments are linked to examples and data
taken from experiment. Summaries are included at the end of each chapter, as

well as problems with wide variations in difficulty.
This book was written for use in a one-semester course at the sophlomore or
iunior
level. The course could be shortened by omitting some topics; for example,
Chapter 7, Chapter 12, Chapters 13 through 15, and Chapter 16 contain blocks
of material which are somewhat independent of each other.
The system of units primarily used throughout is the meter-kilogram-second
system. A table of factors for conversion to other useful units is given in Appen-
dix 4. Atomic mass units are
#defined
with the
C”
atom as
tihe
standard.
We are grateful for the helpful comments of a large number of students, who
used the book in preliminary
term
for a number of years. We also thank our
colleagues and reviewers for their constructive criticism. Finally, we wish to ex-
press our thanks to Mrs. Ruth Wilson for her careful typing of the manuscript.
vii

contents
1 INTRODUCTION
1 .l HISTORICAL SURVEY
1.2 NOTATION AND UNITS
1.3 UNITS OF ENERGY AND MOMENTUM
1.4 ATOMIC MASS UNIT
1.5 PROPAGATION OF WAVES; PHASE AND GROUP SPEEDS

1.6 COMPLEX NUMBERS
2 PROBABILITY
2.1 DEFINITION OF PROBABILITY
2.2 SUMS OF PROBABILITIES
2.3 CALCULATION OF PROBABILITIES BY COUN’TING
2.4 PROBABILITY OF SEVERAL EVENTS OC:CUF!RING TOGETHER
2.5 SUMMARY OF RULES FOR CALCULATINIG PROBABILITIES
2.6 DISTRYBUTION FUNCTIONS FOR COIN FLIPPING
2.7 DISTRIBUTION FUNCTIONS FOR MORE THAN TWO POSSIBLE
OUTCOMES
2.8 EXPECTATION VALUES
2.9 NORMALIZATION
2.10 EXPECTATION VALUE OF THE NUMBER OF HEADS
2.1 1 EXPERIWIENTAL DETERMINATION OF PROBABILITY
2.12 EXPERIMENTAL ERROR
2.13 RMS DEVIATION FROM THE MEAN
2.114 RMS DEVIATION FOR COIN FLIPPING
2.15 ERRORS IN A COIN-FLIPPING EXPERIMENT
2.16 ERRORS IN AVERAGES OF REPEATED EXPERIMENTS
2.17 PROBABILITY DENSITIES
2.18 EXPECTATION VALUES FROM PROBABILITY DENSITIES
2.19 GAUSS1A.N DISTRIBUTION
2.20 EXPECTATION VALUES USING A GAUSS1A.N DISTRIBUTION
SUMh\ARY
PROBLEMS
3 SPECIAL THEORY OF RELATIVITY
3.1 CONFLICT BETWEEN ULTIMATE SPEED AND NEWTON’S LAWS
1
1
3

.4
.5
(6
:3
II
1
:2
1
:3
14
14
1:s
16
19
20
2 ‘I
2 ‘I
22
24
24
25
27
28
30
3:!
34
35
37
3Ei
421

42
ix
3.2 CLASSICAL MOMENTUM AND EINERGY
CONSERVATION-
COINFLICT WITH EXPERIMENT
3.3 CONSERVATION OF
MASS-COlNFLICT
WITH EXPERIMENT
3.4 CORRESPONDENCE PRINCIPLE
3.5 INERTIAL SYSTEMS
3.6 NON-INERTIAL SYSTEMS
3.7 AXES RELATIVE TO FIXED STARS
3.8 GALILEAN TRANSFORMATIONS
3.9 GALILEAN VELOCITY TRANSFORMATIONS
3.10 SECGND LAW OF MOTION UNDER GALILEAN
TRANSFORMATIONS
3.11 THIRD LAW UNDER GALILEAN TRANSFORMATIONS
:3.12
MICHELSON-MORLEY EXPERIMENT
3.13 POSTULATES OF RELATIVITY
3.14 EXPERIMENTAL EVIDENCE FOR THE SECOND POSTULATE
3.15 GALILEAN TRANSFORMATIONS AND THE PRINCIPLE OF
RELATIVITY
3.16 TRANSFORMATION OF LENGTHS PERPENDICULAR TO THE
RELATIVE VELOCITY
3.17 TIME DILATION
3.18 LENGTH CONTRACTION
3.19 LORENTZ TRANSFORMATIONS
3.20 SIMULTANEITY
3.21 TRANSFORMATION OF VELOCITIES

SUMMARY
PROBLEMS
4 RELATIVISTIC MECHANICS AND DYNAMICS
4.1 LORENTZ TRANSFORMATIONS
4.2 DISCREPANCY BETWEEN EXPERIMENT AND NEWTONIAN
MOMENTUM
4.3 MOMENTUM FROM A THOUGHT EXPERIMENT
4.4 EXPERIMENTAL VERIFICATION OF MASS FORMULA
4.5 RELATIVISTIC SECOND LAW OF MOTION
4.6 THIRD LAW OF MOTION AND CONSERVATION OF
MOMENTUM
4.7 RELATIVISTIC ENERGY
4.8 KINETIC ENERGY
4.9 POTENTIAL ENERGY AND CONSERVATION OF ENERGY
4.10 EXPERIMENTAL ‘VERIFICATION OF EQUIVALENCE OF MASS
AND ENERGY
4.11 RELATIONSHIP BETWEEN ENERGY AND MOMENTUM
4:12 REST MASS
(OF

ilo
FROM EXPERIMENT
4.13 TRANSFORMATION PROPERTIES OF ENERGY AND
MOMENTUM
43
44
47
47
49
50

51
52
53
54
54
55
57
59
59
60
64
65
67
71
74
76
79
79
80
81
83
85
85
86
87
88
89
89
90
96

Contents xi
4.14 TRANSFORMATIONS FOR FREQUENCY AND WAVELENGTH
4.15 TRANSVERSE DijPPLER EFFECT
4.16 LONGITUDINAL DOPPLER EFFECT
SUMMARY
PROBLIfMS
5 QUANTUM PROPERTIES OF LIGHT
5.1 ENERGY TRANSFORMATION FOR PARTICLES OF ZERO REST
MASS
5.2 FORM-INVARIANCE OF E =
hv
5.3 THE DUANE-HUNT
L.AW
5.4 PHOTOELECTRIC EFFECT
5.5 COMPTON EFFECT
5.6 PAIR PRODUCTION AND ANNIHILATION
5.7 UNCERTAINTY PRINCIPLE FOR LIGHT WAVES
5.8 MOMENTUM, POSITION UNCERTAINTY
5.9 PROBABILITY INTERPRETATION OF AMPLITUIDES
SUMMARY
PROBLEMS
6 MATTER WAVES
6.1 PHASE OF
.4
PLANE WAVE
6.2 INVARIANCE OF THE PHASE OF .A PLANE WAVE
6.3 TRANSFORMATION EQUATIONS FOR WAVEVECTOR A,ND
FREQUENCY
6.4 PHASE SPEED OF DE BROGLIE WAVES
6.5 PARTICLE INCIDENT ON INTERFACE SEPARATING DIFFERENT

POTENTIAL ENERGIES
6.6 WAVE RELATION AT INTERFACE
6.7 DE BROGLIE RELATIONS
6.8 EXPERIMENTAL DETERMINATION OF A
6.9
BRA.GG
EQUATION
6.10 DIFFRACTION OF ELECTRONS
6.11 UNCERTAINTY PRINCIPLE FOR PARTICLES
6.12 UNCERTAINTY AND SINGLE SLIT DIFFRACTION
6.13 UNCERTAINTY IN BALANCING AN OBJECT
6.14 ENERGY-TIME UNCERTAINTY
6.15 PROBABILITY INTERPRETATION OF VVAVEFUNCTllON
6.16 EIGENFUNCTIONS OF ENERGY AND MOMENTUM
OPERATORS
6.17 EXPECTATION VALUES FOR MOMENTUM IN A PARTICLE
BEAM
6.18 OPERATOR FORMALISM FOR CALCULATION OF MOMENTUM
EXPECTATION VALLJES
6.19 ENERGY OPERATOR AND EXPECTATION VALUES
6.20 SCHRODINGER EQUATllON
99
101
102
104
105
110
111
112
113

115
1119
123
126
128
129
131
13i
136
136
138
139
141
143
144
145
146
147
148
152
152
155
155
156
158
160
162
164
165
xii Contents

6.21 SCHRijDlNGER EQUATION FOR VARIABLE POTENTIAL
6.22 SOLUTION OF THE SCHRijDlNGER EQUATION FOR A
CONSTANT POTENTIAL
6.23’
BOUNDARY CONDITIONS
SUMMARY
PROBLEMS
167
7 EXAMPLES OF THE USE OF SCHRiiDINGER’S EQUATION
7.1 FREE PARTICLE GAUSSIAN WAVE PACKET
7.2 PACKET AT
t
= 0
7.3 PACKET FOR t > 0
7.4 STEP POTENTIAL; HIGH ENERGY E >
V,
7.5 BEAM OF INCIDENT PARTICLES
7.6 TRANSMISSION AND REFLECTION COEFFICIENTS
7.7 ENERGY LESS THAN THE STEP HEIGHT
7.8 TUNNELING FOR A SQUARE POTENTIAL BARRIER
7.9 PARTICLE IN A BOX
7.10 BOUNDARY CONDITION WHEN POTENTIAL GOES TO
INFINITY
169
170
172
175
178
178
180

181
183
185
186
187
188
190
7.11 STANDING WAVES AND DISCRETE ENERGIES
7.12 MOMENTUM AND UNCERTAINTY FOR A PARTICLE
IN A BOX
192
192
7.‘13
LINEAR MOLECULES APPROXIMATED BY PARTICLE IN A BOX
7.14 HARMONIC OSCILLATOR
7.15 GENERAL WAVEFUNCTION AND ENERGY FOR THE
HARMONIC OSCILLATOR
7.16 COMPARISON OF QIJANTUM AND NEWTONIAN
MECHANICS FOR THE HARMONIC OSCILLATOR
7.17 CORRESPONDENCE PRINCIPLE IN QUANTUM THEORY
SUMMARY
PROBLEMS
194
195
196
198
8 HYDROGEN ATOM AND ANGULAR MOMENTUM
8.1 PARTICLE IN A BOX
8.2 BALMER’S EXPERIMENTAL FORMULA FOR THE HYDROGEN
SPECTRUM

204
207
208
209
213
213
8.3 SPECTRAL SERIES FOR HYDROGEN
8.4 BOHR MODEL FOR HYDROGEN
8.5 QUANTIZATION IN THE BOHR MODEL
8.6 REDUCED MASS
8.7 SCHRoDlNGER EQUATION FOR HYDROGEN
8.8 PHYSICAL INTERPRETATION OF DERIVATIVES WITH RESPECT
TO r
215
216
217
218
220
221
8.9 SOLUTIONS OF
THIE
SCHRijDlNGER EQUATION
8.10 BINDING ENERGY AND IONIZATION ENERGY
8.11 ANGULAR MOMENTUM IN QUANTUM MECHANICS
8.12 ANGlJLAR MOMENTUM COMPONENTS IN SPHERICAL
223
225
230
230
COORDINATES

231
Confents
*‘*
XIII
8.13 EIGENFUNCTIONS OF
L,;
AZIMUTHAL QUANTU,M NUMBER
8.14 SQUARE OF THE TOTAL ANGULAR MOMENTUM
8.15 LEGENDRE POILYNOMIALS
8.16
SlJMMARY
OF QUANTUM NUMBERS FOR THE
HYDROGEN ATOM
8.17 ZEEMAN EFFECT
8.18 SPLITTING OF LEVELS IN A MAGNETIC FIELD
8.19 SELECTION RULES
8.20 NORMAL ZEEMAN SPLITTING
8.21 ELECTRON SPIN
8.22 SPIN-ORBIT INTERACTION
8.23 HALF-INTEGRAL SPINS
8.24 STERN-GERLACH EXPERIMENT
8.25 SUMS OF ANGULAR ,MOMENTA
8.26 ANOMALOUS ZEEMAN EFFECT
8.27 RIGID DIATOMIC ROTATOR
SUMMARY
PROBLEMS
9
PAW
EXCLUSION PRINCIPLE AND THE PERIODIC TABLE
9.1 DESIGNATION OF ATOMIC STATES

9.2 NUMBER OF STATES IN AN n SHELL
9.3 INDISTINGUISHABILITY OF PARTICLES
9.4 PAULI EXCLUSION PRINCIPLE
9.5 EXCLUSION PRINCIPLE AND ATOMIC ELECTRON STATES
9.6 ELECTRON CONFIGURATIONS
9.7 INERT GASES
9.8 HALOGENS
9.9
ALKAILI
METALS
9.10 PERIODIC TABLE OF THE ELEMENTS
9.1
11
X-RAYS
9.12 ORTHO- AND
PARA-H’YDROGEN
!jUMMARY
PROBLEMS
10 CLASSICAL STATISTICAL MECHANICS
10.1 PROBABILITY DISTIPIBUTION IN ENERGY FOR SYSTEMS
IN
THERMAL
EQ~UILIBRIUM
10.2 BOLTZMANN DISTRIBUTION
10.3 PROOF THAT P(E) IS OF EXPONENTIAL FORM
10.4
PHA!jE
SPACE
10.5 PHASE SPACE DISTRIBUTION FUNCTIONS
10.6 MAXWELL-BOLTZMANN DISTRIBUTION

10.7 EVALUATION OF
/I
10.8 EVALUATION
OIF

NP(O)p
lo 9
MAXWELL-BOLTZMANN DISTRIBUTION INCLUDING
POTENTIAL ENERGY
10.10 GAS IN A GRAVITATIONAL FIELD
232
233
234
235
236
237
238
239
240
240
241
242
242
243
244
246
249
254
255
256

256
258
260
262
263
265
265
266
270
273
273
275
279
280
281
282
283
285
287
288
291
292
293
xiv
Contenfs
10.11 DISCRETE ENERGIES
10.12 DISTRIBUTION OF THE MAGNITUDE OF MOMENTUM
10.13 EXPERIMENTAL VERIFICATION OF MAXWELL DISTRIBUTION
10.14 DISTRIBUTION OF ONE COMPONENT OF MOMENTUM
10.15 SIMPLE HARMONIC OSCILLATORS

10.16 DETAILED BALANCE
10.17 TIME REVERSIBILITY
SUMMARY
PROBLEMS
11 QUANTUM STATISTICAL MECHANICS
11.1 EFFECTS OF THE EXCLUSION PRINCIPLE ON STATISTICS
OF PARTICLES
11.2 DETAILED BALANCE AND FERMI-DIRAC PARTICLES
11.3 FERMI ENERGY AND FERMI-DIRAC DISTRIBUTION
11.4 ONE DIMENSIONAL DENSITY OF STATES FOR PERIODIC
BOUNDARY CONDITIONS
11.5 DENSITY OF STATES IN THREE DIMENSIONS
11.6 COMPARISON BETWEEN THE CLASSICAL AND QUANTUM
DENSITIES OF STATES
11.7 EFFECT OF SPIN ON THE DENSITY OF STATES
11.8 NUMBER OF STATES PIER UNIT ENERGY INTERVAL
11.9 FREE PARTICLE FERMI ENERGY-NONDEGENERATE CASE
11.10 FREE ELECTRONS IN METALS-DEGENERATE CASE
11.11 HEAT
CAPACIITY
OF AN ELECTRON GAS
11.12 WORK FUNCTION
11
.lm
3 PHOTON DISTRIBUTION
11.14
PLA.NCK
RADIATION FORMULA
11 .15 SPONTANEOUS EMISSION
11.16 RELATIONSHIP BETWEEN SPONTANEOUS AND STIMULATED

EMISSION
11.17 ORIGIN OF THE FACTOR 1 +
II,
IN BOSON TRANSITIONS
1
I
.18 BOSE-EINSTEIN DISTRIBUTION FUNCTION
SUMMARY
PROBLEMS
112 SOLID STATE PHYSICS
12.1 CLASSIFICATION OF CRYSTALS
12.2 REFLECTION
AIND
ROTATION SYMMETRIES
12.3 CRYSTAL BINDING FORCES
12.4 SOUND WAVES IN A CONTINUOUS MEDIUM
12.5 WAVE EQUATION FOR SOUND WAVES IN A DISCRETE
MEDIUM
12.6 SOLUTIONS OF THE WAVE EQUATION FOR THE DISCRETE
MEDIUM
12.7 NUMBER OF SOLUTIONS
12.8 LINEAR CHAIN WITH TWO MASSES PER UNIT CELL
294
295
296
298
300
303
305
306

308
312
313
313
315
316
318
319
320
320
321
323
324
325
326
328
331
332
333
335
336
338
341
341
342
346
347
349
351
352

354
contents

xv
12.9 ACOUSTIC AND ‘OPTICAL BRANCHES
12.10 ENERGY OF LATTICE VIBRATIONS
12.11 ENERGY FOR A SUPERPOSITION OF MODES
12.12 QUANTUM
THIEORY
OF HARMONIC OSCILLATORS AND
LATTICE VIBRATIONS
12.13 PHONONS;
AVEl?AGE
ENERGY PER MODE AS A FUNCTION
OF TEMPERATIJRE
12.14 LATTICE SPECIFIC HEAT OF A SOLID
12.15 ENERGY BANDS OF ELECTRONS IN CRYSTALS
12.16 BLOCH’S THEOREM
12.17 NUMBER OF BLOCH FUNCTIONS PER BAND
12.18 TYPES OF BANDS
12.19 EFFECTIVE MASS IN A BAND
12.20 CONDIJCTORS, INSULATORS, SEMICONDUCTORS
12.21 HOLES
12.2;! n-TYPE AND p-TYPE SEMICONDUCTORS
‘12.23
H.ALL
EFFECT
SUMMARY
PROBLEMS
13 PROBING THE NUCLEUS

13.1 A NUCLEAR MODEL
13.2 LIMITATIONS ON NUCLEAR SIZE FROM ATOMIC
CONSIDERATIONS
13.3 SCATTERING EXPERIMENTS
13.4 CROSS-SECTIONS
13.5 DIFFERENTIAL CROSS-SECTIONS
13.6 NUMBER OF SCATTERERS PER UNIT AREA
13.7 BARN AS A UNIT OF CROSS-SECTION
13.8
a
AND
@
PARTICLES
13.9 RUTHERFORD MODEL OF THE ATOM
13.10 RUTHERFORD THEORY; EQUATION OF ORBIT
113.11 RUTHERFORD SCATTERING ANGLE
13.12 RUTHERFORD DIFFERENTIAL CROSS-SECTION
13.13 MEASUREMENT OF THE DIFFERENTIAL CROSS-SECTION
13.14 EXPERIMENTAL VERIFICATION OF THE RLJTHERFORD
SCATTERING FORMlJLA
13.15 PARTICLE ACCELERATORS
SUMMARY
PROBLEMS
14 NUCLEAR STRUCTURE
14.1
NUCLEC\R
MASSES
14.2 NEUTRONS IN THE NUCLEUS
14.3 PROPERTIES OF THE NEUTRON AND PROTON
14.4 THE DEUTERON

(,H’)
14.5 NUCLEAR FORCES
14.6 YUKAWA FORCES
356
357
359
360
361
362
364
365
366
367
368
369
371
372
373
374
377
381
381
383
385
386
387
390
390
391
393

394
395
397
398
400
402
404
405
408
408
410
411
414
416
418
xvi
Contents
14.7 MODELS OF THE NUCLEUS
SUMMARY
PROBLEMS
15 TRANSFORMsATlON OF THE NUCLEUS
15.1 LAW OF RADIOACTIVE DECAY
15.2 HALF-LIFE
15.3 LAW OF DECAY FOR UNSTABLE DAUGHTER NUCLEI
15.4 RADIOACTIVE SERIES
15.5 ALPHA-PARTICLE DECAY
15.6 THEORY OF ALPHA-DECAY
15.7 BETA DECAY
15.8 PHASE SPACE AND THE: THEORY OF BETA DECAY
15.9 ENERGY IN

p+
DECAY
15.10 ELECTRON CAPTURE
15.11 GA,MMA DECAY AND INTERNAL CONVERSION
‘15.12 LOW ENERGY NUCLEAR REACTIONS
15.13 THRESHOLD ENERGY
15.14 NUCLEAR FISSION AND FUSION
15.15 RADIOACTIVE CARBON DATING
SUMMARY
PROBLEMS
16 ELEMENTARY PARTICLES
16.1 LEPTONS
16.2 MESONS
16.3 BARYONS
16.4 CONSERVATION LAWS
16.5 DETECTION OF PARTICLES
16.6 HYPERCHARGE, ISOTOPIC SPIN PLOTS
16.7 QUARKS
16.8 MESONS IN TERMS OF QUARKS
SUMMARY
PROBLEMS
APPENDICES
APPENDIX 1
APPENDIX 2
APPENDIX 3
APPENDIX 4
BIBLIOGRAPHY
INDEX
421
427

429
431
431
433
433
433
441
443
447
450
452
453
454
454
456
457
458
458
461
464
464
466
467
468
472
473
474
477
478
479

483
491
496
504
505
507
principles of modern physics

1
introduction
I
.1
HISTORICAL SURVEY
The term modern physics generally refers to the study
<of
those facts and theories
developed in this century, that concern the ultimate structure and interactions of
matter, space and time. The three main branches of classical physics-mechanics,
heat and electromagnetism were developed over a period of approximately
two centuries prior to 1900. Newton’s mechanics dealt successfully with the
motions of bodies of macroscopic size moving with low speeds, and provided a
foundation for many of the engineering accomplishments of the eighteenth and
nineteenth centuries. With Maxwell’s discovery of the displacement current and
the completed set of electromagnetic field equations, classical technology re-
ceived new impetus: the telephone, the wireless, electric light and power, and a
host of other applications followed.
Yet the theories of mechanics and electromagnetism were not quite consistent
with each other. According to the
G&lean
principle of relativity, recognized by

Newton, the laws of mecharlics should be expressed in the same mathematical
form by observers in different inertial frames of reference, which are moving with
constant velocity relative to each other. The transformation equations, relating
measurements in two relatively moving inertial frames, were not consistent with
the transformations obtained by Lorentz from
similclr
considerations of
form-
invariance applied to Maxwell’s equations. Furthermore, by around 1900 a
number of phenomena had been discovered which were inexplicable on the basis
of classical theories.
The first
major
step toward a deeper understanding of the Inature of space
and time measurements was due to Albert Einstein, whose special theory of rela-
tivity (1905) resolved the inconsistency between mechanics and electromagnetism
by showing, among other things, that Newtonian mechanics is only a first ap-
proximation to a more general set of mechanical laws; the approximation is,
however, extremely good when the bodies move with speeds which are small
compared to the speed of light. Among the impel-tant results obtained by
Einstein was the equivalence of mass and energy, expressed in the famous
equation E = mc2.
From a logical standpoint, special relativity lies at the heart of modern
physics. The hypothesis that electromagnetic radiaticmn energy is quantized in
bunches of amount hu, where
v
is the frequency and h is a constant, enabled
1
2
introduction

Planck to explain the intensity distribution of black-body radiation. This occurred
several years before Einstein published his special theory of relativity in 1905.
At about this time, Einstein also applied the quantum hypothesis to photons in an
explanation of the photoelectric effect. This hypothesis was found to be con-
sistent with special relativity. Similarly,
Bohr’s postulate-that the electron’s
angular momentum in the hydrogen atom is quantized in discrete
amounts-
enabled him to explain the positions of the spectral lines in hydrogen. These first
guesses at a quantum theory were followed in the first quarter of the century by
a number of refinements and ad hoc quantization rules; these, however, achieved
only limited success. It was not until after 1924, when Louis de Broglie proposed,
on the basis of relativity theory, that waves were associated with material par-
ticles, that the foundations of a correct quantum theory were laid. Following
de Broglie’s suggestion,
Schrodinger
in 1926 proposed a wave equation describ-
ing the propagation of these particle-waves, and developed a quantitative
explanation of atomic spectral line intensities. In a few years thereafter, the
success of the new wave mechanics revolutionized physics.
Following the discovery of electron spin, Pauli’s exclusion principle was rigor-
ously established, providing the explanation for the structure of the periodic
table of the elements and for many of the details of the chemical properties of
the elements. Statistical properties of the systems of many particles were studied
from the point of view of quantum theory, enabling Sommerfeld to explain the
behavior of electrons in a metal. Bloch’s treatment of electron waves in crystals
simplified the application of quantum theory to problems of electrons in solids.
Dirac, while investigating the possible first order wave equations allowed by
relativity theory, discovered that a positively charged electron should exist; this
particle, called a positron, was later discovered. These are only a few of the

many discoveries which were made in the decade from 1925-l 935.
From one point of view, modern physics has steadily progressed toward the
study of smaller and smaller features of the microscopic structure of matter, using
the conceptual tools of relativity and quantum theory. Basic understanding of
atomic properties was in principle achieved by means of Schrodinger’s equation
in 1926. (In practice,. working out the implications of the
Schrodinger
wave
mechanics for atoms and molecules is difficult, due to the large number of
variables which appear in the equation for systems of more than two or three
particles.) Starting
iIn1
1932 with the discovery of the neutron by Chadwick,
properties of atomic nuclei have become known and understood in greater and
greater detail. Nuclear fission and nuclear fusion are byproducts of these studies,
which are still extrernely active. At the present time some details of the inner
structure of protons, neutrons and other particles involved in nuclear inter-
actions are just beginning to be unveiled.
Over fifty of the s’o-called elementary particles have been discovered. These
particles are ordinarily created by collisions between high-energy particles of
some other type, usually nuclei or electrons. Most of the elementary particles are
unstable and decay
illto
other more stable objects in a very short time. The study
7.2 Notation
and
unifs 3
of these particles and their interactions forms an important branch of present-day
research in physics.
It should be emphasized that one of the most important unifying concepts in

modern physics is that of energy. Energy as a conserved quantity was well-known
in classical physics. From the time of Newton until Einstein, there were no funda-
mentally new mechanical laws introduced; however, the famous variational
principles of Hamilton and Lagrange expressed Newtonian lows in a different
form, by working with mathematical expressions for the kinetic and potential
energy of a system. Einstein showed that energy and momentum are closely re-
lated in relativistic transformation equations, and established the equivalence of
energy and mass. De Broglie’s quantum relations connected the frequency and
wavelength of the wave motions associated with particles, with the particle’s
energy and momentum.
S:hrb;dinger’s
wave equation is obtained by certain
mathematical operations performed on the expression for the energy of a system.
The most sophisticated expressions of modern-day relativistic quantum theory are
variational principles, which involve the energy of a system expressed in
quantum-mechanical form. And, perhaps most important, the stable stationary
states of quantum systems are states of definite energy.
Another very important concept used throughout modern physics is that of
probability. Newtonian mechanics is a strictly deterministic theory; with the
development of quantum theory, however, it eventually became clear that
microscopic events could not be precisely predicted or controlled. Instead, they
had to be described in terms of probabilities. It is somewhat ironic that proba-
bility was first introduced into quantum theory by Einstein in connection with his
discovery of stimulated emission. Heisenberg’s uncertainty principle, and the
probability interpretation of the
Schradinger
wavefunction, were sources of
distress to Einstein who, not feeling comfortable with a probabilistic theory, later
declared that he would never believe that “God plays dice with the world.”
As a matter of convenience, we shall begin in Chapter 2 with a brief intro-

duction to the concept of probability and to the rules for combining proba-
bilities. This material will be used extensively in later chapters on the quantum
theory ond on statistical mechanics.
The remainder of the present chapter consists of review and reference material
on units and notation, placed here to avoid the necessity of later digressions.
1.2 NOTATION AND UNITS
The well-known meter-kiloglram-second (MKS) system of units will be used in
this book. Vectors will be denoted by boldface type,
Isuch
as F for force. In these
units, the force on a point charge of Q coulombs, moving with velocity v in meters
per second, at a point where the electric field is E volts per meter and the mag-
netic field is B webers per square meter, is the Lorentz force:
F = Q(E + v x
6)
(1.1)
4 Introduction
where v x B denotes the vector cross-product of v and B. The potential in volts
produced by a point charge Q at a distance r from the position of the charge is
given by Coulomb’s law:
V(r) =
2.;
II
where the constant t0 is given by
I
(4Tto)
-
9 x
lo9
newtons-m2/coulomb2

(‘4
(1.3)
These particular expressions from electromagnetic theory are mentioned here
because they will be used in subsequent chapters.
In conformity with modern notation, a temperature such as “300 degrees
Kelvin” will be denoted by 300K. Boltzmann’s constant will be denoted by
k
,,
, with
k,
= 1.38 x
10mz3
joules/molecule-K
(1.4)
A table of the fundammental constants is given in Appendix 4.
1.3 UNITS OF ENERGY AND MOMENTUM
While in the MKS system of units the basic energy unit is the joule, in atomic and
nuclear physics several other units of energy have found widespread use. Most of
the energies occurring in atomic physics are given conveniently in terms of the
elecfron volt, abbreviated eV. The electron volt is defined as the amount of work
done upon an electron as it moves through a potential difference of one volt.
Thus
1 eV = e x V = e(coulombs) x 1 volt
= 1.602 x
lo-l9
joules
(1.5)
The electron volt is an amount of energy in joules equal to the numerical value
of the electron’s charge in coulombs. To convert energies from joules to eV, or
from eV to joules, one divides or multiplies by e, respectively. For example, for a

particle with the mass of the electron, moving with a speed of 1% of the speed of
light, the kinetic energy would be
1
2
mv’! = 1 9
11
2(.
x
10m3’

kg)(3
x
lo6
m/sec)2
= 4.1 x 1 O-l8
ioules
4.1 x
lo-l8

i
= (1.6 x
lo-I9

i/eV)
= 2.6 eV
(1.6)
In
nuclear physics most energies are of the order of several million electron
volts, leading to the definition of a unit called the MeV:
1.4 Atomic moss unit

5
1 MeV = 1 million eV = 106eV
= 1.6 x
lo-l3

ioules
= (1
06e)joules
(1.7)
For example, a proton of rnass 1.667 x 10mz7 kg, traveling with 10% of the
speed of light, would have a kinetic energy of approximately
1
(1.67 x 10m2’
kg)(3
x
IO7
m/sec)2
;

Mv2

zx

;i

-~-
(1.6 x
lo-l3
i/EheV)
-

= 4.7 MeV
(1.8)
Since energy has units of mass x (speed)2, while momentum has units of
mass x speed, for mony applications in nuclear and elementary particle physics
a unit of momentum called ,UeV/c is defined in such o
sway
that
1 MeV
-

-

lo6

e

kg-m/set
C C
= 5.351 x
lo-l8

kg-m/set
(1.9)
where c and e are the numerical values of the speed of light and electronic
charge, respectively, in
MKS
units. This unit of momentum is particularly con-
venient when working with relativistic relations between energy and momentum,
such as E = pc, for photons. Then if the momenturrl p in MeV/c is known, the
energy in MeV is numerically equal to p. Thus, in general, for photons

E(in
MeV) = p(in MeV/c)
(1.10)
Suppose, for instance, that a photon hos a momentum of
10m2’

kg-m/set.
The
energy would be pc = 3 x
lo-l3
joules = 1.9 MeV, after using Equation (1.7).
On the other hand, if p is expressed in MeV/c, using Equation (1.9) we find that
p

=

10m2’

kg-m/set
= 1.9 MeV/c
The photon energy is then E =
pc

=
(1.9 MeV/c)(c) = 1.9 MeV.
1.4 ATOMIC MASS UNIT
The atomic mass unit, abbreviated amu, is chosen in such a way that the mass
of the most common atom of carbon, containing six protons and six neutrons in a
nucleus surrounded by six electrons, is exactly 12.000000000 . . amu. This unit is
convenient when discussing atomic masses, which are then always very close to

an integer. An older atomic mass unit, based on on atomic mass of exactly
16 units for the oxygen atclm with 8 protons, 8 neutrons, and 8 electrons, is no
longer in use in physics reselzrch. In addition, a slightly different choice of atomic
mass unit is commonly useu in chemistry. All atomic masses appearing in this
book are based on the physical scale, using carbon as the standard.
The conversion from amu on the physical scale to kilograms may be obtained
by using the fact that one gram-molecular weight of a substance contains
6 fntroduction
1.5
PROPAGATION OF WAVES; PHASE AND GROUP SPEEDS
Avogadro’s number,
At,,
= 6.022 x 10z3, of molecules. Thus, exactly 12.000 . . .
grams of C’* atoms contains
N,
atoms, and
1 amu =
+2
x
= 1.660 x 10m2’ kg
(1.11)
In later chapters, many different types of wave propagation will be considered:
the de Broglie probability waves of quantum theory, lattice vibrations in solids,
light waves, and so on. These wave motions can be described by a displacement,
or amplitude of vibration of some physical quantity, of the form
#(x, t) = A cos (kx
ztz

of
+

4)
(1.12)
where A and
4
are constants, and where the wavelength and frequency of the
wave are given by
(1.13)
Here the angular frequency is denoted by
o
= o(k), to indicate that the fre-
quency is determined by the wavelength, or wavenumber k. This frequency-
wavelength relation,
01
= w(k), is called a dispersion relation and arises because
of the basic physical laws satisfied by the particular wave phenomenon under
investigation. For example, for sound waves in air, Newton’s second law of
motion and the adiabatic gas law imply that the dispersion relation is
where v is a constant.
w = vk (1.14)
If the negative sign is chosen in Equation
(1.12),
the resulting displacement
(omitting the phase constant
b,)
is
#(x,
t)
= A cos (kx
-
wt) = A cos

[+

(f)f,]
(1.15)
This represents a wave propagating in the positive x direction. Individual crests
and troughs in the waves propagate with a speed called the phase speed,
given by
w=o
k
(1.16)
In nearly all cases, the wave phenomena which we shall discuss obey the
principle of superposition-namely, that if waves from two or more sources
arrive at the same physical point, then the net displacement is simply the sum of
the displacements from the individual waves. Consider two or more wave trains
propagating in the same direction. If the angular frequency w is a function of
Propagation of waves; phase and group speeds
7
the wavelength or wavenumber, then the phase speed can be a function of the
wavelength, and waves
of
differing wavelengths travel at different speeds.
Reinforcement or destructive interference can then occur as one wave gains on
another of different wavelength. The speed with which the regions of constructive
or destructive interference advance is known as the group speed.
To calculate this speed, consider two trains of waves of the form of Equation
(1.15),
of the same amplitude but of slightly different wavelength and frequency,
such as
I),
= A

<OS
[(k +
%
Ak)x
-
(o
-F

%

AC+]
I,L~
= A
(OS
[(k
-

%
Ak)x
-

(w

Yz
Aw)t]
(1.17)
Here, k and
u
are the central wavenumber and angular frequency, and Ak,
Ao

are the differences between the wavenumbers and angular frequencies of
identity 2 cos A cos B =
the two waves. The resultant displacement, using the
cos
(A + 13) + cos (A
-
B), is
$
q = $1 + ti2
=:
(2 A cos
‘/2
(Akx
-
Awt)
I}

cos
(kx
-
wt) (1.18)
This expression represents
a
wave traveling with phase speed w/k, and with an
amplitude given by
2 A cos % (Akx
-
Awt) = 2 A cos
‘/2

Ik
(1.19)
The amplitude is a cosine curve; the spatial distance between two successive zeros
of this curve at a given instant is
r/Ak,
and is the distance between two suc-
cessive regions of destructive interference.
These regions propagate with the
group speed
vg
, given by
AU
vg
=
-
dw
(k)
Ak
ak=-o
dk
in the limit of sufficiently small Ak.
Thus, for sound waves in air, since w = vk, we derive
“g
d
(vk)
= F”Yw
dk
(1.20)
(1.21)
and the phase and group speeds are equal. On the other hand, for surface

gravity waves in a deep seo, the dispersion relation is
w =
{gk
+
k3J/p)“2
where g is the gravitational acceleration,
J is the surface tension and
p
is the
density. Then the phase speed is
wTw=
k
(1.23)