Principles of Modern Physics

540 Pages·2000·8.65 MB·English

principles of modern physics

principles of

modelrn physics

N E I L A S H B Y

S T A N L E Y C . M I L L E R

University of Colorado

H O L D E N - D A Y , I N C .

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

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1:s

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2 ‘I

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3:!

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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

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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 E F F E C T

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 . 2 0 SCHRODINGER
EQUATllON

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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

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7.11 STANDING WAVES AND DISCRETE ENERGIES
7.12 MOMENTUM AND UNCERTAINTY FOR A PARTICLE
IN A BOX

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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

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8 HYDROGEN ATOM AND ANGULAR MOMENTUM
8.1 PARTICLE IN A BOX
8.2 BALMER’S EXPERIMENTAL FORMULA FOR THE HYDROGEN
SPECTRUM

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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
T O r

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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

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230
230

COORDINATES
231

C o n f e n t s *‘*
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
1 0 C L A S S I C A L S T A T I S T I C A L M E C H A N I C S
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

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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

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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

O F 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

1 2 . 2 1 H O L E S

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
M A S S E S

14.2 NEUTRONS IN THE NUCLEUS

14.3 PROPERTIES OF THE NEUTRON AND PROTON

1 4 . 4 T H E DEUTERON
(,H’)

14.5 NUCLEAR FORCES

14.6 YUKAWA F O R C E S

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