Principles of Modern Physics doc

The material discussed here includes probability, relativity, quantum chanics, atomic physics, statistical mechanics, nuclear physics and elementaryparticles.. 3.2 CLASSICAL MOMENTUM AND

principles of modern physics

principles of

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

H O L D E N - D A Y S E R I E S I N P H Y S I C S McAllister Hull and David S Saxon, Editors

This book is intended as a general introduction to modern physics for science andengineering students It is written at a level which presurnes a prior tull year’scourse in classical physics, and a knowledge of elementary differential andintegral calculus

The material discussed here includes probability, relativity, quantum chanics, atomic physics, statistical mechanics, nuclear physics and elementaryparticles Some of these top&, such as statistical mechanics and probability, areordinarily not included in textbooks at this level However, we have felt that forproper understanding of many topics in modern physics such as quaIlturn me-chanics and its applications this material is essential It is our opilnion thatpresent-day science and engineering students should be able to worlk quanti-tatively with the concepts of modern physics Therefore, we have attempted topresent these ideas in a manner which is logical and fairly rigorous A number oftopics, especially in quantum1 mechanics, are presented in greater depth than iscustomary In many cases, unique ways of presentation are given which greatlysimplify the discussion of there topics However, few of the developments requiremore mathematics than elementary calculus and the algebra of complex nurn-bers; in a few places, familiarity with partial differentiation will be necessary

me-Unifying concepts which halve important applications throughout modernphysics, such as relativity, probability and the laws of conservation, have beenstressed Almost all theoretical developments are linked to examples and datataken from experiment Summaries are included at the end of each chapter, aswell as problems with wide variations in difficulty

This book was written for use in a one-semester course at the sophlomore oriunior 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-secondsystem 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, whoused the book in preliminary term for a number of years We also thank ourcolleagues 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

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 P R O B A B I L I T Y

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 F R O M E X P E R I M E N T 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 T H E D U A N E - H U N T 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

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

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

T O 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

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 E X C L U S I O N P R I N C I P L E A N D T H E P E R I O D I C T A B L E

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 M E T A L S 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

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

x i v 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

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

2 9 5 296

2 9 8 300

3 0 3

3 0 5 306

3 0 8 312

313 313 315 316 318 319

320 320 321

3 2 3 324

3 2 5 326

3 2 8 331

3 3 2

3 3 3

3 3 5 336

3 3 8 341 341

3 4 2 346 347 349

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

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

1 4 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

1 4 6 YUKAWA F O R C E S

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

4 0 8 408 410 411 414 416 418

xvi Contents

14.7 MODELS OF THE NUCLEUS

SUMMARY PROBLEMS

1 5 TRANSFORMsATlON O F T H E N U C L E U S

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

1 6 1 L E P T O N S

1 6 2 M E S O N S

1 6 3 B A R Y O N S 16.4 CONSERVATION LAWS 16.5 DETECTION OF PARTICLES 16.6 HYPERCHARGE, ISOTOPIC SPIN PLOTS

1 6 7 Q U A R K S 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 theoriesdeveloped in this century, that concern the ultimate structure and interactions ofmatter, space and time The three main branches of classical physics-mechanics,heat and electromagnetism -were developed over a period of approximatelytwo centuries prior to 1900 Newton’s mechanics dealt successfully with themotions of bodies of macroscopic size moving with low speeds, and provided afoundation for many of the engineering accomplishments of the eighteenth andnineteenth centuries With Maxwell’s discovery of the displacement current andthe completed set of electromagnetic field equations, classical technology re-ceived new impetus: the telephone, the wireless, electric light and power, and ahost of other applications followed

Yet the theories of mechanics and electromagnetism were not quite consistentwith each other According to the G&lean principle of relativity, recognized byNewton, the laws of mecharlics should be expressed in the same mathematicalform by observers in different inertial frames of reference, which are moving withconstant velocity relative to each other The transformation equations, relatingmeasurements in two relatively moving inertial frames, were not consistent withthe transformations obtained by Lorentz from similclr considerations of form-invariance applied to Maxwell’s equations Furthermore, by around 1900 anumber 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 spaceand 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 proximation to a more general set of mechanical laws; the approximation is,however, extremely good when the bodies move with speeds which are smallcompared to the speed of light Among the impel-tant results obtained byEinstein was the equivalence of mass and energy, expressed in the famousequation E = mc2

ap-From a logical standpoint, special relativity lies at the heart of modernphysics The hypothesis that electromagnetic radiaticmn energy is quantized inbunches of amount hu, where v is the frequency and h is a constant, enabled

1

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 ticles, that the foundations of a correct quantum theory were laid Following

par-de Broglie’s suggestion, Schrodinger in 1926 proposed a wave equation par- 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.

describ-Following the discovery of electron spin, Pauli’s exclusion principle was 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.

rigor-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-dayresearch in physics

It should be emphasized that one of the most important unifying concepts inmodern 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 mentally new mechanical laws introduced; however, the famous variationalprinciples of Hamilton and Lagrange expressed Newtonian lows in a differentform, by working with mathematical expressions for the kinetic and potentialenergy of a system Einstein showed that energy and momentum are closely re-lated in relativistic transformation equations, and established the equivalence ofenergy and mass De Broglie’s quantum relations connected the frequency andwavelength of the wave motions associated with particles, with the particle’senergy and momentum S:hrb;dinger’s wave equation is obtained by certainmathematical operations performed on the expression for the energy of a system.The most sophisticated expressions of modern-day relativistic quantum theory arevariational principles, which involve the energy of a system expressed inquantum-mechanical form And, perhaps most important, the stable stationarystates of quantum systems are states of definite energy

funda-Another very important concept used throughout modern physics is that ofprobability Newtonian mechanics is a strictly deterministic theory; with thedevelopment of quantum theory, however, it eventually became clear thatmicroscopic events could not be precisely predicted or controlled Instead, theyhad 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 hisdiscovery of stimulated emission Heisenberg’s uncertainty principle, and theprobability interpretation of the Schradinger wavefunction, were sources ofdistress to Einstein who, not feeling comfortable with a probabilistic theory, laterdeclared 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 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 quantumtheory ond on statistical mechanics

intro-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 inthis book Vectors will be denoted by boldface type, Isuch as F for force In theseunits, the force on a point charge of Q coulombs, moving with velocity v in metersper 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:

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.;

IIwhere the constant t0 is given by

elecfron volt, abbreviated eV The electron volt is defined as the amount of workdone 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, orfrom eV to joules, one divides or multiplies by e, respectively For example, for aparticle with the mass of the electron, moving with a speed of 1% of the speed oflight, the kinetic energy would be

1.4 Atomic moss unit 5

1 MeV = 1 m i l l i o n 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 thespeed of light, would have a kinetic energy of approximately

E(in MeV) = p(in MeV/c) (1.10)Suppose, for instance, that a photon hos a momentum of 10m2’ kg-m/set Theenergy 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 anucleus surrounded by six electrons, is exactly 12.000000000 amu This unit isconvenient 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 nolonger in use in physics reselzrch In addition, a slightly different choice of atomicmass unit is commonly useu in chemistry All atomic masses appearing in thisbook 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 6 6 0 x 10m2’ k g (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 ( k x ztz of + 4) (1.12)where A and 4 are constants, and where the wavelength and frequency of thewave are given by

(1.13)

Here the angular frequency is denoted by o = o(k), to indicate that the quency is determined by the wavelength, or wavenumber k This frequency-wavelength relation, 01 = w(k), is called a dispersion relation and arises because

fre-of the basic physical laws satisfied by the particular wave phenomenon underinvestigation For example, for sound waves in air, Newton’s second law ofmotion and the adiabatic gas law imply that the dispersion relation is

w=o

In nearly all cases, the wave phenomena which we shall discuss obey theprinciple of superposition-namely, that if waves from two or more sourcesarrive at the same physical point, then the net displacement is simply the sum ofthe displacements from the individual waves Consider two or more wave trainspropagating 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 thewavelength, and waves of differing wavelengths travel at different speeds.Reinforcement or destructive interference can then occur as one wave gains onanother 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 anamplitude given by

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 cessive regions of destructive interference These regions propagate with thegroup speed vg , given by

suc-AU

A k ak=-o dk

in the limit of sufficiently small Ak

Thus, for sound waves in air, since w = vk, we derive

wTw=

of water.

1 6 C O M P L E X N U M B E R ! ;

Because the use of complex numbers is essential in the discussion of the wavelikecharacter of particles, a brief review of the elementary properties of complexnumbers is given here A complex number is of the form # = a + ib, where

u and b are real numbers and i is the imaginary unit, iz = - 1 The real part

of $ is a, and the imaginary part is b:

R e ( a + i b ) = aIm(a + i b ) = b (1.25)

A complex number $ == a + ib can be represented as a vector in two dimensions,with the x component of the vector identified with Re($), and the y component

Figure 1 l Two-dimensional vector representation of o complex number 1c/ = o + ib

of the vector identified with Im (Ic/), as in Figure 1 l The square of the magnitude

of the vector is

( # 1’ = a2 + bZ (1.26)The complex conjugate of $ = a + ib is denoted by the symbol #* and is ob-tained by replacing the imaginary unit i by -i:

I,L* = a - ib

I 6 Complex numbers 9

We can calculate the magnitude of the square of the vector by multiplying $ byits complex conjugate:

I$ 1’ =: #*$ = a2 - (jb)’ = a2 + b2 (1.28)

The complex exponential function, e”, or exp (i@, where 0 is a real function

or number, is of particula* importance; this function may be defined by thepower series

e in8 = co5 f-10 + i sin n0 = (cos 0 t i sin 01”

Since (e”)* = e-j8, we also love the following identities:

Re eia =: ~0s 0 = i (e’” + em”‘)

Im e” = sin (j = + (e” - em’“)

/ e’a 12 =: ,-‘a,‘fj = 10 = 1

_ _ _ = _ x - =.- a ib( a + i b ) a + ib a - ib a2 + b2

The integral of an exponential function of the form ecX is

=(-I +0-l) -2

i = - = .+2;

i

We have ninety chances in a hundred.

Napoleon at Waterloo, 1815

The commonplace meaning of the word “chance ” i:j probably already familiar

to the reader In everyday life, most situations in which we act are characterized

by uncertain knowledge of the facts and of the outcomes of our actions We are

thus forced to make guesses, and to take chances In the theory of probability,

the concepts of probability and chance are given precise meanings The theorynot only provides a systematic way of improving our guesses, it is also anindispensable tool in studying the abstract concepts of modern physics To avoidthe necessity of digressions on probability during the later development ofstatistical mechanics and quantum mechanics, we present here a brief intro-duction to the basic elements of probability theory

When Napoleon utterecl the statement above, he did not mean that if theBattle of Waterloo were fought a hundred times, he would win it ninety times

He was expressing an intuitive feeling about the outcome, which was based onyears of experience and on the facts as he knew them Had he known enemyreinforcements would arrive, and French would not, he would have revised theestimate of his chances downward Probability is thus seen to be a relative thing,depending on the state of knowledge of the observer As another example, astudent might decide to study only certain sectiom of the text for an exam,whereas if he knew what the professor knew-namely, which questions were to

be on the exam-he could probably improve his chances of passing by studyingsome other sections

In physics, quantitative application of the concept of chance is of greatimportance There are several reasons for this For example, it is frequentlynecessary to describe quclntitatively systems with a great many degrees offreedom, such as a jar containing 10z3 molecules; however, it is, as a practicalmatter, impossible to know exactly the positions or velocities of all molecules inthe jar, and so it is impossible to predict exactly whalt will happen to each mole-cule This is simply because the number of molecules is so great It is then neces-sary to develop some approximate, statistical way to describe the behavior of themolecules, using only a few variables Such studies Jorm the subject matter of a

branch of physics called stofistical mechanics.

Secondly, since 1926 the development of quantum mechanics has indicatedthat the description of mechanical properties of elementary particles can only

be given in terms of probclbilities These results frown quantum mechanics have

11

1 2 Probability

profoundly affected the physicist’s picture of nature, which is now conceived andinterpreted using probabilities

Thirdly, experimental measurements are always subject to errors of one sort

or another, so the quantitative measurements we make always have some certainties associated with them Thus, a person’s weight might be measured as176.7 lb, but most scales are not accurate enough to tell whether the weight

un-is 176.72 lb, or 176.68 lb, or something in between All measuring instrumentshave similar limitations Further, repeated measurements of a quantity willfrequently give different values for the quantity Such uncertainties can usually

be best described in telrms of probabilities

2.1 DEFINITION OF PRCIBABILITY

To make precise quaniii,tative statements about nature, we must define the cept of probability in a quantitative way Consider an experiment having anumber of different possible outcomes or results Here, the probability of a par-ticular result is simply the expected fraction of occurrences of that result out of avery large number of repetitions or trials of the experiment Thus, one could ex-perimentally determine the probability by making a large number of trials andfinding the fraction of occurrences of the desired result It may, however, beimpractical to actually repeat the experiment many times (consider for examplethe impossibility of fighting the Battle of Waterloo more than once) We thenuse the theory of probability; that is a mathematical approach based on a simpleset of assumptions, or postulates, by means of which, given a limited amount ofinformation about the situation, the probabilities of various outcomes may becomputed It is hoped that the assumptions hold to a good approximation in theactual physical situatiomn

con-The theory of probability was originally developed to aid gamblers interested

in improving their inc~ome, and the assumptions of probability theory may benaturally illustrated v&th simple games Consider flipping a silver dollar tentimes If the silver dollar is not loaded, on the average it will come down headsfive times out of ten ‘The fraction of occurrences of heads on the average is

‘,‘,, or % Then we say that probability P(heads) of flipping CI head in one try isP(heads) = % S i m i l a r l y , t h e p r o b a b i l i t y o f f l i p p i n g a t a i l i n o n e t r y i sP(tails) = %

In this example, it is assumed that the coin is not loaded This is equivalent tosaying that the two sides of the coin are essentially identical, with a plane ofsymmetry; It I S then reasonable to assume that since neither side of the coin isfavored over the other, on the average one side will turn up as often as the other.This illustrates an important assumption of probability theory: When there areseveral possible alternatives and there is no apparent reason why they should

occur with different frequencies, they are assigned equal probabilities This is sometimes called the postulate of equal a priori probabilities.

2.2 Sums of probabilities 1 3 2.2 SUMS OF PROBABILITIE’S

Some general rules for combining probabilities are also illustrated by the flipping experiment In every trial, it is certain that either heads or tails will turn

coin-up The fraction of occurrences of the result “either heads, or tails” must be unity,and so

P(either heads or tails) = 1 (2.1)

In other words, the probability of an event which is certain is taken to be 1

Further, the fraction of lheads added to the fraction of tails must equal thefraction of “either heads or tails,” and so

P(either heads or tails) = P(heads) + P(tails) (2.2)

In the special case of the fak coin, both P(heads) and P(tails) are ‘/:t, and theabove equation reduces to 1 = % + %

M o r e g e n e r a l l y , i f A , B, C, are events that occur with probabilitiesP(A), P(B), P(C), , then the probability of either A or B occurring will be given

by the sum of the probabilities:

P(either A or B) = P(A) + I’(B)

Similarly, the probability of elither A or B or C occurring will be

(2.3)

P(either A or 6 or C) = P(A) + P(B) + P(C) (2.4)

Here it is assumed that the labels A, 6, C, refer to mutually exclusive tives, so that if the event A occurs, the events B, C, cannot occur, and so on.The above relation for combining probabilities simply amounts to addition of thefractions of occurrences of the various events A, B and C, to find the total frac-tion of occurrences of some one of the events in the set A, 6, C

alterna-These relations may easily be generalized for any number of alternatives Forexample, consider an experiment with six possible outcomes, such as the sixpossible faces of a die which could be turned up wheil the die is thrown Imaginethe faces numbered by an index i that varies from 1 to 6, and let P, be theprobability that face i turns up when the die is thrown Some one face willdefinitely turn up, and so the total probability that some one face will turn up will

be equal to unity, Also, the probability that some one face will turn up is thesame as the probability that either face one, or face two, or face three, or, ,

or face six will turn up This will be equal to the sum of the individual ties P, Mathematially,

probabili-1 =f:P,

In words, this equation expresses the convention that the probability of an eventwhich is certain is equal to I, It also utilizes a generalization of the rule given inEquation (2.3), which says the probability of either A or B is the sum of theprobabilities of A and of B

1 4 Probability

2.3 CALCULATION OF PROBABILITIES BY COUNTING

Given a fair die, there is no reason why the side with the single dot should come

up more often than the side with five dots, or any other side Hence, according tothe postulate of equal a priori probabilities, we may say that P, = P,, and,

indeed, that P, = P2 = P3 = P, = P, =: P, Then ~~=I P, = 6P, = 1, or

PI = ‘/, a n d h e n c e .P, = ‘/, f o r a l l i T h i s s i m p l e c a l c u l a t i o n h a s y i e l d e dthe numerical values of the probabilities P, A general rule which is very useful

in such calculations may be stated as follows:

The probability of a particular event is the ratio of the number of ways this eventcan occur, to the fatal number of ways o/l possible events can occur

Thus, when a die is thrown,, six faces can turn up There is only one face that hastwo dots on it Therefore, the number of ways a two dot face can turn up, divided

by the total number of ways all faces can turn up, is ‘/,

If one card is drawn at random from a pack of cards, what is the probabilitythat it will be the ace of spades? Since the ace of spades can be drawn in onlyone way, out of a total of 52 ways for all possible cards, the answer is

p = (1 ace of spades)(52 possible cards)

o r P = %, Likewise, if one card is drawn from a pack, the probability that itwill be an ace is (4 aces),1(52 possible cards) or P = “/:, = I/,, We can alsoconsider this to be the sum of the probabilities of drawing each of the four aces

2.4 PROBABILITY OF SEVERAL EVENTS OCCURRING TOGETHER

Next we shall consider o slightly more complicated situation: flipping a cointwice What is the probability of flipping two heads in succession? The possibleoutcomes of this experiment are listed in Table 2.1

TABLE 2.1 Different possible

o~utcomes for flipping a coin twice

Krst F l i p Second Flip heads

heads tails tails

heads tails heads tails

Since there are two possible outcomes for each flip, there are two times two orfour possible outcomes for the succession of two coin flips Since there is noreason to assume that one of these four outcomes is more probable than another,

we may assign each of the four outcomes equal probabilities of VI The total

2.5 Calculating probabilities 15

number of outcomes is the product of the number of outcomes on the first flip andthe number of outcomes on the second flip, while the number of ways of gettingtwo heads is the product of the number of ways of getting a head on the firstflip and the number of ways of getting a head on the second flip Thus,

P(two heads in succession)

in any way-then the probability of both A and 6 occurrin’g is

In words, the probability of two independent events both occurring is equal tothe product of the probabilities of the individual events

nple If you throw a six-sided die and draw one card from a pack, the probability thatyou throw a six and pick an ace (any ace) is equal to

Another way to obtain the answer is to divide the number Iof ways of getting thesix and any ace (1 x 4), by the total number of ways of getting all possibleresults (6 x 52), or

in this case

(1x4) 1(6 x 52) = 78

2.5 SUMMARY OF RULES FOR CALCULATING PROBABILITIES

We may summarize the important features of the probability theory disf:ussed sofar in the following rules:

(1) The probability of an event that is certain is equal to 1

(2) In a set of events that can occur in several ways, the probability of aparticular event is the number of ways the particular event may occur, dilvided bythe total number of ways all possible events may occur

1 6 hbobi/;ty

(3) (Postulate of equal a priori probabilities): In the absence of any contraryinformation, equivalent possibilities may be assumed to have equal probabilities.(4) If A and B are mutually exclusive events that occur with probabilities

P(A) and P(6), then the probability of either A or 6 occurring is the sum of the

individual probabilities:

P ( A o r 6) = P ( A ) + P ( B ) (2.9)

(5) If A and 8 are independent events that occur with probabilities P(A)and P(B), then the probability of both A and 6 occurring is the product of theindividual probabilities:

P(A and B) = P(A)P(B) (2.10)

2.6 DISTRIBUTION FUNCTIONS FOR COIN FLIPPING

In order to introduce the idea of a distribution function, we continue with someexamples of coin-tossing Distribution functions are functions of one or more inde-pendent variables which label the outcomes of some experiment; the distributionfunctions themselves are proportional to the probabilities of the various out-comes (in some case’s they are equal to the probabilities) The variables might

be discrete or continuous Imagine, for example, a single experiment consisting

of flipping a coin N times, when N might be some large integer Let nH be thenumber of times heads turns up in a particular experiment If we repeat thisexperiment many times, then nH can vary from experiment to experiment Weshall calculate the probability that n,, heads will turn up out of N flips; thisprobability will be denoted by P,., (rt”) H ere the independent variable is nH;and the quantity P,{n,), which for fixed N is a function of n,,, is an example

of a distribution function In this case, the function only has meaning if nH i s anonegative integer not glreater than N

To get at the problem of finding P,(nH), we define PHI to be the probability

of getting a head in the first toss and PT1 to be the probability of getting a tail(both are % for a fair coin but differ from, % for a weighted coin) Then P,, +

P T1 = 1 Likewise folr the second toss, PHZ $- Pr2 = 1 I f t h e s e t w o e x p r e s s i o n sare multiplied together, we get P HlPHP + PHIPK + PTlPH2 + PT1PT2 = 1.Note that these four ,termls correspond to the four possibilities in Table 1, and thateach term gives the probability of getting the heads and tails in a particularorder

In N tosses,

(PHI + PTI)(PH? + Pn)**-(PHN + PrrJ) = 1 (2.11)and when the products on the left are carried out, the various terms give theprobabilities of getting heads and tails in a particular order For example, inthree tosses, the product of Equation (2.1 1) contains eight terms, one of which isPT,PH2PT3 This is equal to the probability of getting a tail, a head and atail, in that order, in three tosses If we were interested only in the probability of

2.6 Disfribulion functions for coin flipping 1 7

getting a given total number of heads nH in N tosse,j regardless of order, we

%would take all the terms which contain nH factors of the form P,,,, regardless ofthe subscript numbers, and simply find their sum This is equivalent to droppingall numerical subscripts and combining terms with similar powers of P,

If the expression on the left of the equation, (P, t Pr)” = 1, is expanded,the term proportional to (PH)“H(PT)N-nH .I S the probability of getting nH headsond N - n,, tails in N tosses, regardless of order I\ccording to the binomialtheorem

of order, or in our case, the number of ways of getting nH heads in N tosses Thus,

a given term is the total number of different ways of getting nH heads times theprobability, (P,.,)“H(PT)Nm”H, of getting nn heads in a’ne of these ways There-fore, in the special case of a fair coin when P,, = PT = !/2, the probability ofgetting nH heads in N tosses, regardless of order, is

PN(“H) = ;,,!(N - n,)! 2N

In Figures 2.1 through 2.4, the probability P,.,(nH) of Equation 2.13 is plotted

as o function of nH for N = 5, ‘I 0, 30 and 100 It may he seen that as N becomeslarger, the graph approaches a continuous curve with a symmetrical bell-like

,shape The function P,.,(n,) i:, called a probability disfribution function, because

id gives a probability as a function of some parameter, in this case n,,

lple l(a) Consider a coin which is loaded in such a way that the probability PH offlipping a head is PH = 0.3 The probability of flipping a tail is then PT = 0.7

‘What is the probability of flipping two heads in four tries?

lion Use Equation (2.13) with N = 4, nH = 2; the required probability is

2.7 More than two possible oufcomes 19

p/e 2(a) If the probability of getting ail the forms filled out correctly at registration

is 0.1, what is the probability of getting all forms filled out properly only onceduring registrations in three successive terms?

ion The probability of not getting the forms correct is 0.9 each time Then the desiredprobability is

Suppose we consider another experiment in which therl? are four possible results,

A, B, C, and D, in o single tricrl The probabilities for each result in this trial ore,respectively, PA, Pg, PC and Pr, = 1 - PA - Ps - P, If the quantity on the leftside of the equation

(PA + PB + PC + PD)N = 1 (2.14)

is expanded, the term proportional to

is the probability that in N trials result A occurs nA times, 6 occurs ‘n, times,

C occurs nc times and, of course, D occurs no times, with nr, = N -~ nA - ns - nc

A generalized multinomial expansion may be written OS follows:

The generalizotion to the ca’se of any number of alternatives in the results of asingle trial is obvious

92 twice, 93 once, 94 four times, 95 twice Figure 2.5 is a plot of the number,

5-90 91 92 93 94 95 IFigure 2.5 Grade distribution function.

f(n), of times the grade n was made, as a function of n This function f(n) is alsocalled a distribution function, but it is not a probability distribution function,

since f(n) is the number of occurrences of the grade n, rather than the

proba-bility of occurrences of the grade n To compute the average grade, one mustadd up all the numerlical grades and divide by the total number of grades Using

the symbol ( n ) to d enote the average of n, we have

may be written as (1 x 91) + (2 x 92) + (1 x 93) + (4 x 94) + (2 x 95) or,

in terms of n and f(n), the numerator is c n f(n), where the summation is overall possible n In the denominator, there is a 1 for each occurrence of an exam.The denominator is then the total number of exams or the sum of all the f(n).Thus, a formula for the denominator is c f(n), summed over all n Now we can

As a further example, suppose you made grades of 90, 80, and 90 on threeexaminations The expectation value of your grade Nould be (80 + 2 x 90)/( 1 + 2 ) = 8 6 6 7

2 9 NORMAUZATION

For any distribution function f(n), the value of the reciproc:al of the sum c f(n) iscalled the normalization of the distribution function It 1: f(n) == N, we say thatf(n) is normalized to the value N, and the normalization is l/N Since the sum

of the probabilities of all events is unity, when f(n) is a probability distributionfunction, it is normalllzed to ‘Jnity:

Cf(n) = 1

Equation (2.18) refers to the expectation of the ndependent variable, (n).However, in some applications it might be necessary to know the expectationvalues of n2, or n3, or of some other function of n In general, to find the average

or expectation value of a function of n, such as A(n), one rnay use the equation:

For a more detailed example of an expectation valut: calculation, we return tothe flipping of a coin As was seen before, if a number of experiments are per-formed in each of which the coin is flipped N times, we would expect that, on theaverage, the number of heads would be N/2, or (17~) = N/2 To obtain thisresult mathematically using Equation (2.18), we shall evaluate the sum

(nH) = 2 hP,(nH)

‘The result is indeed N/:2 The reader who is not interested in the rest of the details

of the calculation can skip to Equation (2.26).

W e h a v e t o e v a l u a t e t h e s u m m a t i o n i n (n,+) = ~~~=on,N!/[2NnH!(N - n,)!].

We can calculate this by a little bit of relabeling First, note that the term

corre-s p o n d i n g t o nH = 0 doecorre-s not contribute to the corre-sum becaucorre-se the factor nH icorre-s incorre-side the sum, and in the denominator there is O!, which is defined to be 1 Therefore, instead of going from 0 to N, the sum goes effectively from 1 to N It is easily verified that after usinlg the following identities:

N tosses many times.

2.11 EXPERIMENTAL DETERMINATION OF PROBABILITY

O u r p r e v i o u s d i s c u s s i o n h a s s u g g e s t e d t h a t w e c o u l d e x p e r i m e n t a l l y m e a s u r e the probability of some particular result by repeating the experiment many times That is, the probability of an event should be equal to the fractional number of times it occurs in a series of trials For example, if you know a coin is loaded, you

c a n n o t a s s u m e t h a t P(heads) = P(tails), a n d i t m i g h t b e d i f f i c u l t t o c a l c u l a t e these probabilities theoretically One way to find out what P(heads) is, would be