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 byHolden-Day, Inc.,

500 Sansome StreetSan Francisco, California

All rights reserved.No part of this book

may be reproduced in any form,by mimeograph or any

other means, withoutpermission in writing from

the publisher.library of Congress Catalog

Card Number: 71-l 13182Manufactured in

the United States of America

HOLDEN-DAY SER IES IN PHYS ICS

McAllister Hull and David S. Saxon, Editors

preface

This book is intended as a general introduction to modern physics for science andengineering students. I t i s wr itten at a level which presurnes a pr ior tull yearscourse in classical physics, and a knowledge of elementary differential andintegral calculus.

The mater ial discussed here includes probabi l i ty, relat iv ity, quantum me-chanics, atomic physics, statist ical mechanics, nuclear physics and elementaryparticles. Some of these top&, such as statistical mechanics and probability, areordinar i ly not included in textbooks at th is level . However, we have felt that forproper understanding of many topics in modern physics--such as quaIlturn me-chanics and its appl ications--this mater ial is essential . I t i s our opilnion thatpresent-day science and engineering students should be able to worlk quant i -tat ively with the concepts of modern physics. Therefore, we have attempted topresent these ideas in a manner which is logical and fair ly r igorous. A number oftopics, especial ly in quantum1 mechanics, are presented in greater depth than iscustomary. In many cases, unique ways of presentat ion are given which greatlys implify 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.

Unify ing concepts which ha lve important appl ications throughout modernphys ics, such as re lat iv i ty, probabi l i ty and the laws of conservat ion, have beenstressed. Almost al l theoretical developments are l inked 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 blocksof material which are somewhat independent of each other.

The system of units primari ly used throughout is the meter-ki logram-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 prel iminary term for a number of years. We also thank ourcol leagues and reviewers for their constructive cr it icism. F inal ly, we wish to ex-press our thanks to Mrs. Ruth Wilson for her careful typing of the manuscript.

vii

contents

1 INTRODUCTION1 .l HISTORICAL SURVEY

1.2 NOTATION AND UNITS1.3 UNITS OF ENERGY AND MOMENTUM

1.4 ATOMIC MASS UNIT1.5 PROPAGATION OF WAVES; PHASE AND GROUP SPEEDS

1.6 COMPLEX NUMBERS2 P R O B A B I L I T Y

2.1 DEFINITION OF PROBABILITY2.2 SUMS OF PROBABILITIES

2.3 CALCULATION OF PROBABILITIES BY COUNTING2.4 PROBABILITY OF SEVERAL EVENTS OC:CUF!RING TOGETHER

2.5 SUMMARY OF RULES FOR CALCULATINIG PROBABILITIES2.6 DISTRYBUTION FUNCTIONS FOR COIN FLIPPING

2.7 DISTRIBUTION FUNCTIONS FOR MORE THAN TWO POSSIBLEOUTCOMES

2.8 EXPECTATION VALUES2.9 NORMALIZATION

2.10 EXPECTATION VALUE OF THE NUMBER OF HEADS2.1 1 EXPERIWIENTAL DETERMINATION OF PROBABILITY

2.12 EXPERIMENTAL ERROR2.13 RMS DEVIATION FROM THE MEAN

2.114 RMS DEVIATION FOR COIN FLIPPING2.15 ERRORS IN A COIN-FLIPPING EXPERIMENT

2.16 ERRORS IN AVERAGES OF REPEATED EXPERIMENTS2.17 PROBABILITY DENSITIES

2.18 EXPECTATION VALUES FROM PROBABILITY DENSITIES2.19 GAUSS1A.N DISTRIBUTION

2.20 EXPECTATION VALUES USING A GAUSS1A.N DISTRIBUTIONSUMh\ARYPROBLEMS

3 SPECIAL THEORY OF RELATIVITY3.1 CONFLICT BETWEEN ULTIMATE SPEED AND NEWTONS LAWS

113.4.5(6:3

II1 :21 :314141:s16

19202 I2 I222424252728303:!3435373Ei42142

ix

3.2 CLASSICAL MOMENTUM AND EINERGY CONSERVATION-COINFLICT WITH EXPERIMENT

3.3 CONSERVATION OF MASS-COlNFLICT WITH EXPERIMENT3.4 CORRESPONDENCE PRINCIPLE

3 .5 INERT IAL SYSTEMS3 . 6 N O N - I N E R T I A L S Y S T E M S

3.7 AXES RELATIVE TO FIXED STARS3.8 GALILEAN TRANSFORMATIONS

3.9 GALILEAN VELOCITY TRANSFORMATIONS3.10 SECGND LAW OF MOTION UNDER GALILEAN

TRANSFORMATIONS3.11 THIRD LAW UNDER GALILEAN TRANSFORMATIONS

:3.12 MICHELSON-MORLEY EXPERIMENT3.13 POSTULATES OF RELATIVITY

3.14 EXPERIMENTAL EVIDENCE FOR THE SECOND POSTULATE3.15 GALILEAN TRANSFORMATIONS AND THE PRINCIPLE OF

RELATIVITY3.16 TRANSFORMATION OF LENGTHS PERPENDICULAR TO THE

RELATIVE VELOCITY3.17 TIME DILATION

3.18 LENGTH CONTRACTION3.19 LORENTZ TRANSFORMATIONS

3 .20 S IMULTANE I TY3.21 TRANSFORMATION OF VELOCITIES

SUMMARYPROBLEMS

4 RELATIVISTIC MECHANICS AND DYNAMICS4.1 LORENTZ TRANSFORMATIONS

4.2 DISCREPANCY BETWEEN EXPERIMENT AND NEWTONIANMOMENTUM

4.3 MOMENTUM FROM A THOUGHT EXPERIMENT4.4 EXPERIMENTAL VERIFICATION OF MASS FORMULA

4.5 RELATIVISTIC SECOND LAW OF MOTION4.6 THIRD LAW OF MOTION AND CONSERVATION OF

MOMENTUM4.7 RELATIVISTIC ENERGY

4.8 KINETIC ENERGY4.9 POTENTIAL ENERGY AND CONSERVATION OF ENERGY

4.10 EXPERIMENTAL VERIFICATION OF EQUIVALENCE OF MASSAND ENERGY

4.11 RELATIONSHIP BETWEEN ENERGY AND MOMENTUM4:12 REST MASS (OF ilo F R O M E X P E R I M E N T

4.13 TRANSFORMATION PROPERTIES OF ENERGY ANDMOMENTUM

4344474749505152

5354545557

59

59606465677174767979

80818385

85868788

898990

96

Contents xi

4.14 TRANSFORMATIONS FOR FREQUENCY AND WAVELENGTH4.15 TRANSVERSE DijPPLER EFFECT

4.16 LONGITUDINAL DOPPLER EFFECTSUMMARYPROBLIfMS

5 QUANTUM PROPERTIES OF LIGHT5.1 ENERGY TRANSFORMATION FOR PARTICLES OF ZERO REST

MASS5.2 FORM-INVARIANCE OF E = hv

5 . 3 T H E D U A N E - H U N T L.AW5.4 PHOTOELECTRIC EFFECT

5 . 5 COMPTON E F F E C T5.6 PAIR PRODUCTION AND ANNIHILATION

5.7 UNCERTAINTY PRINCIPLE FOR LIGHT WAVES5.8 MOMENTUM, POSITION UNCERTAINTY

5.9 PROBABILITY INTERPRETATION OF AMPLITUIDESSUMMARYPROBLEMS

6 M A T T E R W A V E S6.1 PHASE OF .4 PLANE WAVE

6.2 INVARIANCE OF THE PHASE OF .A PLANE WAVE6.3 TRANSFORMATION EQUATIONS FOR WAVEVECTOR A,ND

FREQUENCY6.4 PHASE SPEED OF DE BROGLIE WAVES

6.5 PARTICLE INCIDENT ON INTERFACE SEPARATING DIFFERENTPOTENTIAL ENERGIES

6.6 WAVE RELATION AT INTERFACE6.7 DE BROGLIE RELATIONS

6.8 EXPERIMENTAL DETERMINATION OF A6 . 9 BRA.GG EQUAT ION

6.10 DIFFRACTION OF ELECTRONS6.11 UNCERTAINTY PRINCIPLE FOR PARTICLES

6.12 UNCERTAINTY AND SINGLE SLIT DIFFRACTION6.13 UNCERTAINTY IN BALANCING AN OBJECT

6.14 ENERGY-TIME UNCERTAINTY6.15 PROBABILITY INTERPRETATION OF VVAVEFUNCTllON

6.16 EIGENFUNCTIONS OF ENERGY AND MOMENTUMOPERATORS

6.17 EXPECTATION VALUES FOR MOMENTUM IN A PARTICLEBEAM

6.18 OPERATOR FORMALISM FOR CALCULATION OF MOMENTUMEXPECTATION VALLJES

6.19 ENERGY OPERATOR AND EXPECTATION VALUES6 . 2 0 SCHRODINGER EQUATllON

99101102104105110

111112113115111912312612812913113i136136138

139141

143144145146147148152152155155156

158

160

162164165

xi i Contents

6.21 SCHRijDlNGER EQUATION FOR VARIABLE POTENTIAL6.22 SOLUTION OF THE SCHRijDlNGER EQUATION FOR A

CONSTANT POTENTIAL6.23 BOUNDARY CONDITIONS

SUMMARYPROBLEMS

167

7 EXAMPLES OF THE USE OF SCHRiiDINGERS EQUATION7.1 FREE PARTICLE GAUSSIAN WAVE PACKET

7.2 PACKET AT t = 07.3 PACKET FOR t > 0

7.4 STEP POTENTIAL; HIGH ENERGY E > V,7.5 BEAM OF INCIDENT PARTICLES

7.6 TRANSMISSION AND REFLECTION COEFFICIENTS7.7 ENERGY LESS THAN THE STEP HEIGHT

7.8 TUNNELING FOR A SQUARE POTENTIAL BARRIER7.9 PARTICLE IN A BOX

7.10 BOUNDARY CONDITION WHEN POTENTIAL GOES TOINFINITY

169170172175178178180181183185186187188190

7.11 STANDING WAVES AND DISCRETE ENERGIES7.12 MOMENTUM AND UNCERTAINTY FOR A PARTICLE

IN A BOX

192192

7.13 LINEAR MOLECULES APPROXIMATED BY PARTICLE IN A BOX7.14 HARMONIC OSCILLATOR

7.15 GENERAL WAVEFUNCTION AND ENERGY FOR THEHARMONIC OSCILLATOR

7.16 COMPARISON OF QIJANTUM AND NEWTONIANMECHANICS FOR THE HARMONIC OSCILLATOR

7.17 CORRESPONDENCE PRINCIPLE IN QUANTUM THEORYSUMMARYPROBLEMS

194195196

198

8 HYDROGEN ATOM AND ANGULAR MOMENTUM8.1 PARTICLE IN A BOX

8.2 BALMERS EXPERIMENTAL FORMULA FOR THE HYDROGENSPECTRUM

204207208209213213

8.3 SPECTRAL SERIES FOR HYDROGEN8.4 BOHR MODEL FOR HYDROGEN

8.5 QUANTIZATION IN THE BOHR MODEL8.6 REDUCED MASS

8.7 SCHRoDlNGER EQUATION FOR HYDROGEN8.8 PHYSICAL INTERPRETATION OF DERIVATIVES WITH RESPECT

T O r

215216217218220221

8.9 SOLUTIONS OF THIE SCHRijDlNGER EQUATION8.10 BINDING ENERGY AND IONIZATION ENERGY

8.11 ANGULAR MOMENTUM IN QUANTUM MECHANICS8.12 ANGlJLAR MOMENTUM COMPONENTS IN SPHERICAL

223225230230

COORDINATES 231

C o n f e n t s **XIII

8.13 EIGENFUNCTIONS OF L,; AZIMUTHAL QUANTU,M NUMBER8.14 SQUARE OF THE TOTAL ANGULAR MOMENTUM

8.15 LEGENDRE POILYNOMIALS8.16 SlJMMARY OF QUANTUM NUMBERS FOR THE

HYDROGEN ATOM8.17 ZEEMAN EFFECT

8.18 SPLITTING OF LEVELS IN A MAGNETIC FIELD8.19 SELECTION RULES

8.20 NORMAL ZEEMAN SPLITTING8.21 ELECTRON SPIN

8.22 SPIN-ORBIT INTERACTION8.23 HALF-INTEGRAL SPINS

8.24 STERN-GERLACH EXPERIMENT8.25 SUMS OF ANGULAR ,MOMENTA

8.26 ANOMALOUS ZEEMAN EFFECT8.27 RIGID DIATOMIC ROTATOR

SUMMARYPROBLEMS

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 E9.1 DESIGNATION OF ATOMIC STATES

9.2 NUMBER OF STATES IN AN n SHELL9.3 INDISTINGUISHABILITY OF PARTICLES

9.4 PAULI EXCLUSION PRINCIPLE9.5 EXCLUSION PRINCIPLE AND ATOMIC ELECTRON STATES

9.6 ELECTRON CONFIGURATIONS9.7 INERT GASES

9.8 HALOGENS9 . 9 ALKAILI METALS

9.10 PERIODIC TABLE OF THE ELEMENTS9.1 11 X-RAYS

9.12 ORTHO- AND PARA-HYDROGEN!jUMMARYPROBLEMS

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 S10.1 PROBABILITY DISTIPIBUTION IN ENERGY FOR SYSTEMS IN

THERMAL EQ~UILIBRIUM10.2 BOLTZMANN DISTRIBUTION

10.3 PROOF THAT P(E) IS OF EXPONENTIAL FORM10.4 PHA!jE SPACE

10.5 PHASE SPACE DISTRIBUTION FUNCTIONS10.6 MAXWELL-BOLTZMANN DISTRIBUTION

10.7 EVALUATION OF /I10.8 EVALUATION OIF NP(O)p

lo..9 MAXWELL-BOLTZMANN DISTRIBUTION INCLUDINGPOTENTIAL ENERGY

10.10 GAS IN A GRAVITATIONAL FIELD

232233234

235236237238239240240241242242243244246249254255256256258260262263265265266270273273275279

280281282283285287288291

292293

xiv Contenfs

10.11 DISCRETE ENERGIES10.12 DISTRIBUTION OF THE MAGNITUDE OF MOMENTUM

10.13 EXPERIMENTAL VERIFICATION OF MAXWELL DISTRIBUTION10.14 DISTRIBUTION OF ONE COMPONENT OF MOMENTUM

10.15 SIMPLE HARMONIC OSCILLATORS10.16 DETAILED BALANCE10.17 TIME REVERSIBILITY

SUMMARYPROBLEMS

11 QUANTUM STATISTICAL MECHANICS

11.1 EFFECTS OF THE EXCLUSION PRINCIPLE ON STATISTICSOF PARTICLES

11.2 DETAILED BALANCE AND FERMI-DIRAC PARTICLES11.3 FERMI ENERGY AND FERMI-DIRAC DISTRIBUTION

11.4 ONE DIMENSIONAL DENSITY OF STATES FOR PERIODICBOUNDARY CONDITIONS

11.5 DENSITY OF STATES IN THREE DIMENSIONS11.6 COMPARISON BETWEEN THE CLASSICAL AND QUANTUM

DENSITIES OF STATES11.7 EFFECT OF SPIN ON THE DENSITY OF STATES

11.8 NUMBER OF STATES PIER UNIT ENERGY INTERVAL11.9 FREE PARTICLE FERMI ENERGY-NONDEGENERATE CASE

11.10 FREE ELECTRONS IN METALS-DEGENERATE CASE11.11 HEAT CAPACIITY OF AN ELECTRON GAS

11.12 WORK FUNCTION11 .lm 3 PHOTON DISTRIBUTION

11.14 PLA.NCK RADIATION FORMULA11 .15 SPONTANEOUS EMISSION

11.16 RELATIONSHIP BETWEEN SPONTANEOUS AND STIMULATEDEMISSION

11.17 ORIGIN OF THE FACTOR 1 + II, IN BOSON TRANSITIONS1 I .18 BOSE-EINSTEIN DISTRIBUTION FUNCTION

SUMMARYPROBLEMS

112 SOLID STATE PHYSICS

12.1 CLASSIFICATION OF CRYSTALS12.2 REFLECTION AIND ROTATION SYMMETRIES

12.3 CRYSTAL BINDING FORCES12.4 SOUND WAVES IN A CONTINUOUS MEDIUM

12.5 WAVE EQUATION FOR SOUND WAVES IN A DISCRETEMEDIUM

12.6 SOLUTIONS OF THE WAVE EQUATION FOR THE DISCRETEMEDIUM

12.7 NUMBER OF SOLUTIONS12.8 LINEAR CHAIN WITH TWO MASSES PER UNIT CELL

294295

296298300

303305

306308

312

313313315

316318

319320320321

323324325326

328331

332333335336338

341

341342346347

349

351352354

contents xv

12.9 ACOUSTIC AND OPTICAL BRANCHES12.10 ENERGY OF LATTICE VIBRATIONS

12.11 ENERGY FOR A SUPERPOSITION OF MODES12.12 QUANTUM THIEORY OF HARMONIC OSCILLATORS AND

LATTICE VIBRATIONS12.13 PHONONS; AVEl?AGE ENERGY PER MODE AS A FUNCTION

O F TEMPERATIJRE12.14 LATTICE SPECIFIC HEAT OF A SOLID

12.15 ENERGY BANDS OF ELECTRONS IN CRYSTALS12.16 BLOCHS THEOREM

12.17 NUMBER OF BLOCH FUNCTIONS PER BAND12.18 TYPES OF BANDS

12.19 EFFECTIVE MASS IN A BAND12.20 CONDIJCTORS, INSULATORS, SEMICONDUCTORS

1 2 . 2 1 H O L E S12.2;! n-TYPE AND p-TYPE SEMICONDUCTORS

12.23 H.ALL EFFECTSUMMARYPROBLEMS

13 PROBING THE NUCLEUS13.1 A NUCLEAR MODEL

13.2 LIMITATIONS ON NUCLEAR SIZE FROM ATOMICCONSIDERATIONS

13.3 SCATTERING EXPERIMENTS13.4 CROSS-SECTIONS

13.5 DIFFERENTIAL CROSS-SECTIONS13.6 NUMBER OF SCATTERERS PER UNIT AREA

13.7 BARN AS A UNIT OF CROSS-SECTION13.8 a AND @ PARTICLES

13.9 RUTHERFORD MODEL OF THE ATOM13.10 RUTHERFORD THEORY; EQUATION OF ORBIT

113.11 RUTHERFORD SCATTERING ANGLE13.12 RUTHERFORD DIFFERENTIAL CROSS-SECTION

13.13 MEASUREMENT OF THE DIFFERENTIAL CROSS-SECTION13.14 EXPERIMENTAL VERIFICATION OF THE RLJTHERFORD

SCATTERING FORMlJLA13.15 PARTICLE ACCELERATORS

SUMMARYPROBLEMS

14 NUCLEAR STRUCTURE1 4 . 1 NUCLEC\R M A S S E S

14.2 NEUTRONS IN THE NUCLEUS14.3 PROPERTIES OF THE NEUTRON AND PROTON

1 4 . 4 T H E DEUTERON (,H)14.5 NUCLEAR FORCES1 4 . 6 YUKAWA F O R C E S

356357359

360

361362364365366367368369371372373374377

381381

383385386387390390391393394395397398

400402404405408408410411414416418

xvi Contents

14.7 MODELS OF THE NUCLEUSSUMMARYPROBLEMS

1 5 TRANSFORMsATlON O F T H E N U C L E U S15.1 LAW OF RADIOACTIVE DECAY

15.2 HALF-LIFE15.3 LAW OF DECAY FOR UNSTABLE DAUGHTER NUCLEI

15.4 RADIOACTIVE SERIES15.5 ALPHA-PARTICLE DECAY

15.6 THEORY OF ALPHA-DECAY1 5 . 7 B E T A D E C A Y

15.8 PHASE SPACE AND THE: THEORY OF BETA DECAY15.9 ENERGY IN p+ DECAY

15.10 ELECTRON CAPTURE15.11 GA,MMA DECAY AND INTERNAL CONVERSION

15.12 LOW ENERGY NUCLEAR REACTIONS15.13 THRESHOLD ENERGY

15.14 NUCLEAR FISSION AND FUSION15.15 RADIOACTIVE CARBON DATING

SUMMARYPROBLEMS

16 ELEMENTARY PARTICLES1 6 . 1 L E P T O N S1 6 . 2 M E S O N S

1 6 . 3 B A R Y O N S16.4 CONSERVATION LAWS

16.5 DETECTION OF PARTICLES16.6 HYPERCHARGE, ISOTOPIC SPIN PLOTS

1 6 . 7 Q U A R K S16.8 MESONS IN TERMS OF QUARKS

SUMMARYPROBLEMS

APPENDICESAPPENDIX 1APPENDIX 2APPENDIX 3APPENDIX 4

BIBLIOGRAPHYINDEX

421427429

431431433433433441443447450452453454454456457458458461464464466467468472473474477478479

483491496504505507

principles of modern physics

1 introduction

I .1 HISTORICAL SURVEY

The term modern physics general ly refers to the study

2 introduction

Planck to explain the intensity distr ibution of black-body radiation. This occurred

several years before Einstein published his special theory of relativi ty in 1905.

At about this time, Einstein also applied the quantum hypothesis to photons in an

explanation of the photoelectr ic effect. This hypothesis was found to be con-

sistent with special relativity. Similarly, Bohrs postulate-that the electrons

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 fol lowed in the f i rst quarter of the century by

a number of refinements and ad hoc quantization rules; these, however, achieved

only l imited success. I t was not unti l after 1924, when Louis de Broglie proposed,

on the basis of relativity theory, that waves were associated with material par-

t icles, that the foundations of a correct quantum theory were laid. Fol lowing

de Broglies suggestion, Schrodinger in 1926 proposed a wave equation describ-

ing the propagation of these particle-waves, and developed a quantitative

explanation of atomic spectral l ine intensit ies. In a few years thereafter, the

success of the new wave mechanics revolutionized physics.

Fol lowing the discovery of electron spin, Paulis exclusion pr inciple was r igor-

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. Statist ical 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. Blochs treatment of electron waves in crystals

simplif ied the application of quantum theory to problems of electrons in solids.

Dirac, while investigating the possible f i rst order wave equations al lowed by

relativity theory, discovered that a posit ively charged electron should exist; this

part icle, 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 Schrodingers 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.) Start ing 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 st i l l extrernely active. At the present t ime some details of the inner

structure of protons, neutrons and other part icles involved in nuclear inter-

actions are just beginning to be unveiled.

Over f i f ty of the so-called elementary particles have been discovered. These

part icles are ordinari ly created by coll is ions between high-energy part icles 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.

I t should be emphasized that one of the most important unify ing concepts in

modern physics is that of energy. Energy as a conserved quantity was well-knownin classical physics. From the t ime of Newton unti l E instein, there were no funda-mental ly new mechanical laws introduced; however, the famous var iat ional

pr inciples of Hamilton and Lagrange expressed Newtonian lows in a differentform, by working with mathematical express ions 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 of

energy and mass. De Brogl ies quantum relations connected the frequency andwavelength of the wave motions associated with part ic les, with the part ic lesenergy and momentum. S:hrb;dingers wave equat ion i s obtained by certainmathematical operations performed on the expression for the energy of a system.

The most sophist icated express ions of modern-day relat iv ist ic quantum theory arevar iat ional pr inciples, which involve the energy of a system expressed inquantum-mechanical form. And, perhaps most important, the stable stat ionary

states of quantum systems are states of definite energy.Another very important concept used throughout modern physics is that of

probabi l i ty. Newtonian mechanics is a str ict ly determinist ic theory; with thedevelopment of quantum theory, however, i t eventual ly became clear thatmicroscopic events could not be precisely predicted or control led. Instead, theyhad to be descr ibed in terms of probabi l i t ies. I t i s somewhat i ronic that proba-bi l i ty was f i rst introduced into quantum theory by Einstein in connection with hisdiscovery of st imulated emission. Heisenbergs uncertainty pr inciple, and the

probabi l i ty interpretation of the Schradinger wavefunction, were sources ofdistress to Einstein who, not feel ing comfortable with a probabil ist ic theory, later

declared that he would never believe that God plays dice with the world.As a matter of convenience, we shal l begin in Chapter 2 with a br ief intro-

duction to the concept of probabi l i ty and to the rules for combining proba-bi l i t ies. This mater ial wi l l be used extensively in later chapters on the quantumtheory 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 wel l-known meter-ki loglram-second (MKS) system of units wi l l 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 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 potent ia l in voltsproduced by a point charge Q at a distance r from the position of the charge isgiven by Coulombs 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 herebecause they will be used in subsequent chapters.

In conformity with modern notation, a temperature such as 300 degreesKelv in wi l l be denoted by 300K. Boltzmanns constant wi l l be denoted byk ,, , with

k, = 1 . 3 8 x 10mz3 j o u l e s / m o l e c u l e - K (1.4)

A table of the fundammental constants is given in Appendix 4.

1 . 3 U N I T S O F E N E R G Y A N D M O M E N T U M

While in the MKS system of units the basic energy unit is the joule, in atomic andnuclear physics several other units of energy have found widespread use. Most ofthe energies occurr ing in atomic physics are given conveniently in terms of theelecfron volt, abbreviated eV. The electron volt is defined as the amount of work

done upon an electron as i t moves through a potential dif ference 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 valueof the electrons charge in coulombs. To convert energies from joules to eV, or

from eV to joules, one divides or mult ipl ies 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

2mv! = 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 mil l ion electron

volts, leading to the definition of a unit called the MeV:

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, t ravel ing with 10% of thespeed of light, would have a kinetic energy of approximately

1 ( 1 . 6 7 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, whi le momentum has units ofmass x speed, for mony appl ications in nuclear and elementary part icle physicsa unit of momentum called ,UeV/c is defined in such o sway that

1 MeV- - lo6 e kg-m/set

C C

= 5 . 3 5 1 x lo-l8 kg-m/set (1.9)

where c and e are the numerical values of the speed of l ight and electroniccharge, respectively, in MKS units. This unit of momentum is part icular ly con-venient when working with relat iv ist ic relat ions between energy and momentum,such as E = pc, for photons. Then if the momenturrl p in MeV/c is known, theenergy 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 massof the most common atom of carbon, containing six protons and six neutrons in a

nucleus surrounded by s ix electrons, is exactly 12.000000000 . . amu. This unit isconvenient when discuss ing atomic masses, which are then always very close toan integer. An older atomic mass unit, based on on atomic mass of exactly16 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 atomicmass unit is commonly useu in chemistry. Al l 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 ki lograms 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

Avogadros 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 dif ferent types of wave propagation wi l l be considered:the de Brogl ie probabi l i ty waves of quantum theory, latt ice vibrat ions in sol ids,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 z t z of + 4) (1.12)

where A and 4 are constants, and where the wavelength and f requency of thewave are given by

(1.13)

Here the angular f requency is denoted by o = o(k), to indicate that the fre-

quency is determined by the wavelength, or wavenumber k. Th is frequency-wavelength re lat ion, 01 = w(k), is cal led a dispers ion relation and arises becauseof the basic phys ical laws sat is f ied by the part icular wave phenomenon underinvest igation. For example, for sound waves in air , Newtons second law ofmotion and the adiabatic gas law imply that the dispersion relation is

where v is a constant.

w = vk (1.14)

I f the negative s ign is chosen in Equation (1.12), the result ing displacement

(omitting the phase constant b,) is

#(x, t) = A c o s ( k x - w t ) = A c o s[+ - - (f)f,]

(1.15)

This represents a wave propagating in the pos i t ive x di rect ion. Indiv idual crestsand t roughs in the waves propagate with a speed cal led the phase speed,

given by

w=ok

(1.16)

In near ly al l cases, the wave phenomena which we shal l discuss obey theprinciple of superposit ion-namely, that i f waves from two or more sources

arrive 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. I f 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 thewave length , and waves of dif fer ing wavelengths t ravel at dif ferent speeds.

Reinforcement or destructive interference can then occur as one wave gains onanother of different wavelength. The speed with which the regions of constructiveor 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

8 Introduction

whereas the group speed is

dw 1 g + 3k2J/pcl=-=-d k 2 (gk + k3J/p]2

(1.24)

I f the phase speed is a decreasing function of k, or an increasing function ofwavelength, then the phase speed is greater than the group speed, and individ-ual crests within a region of constructive interference-i .e. within a group ofwaves-t ravel f rom remcrr to front, crests disappearing at the front and reappear-ing at the rear of the group. This can easi ly be observed for waves on a poolof 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 part icles, a brief review of the elementary properties of complex

numbers is given here. A complex number is of the form # = a + ib, whereu and b are real numbers and i is the imaginary unit, iz = - 1. The real partof $ is a, and the imaginary part is b:

R e ( a + i b ) = a

Im(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 i s 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 $ by

its complex conjugate:

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

The complex exponential function, e, or exp (i@, where 0 is a real functionor number, is of particula* importance; this function may be defined by thepower series

eIa = , + (j(q) + 0 + 03 + . . .

2! 3!

=kmn=O n!

(1.29)

Then, replacing i2 everywhere that i t appears by -- 1 and col lect ing real andimaginary terms, we find that

e i6 = 1 +s+.. .+;o-$+li)+...( 5! )

= cos 0 + i sin /3

S ince {e} = eIns, we have de Moivres theorem:

ein8 = co5 f-10 + i sin n0 = (cos 0 t i sin 01

Since (e)* = e-j8, we also love the fol lowing identit ies:

Re eia =: ~0s 0 = i (e + em)

Im e = sin (j = + (e - em)

/ ea 12 =: ,-a,fj = 10 = 1

1 1 a - ib_ _ _ = ___ x - =a -- ib.-__( a + i b ) a + ib a - ib a2 + b2

The integral of an exponential function of the form ecX is

1ecx = C + c o n s t a n t

L C

and this is also valid when c is complex. For example,

s

*$8 * e I*

edfj =: % = ~- e

0 i 0 i

(Cos x + i sin T - 1)

(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

(1.35)

(1.36)

(1.37)i

=(-I +0-l) -2i__ = - = .+2;

i

IO introduction

The complex exponential function is a periodic function with period 2~. Thus

etszr) = c o s (19 + 2 7 r ) + i s i n (0 + 27r)

z cos 0 + i sin 0

18 (1.38)

More generally, if n is any positive integer or negative integer,

e i(o+zr) = eo (1.39)

or exp (2nlri) = 1. Conversely, i f exp (i0) = 1, the only possible solut ions for

0 are

B = 2rn, n = 0,*1,~2,&3 ,... (1.40)

probabilityWe have ninety chances in a hundred.

Napoleon at Waterloo, 1815

The commonplace meaning of the word chance i:j probably al ready fami l iar

to the reader. In everyday life, most situations in which we act are characterizedby uncertain knowledge of the facts and of the outcomes of our actions. We arethus forced to make guesses, and to take chances. In the theory of probabi l i ty,the concepts of probabi l i ty and chance are given precise meanings. The theorynot only provides a systematic way of improving our guesses, i t i s also anindispensable tool in studying the abstract concepts of modern physics. To avoidthe necessity of digressions on probabil i ty during the later development of

stat ist ical mechanics and quantum mechanics, we present here a br ief intro-duction to the basic elements of probability theory.

When Napoleon utterecl the statement above, he did not mean that i f the

Batt le of Waterloo were fought a hundred t imes, he would win i t ninety t imes.He was express ing an intuit ive feel ing about the outcome, which was based onyears of experience and on the facts as he knew them. Had he known enemyreinforcements would arr ive, 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 i f he knew what the professor knew-namely, which quest ions were tobe on the exam-he could probably improve his chances of pass ing by studyingsome other sections.

In physics, quantitat ive appl icat ion of the concept of chance is of greatimportance. There are several reasons for this. For example, it is frequently

necessary to describe quclntitatively systems with a great many degrees offreedom, such as a jar containing 10z3 molecules; however, i t i s , as a practicalmatter, impossible to know exactly the posit ions or velocit ies of al l 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, us ing only a few variables. Such studies Jorm the subject matter of a

branch of physics called stofistical mechanics.Secondly, s ince 1926 the development of quantum mechanics has indicated

that the description of mechanical propert ies of elementary part icles can onlybe given in terms of probclbilities. These results frown quantum mechanics have

11

1 2 Probability

profoundly affected the physicists picture of nature, which is now conceived andinterpreted using probabil i t ies.

Thirdly, experimental measurements are always subject to errors of one sortor another, so the quant i tat ive measurements we make always have some un-certaint ies associated with them. Thus, a persons weight might be measured as

176.7 lb, but most scales are not accurate enough to tel l whether the weightis 176.72 lb, or 176.68 lb, or something in between. Al l measuring instrumentshave s imi lar l imitations. Further, repeated measurements of a quant i ty wi l l

f requently give dif ferent values for the quantity. Such uncertaint ies can usual lybe best described in telrms of probabilities.

2.1 DEFINITION OF PRCIBABILITY

To make precise quaniii,tative statements about nature, we must def ine the con-cept of probabi l i ty in a quant i tat ive way. Cons ider an exper iment having a

number of different possible outcomes or results. Here, the probabil ity of a par-ticular result is simply the expected fraction of occurrences of that result out of avery large number of repetit ions or tr ials of the experiment. Thus, one could ex-perimental ly determine the probabil i ty by making a large number of tr ials andfinding the fraction of occurrences of the desired result. I t may, however, beimpractical to actual ly repeat the experiment many t imes (consider for examplethe impossibi l i ty of f ighting 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 l imited amount of

information about the s i tuation, the probabi l i t ies of var ious outcomes may becomputed. It is hoped that the assumptions hold to a good approximation in the

actual physical situatiomn.The theory of probabi l i ty was or iginal ly developed to aid gamblers interested

in improving their inc~ome, and the assumptions of probabi l i ty theory may benatural ly i l lustrated v&th simple games. Consider f l ipping a s i lver dol lar tent imes. I f the s i lver dol lar i s not loaded, on the average it wi l l come down headsfive t imes out of ten. The fraction of occurrences of heads on the average is

,,, or % Then we say that probabi l i ty P(heads) of f l ipping 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 s ides of the coin are essential ly identical, with a plane of

symmetry; It I S then reasonable to assume that s ince neither s ide of the coin isfavored over the other, on the average one side will turn up as often as the other.This i l lustrates an important assumption of probabi l i ty theory: When there areseveral poss ible alternat ives and there is no apparent reason why they shouldoccur with different frequencies, they are assigned equal probabil i t ies. This issometimes called the postulate of equal a priori probabilities.

2.2 Sums of probabilities 1 3

2.2 SUMS OF PROBABILITIES

Some general rules for combining probabi l i t ies are also i l lustrated by the coin-f l ipping experiment. In every tr ial , i t i s certain that either heads or tai ls wi l l turnup. The fraction of occurrences of the result either heads, or tails must be unity,and so

P(either heads or tai ls) = 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 tai ls must equal thefraction of either heads or tails, and so

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

In the special case of the fak coin, both P(heads) and P(tails) are /:t, and theabove equat ion 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 probabi l i t iesP(A), P(B), P(C), . . . , then the probability of either A or B occurring will be givenby the sum of the probabilities:

P(either A or B) = P(A) + I(B)

Similar ly, the probabil i ty of elither A or B or C occurr ing wil l 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 alterna-t ives, so that i f the event A occurs, the events B, C, .cannot occur, and so on.The above relat ion for combining probabi l i t ies s imply amounts to addit ion of thefractions of occurrences of the various events A, B and C, to f ind the total f rac-tion of occurrences of some one of the events in the set A, 6, C.

These relations may easi ly be general ized for any number of alternatives. Forexample, consider an experiment with s ix possible outcomes, such as the s ixpossible faces of a die which could be turned up wheil the die is thrown. Imaginethe faces numbered by an index i that var ies f rom 1 to 6, and let P, be theprobabi l i ty that face i turns up when the die is thrown. Some one face wi l l

definitely turn up, and so the total probability that some one face will turn up willbe equal to unity, Also, the probabi l i ty that some one face wi l l turn up is thesame as the probability that either face one, or face two, or face three, or,. . . ,or face s ix wi l l turn up. This wi l l be equal to the sum of the individual probabi l i -t ies P,. Mathematial ly ,

1 =f:P,,=I

(2.5)

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 in

Equat ion (2.3), which says the probabil i ty of either A or B is the sum of theprobabil it ies 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 comeup more often than the side with five dots, or any other side. Hence, according to

the postulate of equal a pr ior i probabi l i t ies, we may say that P, = P , , and,

indeed, that P, = P2 = P3 = P, = P, =: P,. Then ~~=I P, = 6P, = 1, orPI = /, 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 probabi l i t ies P,. A general rule which is very usefulin 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, dividedby the total number of ways all faces can turn up, is /, .

I f one card is drawn at random from a pack of cards, what is the probabi l i tythat 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, i f one card is drawn from a pack, the probabi l i ty that i twi l l 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 shal l consider o s l ightly more complicated situation: f l ipping a cointwice. What is the probabi l i ty of f l ipping two heads in succession? The possibleoutcomes of this experiment are listed in Table 2.1.

TABLE 2.1 Different possibleo~utcomes for flipping a coin twice.

Krst F l i p Second Flip

headsheadstai lstai ls

headstai lsheadstai ls

Since there are two possible outcomes for each f l ip, there are two t imes two orfour possible outcomes for the succession of two coin f l ips. 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 probabil i t ies of VI. The tota l

2.5 Calculating probabil i t ies 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 gett ing a head on the f i rstflip and the number of ways of getting a head on the second flip. Thus,

P(two heads in succession)

=t

# of ways for head,; on flip 1--~ x# of outcomes on flip 1 I t

# of ways for heads on flip 2_-# of 0Jtcomes on flip 2

= P(heads on flip 1) x P(heads on flip 2)

1 1 1=- X-=-2 2 4

(2.7)

lp/e I f a die is rol led twice in suc:ession, what is the probabil i ty of rol l ing the snake

eye both times?

t;on P(snake eye twice) = (/,) x (I/,) = /,b.

T h e s e r e s u l t s i l l u s t r a t e allother g e n e r a l p r o p e r t y o f p r o b a b i l i t i e s : I f t w oevents A and 6 are independent-that is , i f they do not inf luence each otherin any way-then the probabi l i ty of both A and 6 occurring is

P(A and 6) = P(A)P(B) (2.8)

In words, the probabi l i ty of two independent events both occurr ing 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 thes ix and any ace (1 x 4), by the total number of ways of gett ing al l possible

results (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 probabil i ty theory disf:ussed so

far in the fol lowing 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 probabi l i ty of a

particular 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 pr ior i probabi l i t ies): In the absence of any contraryinformation, equivalent poss ibi l i t ies may be assumed to have equal probabi l i t ies.

(4) I f A and B are mutual ly exclus ive events that occur with probabi l i t ies

P(A) and P(6), then the probabil i ty of either A or 6 occurring is the sum of theindiv idual probabi l i t ies :

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

(5) I f A and 8 are independent events that occur with probabi l i t ies P(A)and P(B), then the probabi l i ty of both A and 6 occurr ing is the product of theindiv idual probabi l i t ies :

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 distr ibut ion function, we continue with someexamples of coin-tossing. Distr ibution functions are functions of one or more inde-pendent variables which label the outcomes of some experiment; the distributionfunctions themselves are proport ional to the probabi l i t ies of the var ious out-comes ( in some cases they are equal to the probabil i t ies). The variables mightbe discrete or continuous. Imagine, for example, a s ingle experiment consist ing

of f l ipping a coin N times, when N might be some large integer. Let nH be thenumber of t imes heads turns up in a part icular experiment. I f we repeat thisexperiment many t imes, then nH can vary f rom exper iment to exper iment. We

shal l calculate the probabi l i ty that n,, heads wi l l turn up out of N f l ips; thisprobabi l i ty wi l l be denoted by P,., (rt). H ere the independent var iable i s nH;and the quant i ty P,{n,), which for f ixed N is a function of n,,, i s an example

of a distr ibution 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 f inding P,(nH), we define PHI to be the probabi l i tyof gett ing a head in the f i rst toss and PT1 to be the probabi l i ty of gett ing a tai l(both are % for a fair coin but differ from, % for a weighted coin). Then P,, +P T1 = 1. Likewise folr the second toss, P HZ $- Pr2 = 1 . I f t h e s e t w o e x p r e s s i o n s

are mult ipl ied 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 probabi l i ty of gett ing the heads and tai ls in a part icular

order.In N tosses,

(PHI + PTI)(PH? + Pn)**-(PHN + PrrJ) = 1 (2.11)

and when the products on the left are carr ied out, the var ious terms give theprobabil i t ies of gett ing heads and tai ls in a part icular order. For example, inthree tosses, the product of Equation (2.1 1) contains eight terms, one of which isPT,PH2PT3. This i s equal to the probabi l i ty of gett ing a tai l , a head and a

tail, in that order, in three tosses. If we were interested only in the probability of

2.6 Disfribulion functions for coin flipping 1 7

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

%would take al l the terms which contain nH factors of the form P,,,, regardless ofthe subscr ipt numbers, and s imply f ind their sum. This i s equivalent to droppingall numerical subscripts and combining terms with s imi lar powers of P,.

I f the express ion on the left of the equation, (P, t Pr) = 1 , i s expanded,the term proport ional to (PH)H(PT)N-nH .I S the probabil i ty of gett ing nH heads

ond N - n,, tails in N tosses, regardless of order. I\ccording to the binomialtheorem.

(PH + Pry = 2t

N !

1

N-H

H=o [rlH!(N - rlH)!] PH pr (2.12)

where zero factor ial (O!) is defined to be one and n! = n(n - l)(n - 2) * * *3.2~1. The reader may recognize the binomial coefficient N!/n,!(N - n,) ! asthe number of ways of selecting n,., objects from a total of N objects, regardlessof 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 gett ing nH heads t imes theprobabi l i ty, (P,.,)H(PT)NmH, of getting nn heads in ane of these ways. There-fore, in the special case of a fair coin when P,, = PT = !/2, the probabi l i ty of

getting nH heads in N tosses, regardless of order, is

N ! 1PN(H) = ;,,!(N - n,)! 2N

In F igures 2.1 through 2.4, the probabil i ty P,.,(nH) of Equation 2.13 is plottedas 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 symmetr ical bel l - l ike,shape. The function P,.,(n,) i:, cal led a probabi l i ty disfribution f u n c t i o n , 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 probabi l i ty PH of

f l ipping a head is PH = 0.3. The probabil i ty of f l ipping a tai l i s 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 probabi l i ty is

;I& (PH)(P,)* = 0 . 2 6 4 6

IPI~ 1 (b) What is the probability

.3

P5 fflH 1

.2

0 2 4 6 aFigure 2.1. Probability of getting nH heads Figure 2.2. Probability of getting nH headsin 5 tosses. in 10 tosses.

.3 .3

P30 (""I P 100 In,)

.2 .2

0 I I I I-:A.rl.l I I I I

0 6 12 ia 24 30 0 20 40 60 a0 1 0 0Figure 2.3. Probability of getting nH heads Figure 2.4. Probobility of getting nH headsi n 30 tosses. in 100 tosses.

2.7 More than two poss ib le oufcomes 19

p/e 2(a) I f the probabil i ty of gett ing ail the forms f i l led out correctly at registrationis 0.1, what is the probabil i ty of gett ing al l forms f i l led 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 desiredprobabi l i ty is

{is (0.1)(0.9) = 0 . 2 4 3

p/e 2(b) What is the probabil ity of f i l l ing out the forms correctly in one or more ofthe three registrations?

ion This is one minus the probability of doing it incorrectly every time or

1 - (0.9)3 = 0 . 2 7 1

!.7 DISTRIBUTION FUNCTIONS FOR MORE THAN TWO POSSIBLEOUTCOMES

Suppose we consider another exper iment in which therl? are four possible results,A, B, C, and D, in o single tricrl. The probabil i t ies for each result in this tr ial ore,respectively, PA, Pg, PC and Pr, = 1 - PA - Ps - P,. I f the quantity on the leftside of the equation

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

is expanded, the term proportional to

is the probabil i ty that in N tr ials result A occurs nA times, 6 occurs n, t imes,C occurs nc t imes and, of course, D occurs no t imes, with nr, = N -~ nA - ns - nc.

A general ized mult inomial expansion may be written OS fol lows:

(x + y + z + W)N =yA

, [

N ! 1p!q!r!(N - p - q - r)! xpyqzw N - p - q - ,p+pdT,rrlN(2.15)

T h e p r o b a b i l i t y t h a t A occclrs nA t i m e s , 6 o c c u r s n,, t i m e s , a n d C o c c u r s nctimes in N tr ials is therefore

PN(nA,nBtnC) =N!

nA!nB!nc!l:N - flA - nn - nc)! 1(2.16)

The general izotion to the case of any number of alternat ives in the results of a

single tr ial is obvious.

2 0 Probobi/ify

2 . 8

In throwing a die three t imes, with s ix poss ible outcomes on each throw, theprobability of throwling two fours and a three is

E X P E C T A T I O N V A L U E S

One of the important uses of a probabil i ty distr ibution function ar ises in thecomputation of averages. We shal l obtain a general formula for the computa-t ion of an average us ing a dist r ibut ion funct ion. Suppose that over severalmonths a student took ten examinations and made the following grades: 91 once,92 twice, 93 once, 94 four t imes, 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 alsocal led a distr ibut ion function, but i t i s not a probabi l i ty distr ibut ion function,since f (n) i s the number of occurrences of the grade n, rather than the proba-bil ity 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. Usingthe symbol ( n ) to d enote the average of n, we have

91 + 92 t- 92 + 93 + 94 + 94 + 94 + 94 + 95 + 95(n) = -__1+1+1+1+1+1+1+1+1+1

(2.17)

In the numerator, the grade 91 occurs once, the grade 92 occurs twice, 94 occursfour t imes, and, in general, the grade n occurs f(n) t imes. Thus, the numeratormay 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 overal l 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 al l the f(n).

Thus, a formula for the denominator is c f(n), summed over al l n. Now we can

2.9 Normolizotion 2 1

write a general expression iq terms of n and f(n) for the average value of n. It is

c n f(n)

( = 5 f ( n )(2.18)

In this case, the average grtgde turns out to be 93.4. If the student were to takeseveral more examinations, then, on the basis of past (experience, i t could beexpected that the average grade on these new examinat ions would be 93.4.For this reason, the average, (n), .I S a so called the expectafion value. Expecta-Ition values are of considerable importance in quonturn mechanics.

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) iscal led the normalization of the distr ibution function. I t 1: f(n) == N, we say thatf(n) i s normal ized to the value N, and the normalization is l/N. S ince the sumof the probabi l i t ies of al l events is unity, when f(n) is a probabi l i ty distr ibut ionfunction, it is normalllzed to Jnity:

Cf(n) = 1

Equation (2.18) refers to the expectat ion of the ndependent var iable, (n).However, in some appl ications i t 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:

c n n( A ( n ) ) = $p (2.20)

.lO EXPECTATION VALUE OIF THE NUMBER OF HEADS

For a more detai led example of an expectation valut: calculat ion, we return tothe fl ipping 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 the

average, 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)TO(2.21)

2 2 Probobi/ify

Herexf(n) = cP+,(n,,) = 1 , since P,.,(n,) is a probability distribution function

wi th a normal izat ion Iof un i ty . There fore , the denominator has been omi t ted .

F r o m E q u a t i o n (2.13), PN(n,) = N!/[2N n,!(N - n,,)!! for a fa ir coin. Hence,

n,N!(nH) = C ~

[2NnH!(N - nH)!j(2.22)

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).

We have to evaluate the summat ion in (n,+) = ~~~=on,N!/[2NnH!(N - n,)!].

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

sponding to nH = 0 does not contribute to the sum because the factor nH is inside

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! = N(N - I)!; s = ; (N - n,)! = (N - 1 - [nH - II)!nH (nH - l)!

a n d

(2.23)

p = 2.-p (2.24)

Then factoring out an N/2, we get

() = :i N $, [~N-Q,, - ,&-m;! - [nH - I])!](2.25)

Then, for m = n,, - 1 , the summat ion over nH f rom 1 to N can be rewr i t ten

as follows:

N - l

(n,,) = i N g0 F-$~--),! _ m)!, = i N(i + 1) = i N (2.26)

This result agrees with our intuitive idea of an expectation value. The result does

not mean that , in an actual exper iment , heads wi l l come up exact ly % N t imes ,

but heads wi l l only c:ome up /2 N t imes on the uverage af ter repeat ing the

N tosses many times.

2.11 EXPERIMENTAL DETERMINATION OF PROBABILITY

Our prev ious d iscuss ion has suggested that we could exper imenta l ly measure

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

cannot assume that P(heads) = P(tails), and i t might be dif f icult to calculate

these probabilities theoretically. One way to find out what P(heads) is, would be

2.1 I Experimenftrl de)erminofion of probability 2 3

to flip the coin many times, compute n,/N, and set the result equal to P(heads).Thus, if N is very large, we should find that

!k (nH) - /(heads)=--N N

(2.27)

0.4

%N

ti

0.6 -

0.5

0.3

N

Figure 2.6. Graph of fractional number of t eads in N tosses.

Figure 2.6 is a graph of n,/N as a function of N in an actual experiment. Notethe logarithmic horizontal scale. From the graph we see that for

N=l, =o;N

N = 1 0 , H=O.3;N

N = 100, = 0.52N

As N becomes very large, it is seen that n,/N tends to % . In this case, therefore,

P(heods) = (n,> = lim !k = 1N N-X N 2

(2.28)

Although, as N - ~5, one would obtain a unique value for P(heads), one maysee from the graph that in actual practjce the value of n,/N for any finite N may

be greater or less thman I?, and general ly could osci l late about % in some ran-

2 4 P r o b a b i l i t y

dom fashion; the amplitude of these osci l lat ions should decrease, however, as Nincreases.

2.12 EXPERIMENTAL ERROR

Since, in practice, we have to stop an experiment after some f inite number ofrepetit ions of the measurements, we would l ike to know how much error we aremaking on the average when we do this . In F igures 2.7 and 2.8 are given the

l&l/N N=z:4 n,/N N - 1 0

l.O--1

01 2 3 4 5 6 7

Experiment number Experiment number

Figure t2.7. F igure 2 .8 .

actual f ractions n,/N, obtained in several repetit ions of an exper iment in whicha coin was f l ipped I N t imes. In the f i rst experiment, N = 4; in the second,N = 10. One can see qual i tat ively that the points in the N = 10 case l ie gen-

eral ly closer to the mean value of % than they do in the N = 4 case. Judgingroughly from the scatter of the values of n,,lN in Figure 2.7, one might expect

the error made in stopping at N = 4 to be about 0.2, whereas in F igure 2.8 i twould be sl ightly smaller. Thus, general ly speaking, we expect the error to

decrease as the number of repetitions of the measurement increases.

2.13 RMS DEVIATION FROM THE MEAN

How can we define a precise numerical measure of the error? One way would beto average the distances of the points n,/N from the mean value (nH)/N. In usingsuch a measure we would have to be careful to take the magnitude of thedistances; if we took !some distances as positive and others as negative, we might

calculate that the error was zero, which is not reasonable. A s imilar measure oferror which is better ior many purposes is the square root of the average of thesquared differences of the points from the mean value. This is cal led the rootmean squared deviation from the mean.

To i l lustrate what is meant, let us imagine that a coin is f l ipped N t imes. Themean for a large number of experiments N should be M N. Consider the

difference nH - (nH) for a single experiment with N f l ips. This difference is

called the deviafion from the mean. The squared deviation from the mean would

b e j u s t (nH - (nH)). H e r e (nH), as usual, i s the average of nH over many

2.14 RMS deviation for coin ffipping 2 5

experiments, and II,, i s the result for any one experiment. I f we repeated theexper iment many t imes and averaged this squared deviat ion f rom the mean,(n,, - (n))*, over the exper iments to obtain ((n,, - (nH:))), then this averaged

s q u a r e d d e v i a t i o n f r o m t h e m e a n w o u l d b e a m e a s u r e o f t h e s q u a r e o ft h e e x p e c t e d e r r o r . T h u s , o m e a s u r e o f t h e e x p e c t e d e r r o r w o u l d b e

V/h - (n,,))), t h e r o o t mean s q u a r e d deviaticn f r o m t h e m e a n , o r r m serror for short.

The mean squared error may be reduced to another form which is also some-times useful. First, write out the square as follows:

1 nH - (n,)) = fli - 2n,(n,) i- (n,J (2.29)

I f we take the average of tloth s ides of this equation, then, s ince the averageof a sum of terms may be computed term by term, we have

(2.30)

But f rom Equation 2.20, the average of a constant is iust the same constant,s o ((#) = (H)Z. A l so f r o m E q u a t i o n 2 . 2 0 , f o r a n y c o n s t a n t C w e h a v e(CnH) = C(n,)and hence (?n,,(n,)) = 2(n,)(n,). Combining these results, weobtain

((h - - (%/))2) = (4) - (q (2.31)

This result is quite general, for the mean squared errsr of any quant i ty ; i t wasder ived here us ing the variable I-I,,, but the der ivat ion would be the same forany other var iable. l-he equat ion states that the mean squared deviat ion f rom

the mean is equal to the average of the square of the variable, minus the squareof the average of the variable.

.14 R M S D E V I A T I O N F O R C O I N F L I P P I N G

To i l lustrate the use of rms error as a measure of el.ror, we shal l consider thecase of coin flipping with a fair coin, and use the probability

P,i,nff) =N ! 1

n,!(N - n,.,)! p

to calculate the rms error as CI function of N. We knolv that (nH) = N/2; hence,in this case, (n,.,) = N2/4. To calculate (II;), we need to find

go n,!(Z!,,)! $

The result of the calculation IS ni = VI N2 + % N. Anyorle not interested in the

details of this calculation should skip to the result in Equation 2.38.As in the evaluation of nH previously, we shal l use o relabel l ing tr ick to evalu-

ate the sum. We write n,?, := nH(nH - 1) + nH, arid use the fact that the

average of a sum is the sum of averages. Then (nj!+:) = (nH(nH - 1)) + (nH).

2 6 Probability

S i n c e w e a l r e a d y k.now t h a t (nn) = Yz N , w e n e e d t o c a l c u l a t e o n l y

(nH(nH - 1 ) ) . T h i s i s

(nH(nH - - 1 ) ) = 2 nH(nH - l)PN(nti)=O

(2.33)

The terms corresponding to n,, = 0 and nH = 1 do not contr ibute because ofthe presence of the fac:tor nH(nH - 1) in the numerator. The sum then goes onlyover values of n,, from 2 through N. Now we wil l use the fol lowing identit ies to

reduce the sum to something we can evaluate:

nH(nH - 1) 1n , ! (nH - - 2)!

; N ! = N ( N - l)(N - 2)!;

(N - n,)! = (N - 2 - [II,, - 2])!; 2N = 4*2N-2 (2.34)

Factoring out /4 N(N - l), we get

(nH(nH - I)) ahf(N ( N - 2)!= - 1) 2

nH=2 [2N-2(nH - - 2)!(N -- 2 - In, - 2])!]

(2.35)The sum may be evaluated by letting m = nH - 2. Then

(2.36)

Collecting the results, we have

(ni) = (nH(nH - 1 ) ) + (nH) = $N(N - 1) + fN = ~N(N + 1) (2.37)

Finally, the root mean squared deviation from the mean is

Gy (nH))) =z VT;) - (n,.,) = .{s+ 1) - aN2 = k fi

(2.38)

This rms deviat ion from the mean is the approximate number of heads bywhich we could usual ly expect the observat ion of nH to differ from the expecta-t i o n v a l u e , (nH) = N / 2 , in one series of N f l ips. We could call this the ap-proximate error in f inding the expectation value experimental ly i f we do oneexperiment with N f l ips. The fractional error in N tosses, i .e. the error in f inding(nH)/N, is then the error divided by the number of tosses, or l/N t imes the rms

deviation. The fractional error is therefore Yz m/N = 1/(2~%). Thus, in at-

tempting to measure a probability such as P(heads), we would have to say thatafter N flips in which nH heads turned up, the probability would be equal to the

2.15 Ernxs in a coin-ffipping experimenf 2 7

fraction nH/N but with a fractional error l/(26). Hence, we would wr i teP ( H e a d s ) = (nH/N) f l/(2%%).

mpk 1. A f t e r o n e f l i p o f a f a i r c#Din, what would be the rms error in the measuredprobabi l i ty?

ution Yz /vx = % ( 1 ) = 0 . 5 .

mp/e 2. How many t imes would you have to f l ip the coirl to reduce the uncertaintyin the measured probability from 0.5 to 0.05?

ufion 0 . 0 5 = % /vTi o r N = /I /(0.05) = 1 0 0 f l i p s .

In F igure 2.6 the dashed l ines are drawn at % =I= 1/2/.fl to give an idea ofthe limits within which one could expect the graph to vary. This square root type

behavior of an error occurs in many other places in physics. For example, inexperiments in which the rate of decay of radioactive substances is measured,one simply counts the number N of decays in some time t. The measured countingrate is then N/f, and it can be shown by arguments very similar to those for coinf l ipping that the f ract ional terror in the measured rate is of order of magnitudel/ v% Thus, to obtain good stat ist ics, i .e. low error, in counting experiments,it is necessary to take large numbers of counts. To get the counting rate correctto three signif icant f igures or a fractional error of 0.001, one would need

around one mil l ion cfounts.

1.15 ERRORS I N A COIN-FLIPIPING E X P E R I M E N T

We may now compare this theory of the rms error with the experiments depictedin Figures 2.7 and 2.8. In Figure 2.7, each experiment (corresponds to N = 4.F o r t h i s v a l u e o f I V , t h e t h e o r y g i v e s t h e f r a c t i o n a l r m s d e v i a t i o n t o b e% /fi = 0.25. Next, we wi l l use the data of F igure 2.7 to f ind the experimentalrms fractional deviation for this part icular set of trai ls . To do this, we simply cal-culate the square root of ,the average of [(nH/N) - (n,/N)] over the sevenexperiments. The exipectation va lue (nH/N) i s just the average of the results ofthese experiments and is 0.571. We may then obtain roblIe 2.2:

TABLE 2.2~--

Experiment Number Deviation (Deviation) ~--

1 0.179 0.03202 0.179 0.03203 - 0 . 0 7 1 0.00504 -0.071 0.00505 0.179 0.03206 -0.321 0.10307 0.071 0.0050

!ium = 0.2140

28 Probabi l i ty

The average deviat ion squared i s then 0.21417 = 0.0306, and the rms deviat ion

is -06 = 0.175. Based on these seven experiments, the result of the f i rstexperiment could the11 be expressed as

nH077 = 0 . 7 5 0 f 0 . 1 7 5 (2.39)

Likewise for the third experiment, (n,/N) = 0.500 & 0.175. This is in reason-able agreement with the theory, which supposes a very large number of experi-ments instead of seven and gives for the case N = 4,

= 0 . 5 0 0 + 0 . 2 5

The reader can perform simi lar calculations for the data given in Figure 2.8in the case N = 10. Here the experimental result for the rms deviation from themean is 0.105. The theoretical result is

= 0 . 5 0 0 zt 0 . 1 5 8

2.16 ERRORS IN AVERA.GES OF REPEATED EXPERIMENTS

Errors of the type we have descr ibed, depending on l/a, also ar ise when

repeating exper iments which measure the average value of some physicalquantity, such as the diameter of a cylinder or the length of an object. Supposethat an experiment is performed in which the quantity x is measured N t imes. I tmust be kept in mind here that a s ingle experiment means N measurements ofthe value of the quantity, and the result of an experiment is the average valueof the quantity for these N measurements. Then we ask: What is the error in theaverage value? If the experiment is repeated, it should give a new average value

that does not differ from the previous one by much more than the error.What is being measured here is an average itself. This average is not the same

as the expectation vallue. The expectation value would be equal to the averageif the number of mearurements, N, approached infinity.

Suppose that the ,N individual measurements in one experiment are x1 ,x2,. . , x,.,. The result of the experiment is then (x1 + x2 + * * * + x,.,)/N. sup-

pose the true, or expectation, value of the quantity is X. This would be the aver-age of an extremely large number of measurements. The deviation from the truevalue X in a particular experiment is

x, + x2 + *-* + XN- ~- - XN

We can then get a measure of the experimental error by computing the rms error,averaged over many experiments of N measurements each. Cal l the error EN.

2.16 Errors in overages of repeated exper iments 2 9

Then

To i l lustrate how this may be worked out, we shal l take the case N == 2, onlytwo measurements in an experiment. Then

EN =

= (((x, - X)) + 2((.K, - X)(x, - x,) + ((x2 - X))]----1N2

f o r N = 2 . C o n s i d e r t h e t e r m ((x1 - X ) ( x , - X ) ) . W h e r e a s c~ t e r m l i k e( ( x , - X ) ) i s a l w a y s positive, (x1 - X ) is negative about as often as i t i sp o s i t i v e , a n d s o i s (x2 - X). Since the values of (x, - X) and (x2 - X)are independent of each other,, their product will also be negative as often as it

i s pos i t ive, and the expecta,tion va lue ((x1 - X)(x, - X)) wi l l be zero. Hencethe only remaining terms produce:

E, =1/

& - w> + ((x2 - Xl)1- - -N2

(2.44)

This was for the case N = :2. However i t i s easy to see that a silmilar argu-ment applies for any N; 011 the expectation values of the cross-terms whicha r i s e i n t h e s q u a r e [(x1 - X ) + (x2 - X ) + * . . + (x,., - X)] w i l l b e n e a r l yzero. Therefore,, for any N, we can say

li

-~_~~__

EN = I((+ - X)) t ((x2 --x,) + . . * + ((XN - X))}~__-. ~.N2

(2.45)

However, since the svbscrip-ts on the xs denote nothing more than the order inwhich the measurements are made, we expect that, on the average, .the quantity

((x; - X)) wil l be the same for ail x, , or

((x, - X)) = ((x2 - x,) = ((XN - Jo) = E: (2.46)

We call this average ET, since it is the mean squared error of a singlle measure-ment, averaged over many exper iments. That i s , E, i s the rms deviat ion i f weconsider that the experiment consists of one measurement rather than N rneasure-ments. Then, s ince there are N terms l ike ((x, - X)),

(2.47)

3 0 Probobil;ty

Thus, the error in the result of an N-measurement experiment is less than theerror in the result of (1 one-measurement experiment by CI factor of l/-\/N. Tosee how this works in practice, Table 2.3 gives the actual results of 24 measure-

T A B L E 2 . 3 Results of Six Experiments, Each Consisting of Four Measurements of theLellgth of CI Cylinder. Distances in Cenfimeters.

(1) (2) (3) (4) (5) (5)4.11 4 . 0 7 4 . 0 8 4 . 0 5 4 . 0 9 4 . 0 64 . 0 6 4 . 0 5 4 . 1 0 4 . 0 6 4 . 0 8 4 . 1 04 . 0 6 4 . 0 6 4 . 0 9 4 . 0 9 4 . 0 6 4 . 0 74 . 0 8 4 . 0 8 4 . 0 9 4 . 1 0 4 . 0 4 4 . 0 8-_-.

A v . = 4 . 0 7 7 5 A v . = 4 . 0 6 5 A v . = 4 . 0 9 0 A v . = 4 . 0 7 5 A v . = 4 . 0 6 7 5 A v . = 4 . 0 7 7 5Overal l average of the results = 4.0754 cm

ments of the diameter, in centimeters, of Q cyl inder, using vernier calipers. I f weregard these as 24 separate experiments in which N = 1, then we can computethe mean value and the error for these 24 experiments. The mean of the 24measurements, which we shal l take OS the t rue value, i s X = 4.0754 cm and therms error El for one measurement is

li, =: x(deviations)= 0.018 cm

2 4

Let us next regard the data as s ix experiments of four measurements each,in which the quantity being measured is the average of four measurements. In

this case, N = 4, so ,the error in the average of the four measurements shouldb e a b o u t E4 = E,/-6 = 0 . 0 0 9 c m . By subtract ing the overal l average,

4.0754 cm, from each of the averages of the s ix experiments, we can f ind theexper imental deviat ions of the averages f rom the mean. Then the exper imental

E4 i s

.E., =

,,[Bevlation;f averages)l =0.0081 cm (2 .49)

This compares favorably with the result , 0.009 cm, obtained using Equation(2.47). Actually, while we used the mean of 24 experiments as the true value, this

itself has an rms error associated with it. An estimate of this error, again using

E q u a t i o n (2.47), i s El/v/24 = 0.018/4.90 =: .0037 c m . T h e r e a d e r m a y w e l lreflect that these differing measures of error for the same data look somewhatsuspicious; however, this s imply means that quoted errors often depend on themethod of data handling.

2 . 1 7 P R O B A B I L I T Y D E N S I T I E S

So far, we have consi,dered distribution functions which are functions of a discrete

variable. In many cases, the independent variables are continuous. Consider, for

2.17 Probability densities 3 1

example, a thin circular disc on a horizontal axle. I f the disc is given a spin

and then allowed io come to rest, what is the probability that some one point on

the rim of the disc will be exS3ctly on top? Obviously, since one point is only one

of an uncountable inf ini ty o,f points along the r im, the probabil i ty wil l be zero.

However, let us introduce a coordinate system fixed in the wheel, and describe

points on the rim in terms of bcrn angle 0, with 0 varying continuously from 0 to 2s

to describe all dif ferent points on the r im. I f there is no reason ,why one port ion

of the disc should come to the top more often than any other portion, then the

probability that some portion in the infinitesimal range d0 will come up,, denotedby P,,o, is P.,o = d8/2rr. The factor 27r in the denominator is chosen so that thetotal probability that some point (any point) on the rim comes to the top is unity.

We can check this because

(2.50)

Once an inf initesimal probaloi l i ty of this nature is known, i t can be used to

find the probabil i ty that an event takes place in a given range.. Thus, the

probabil i ty that some point in the portion of the rim between ?r/6 alld s wi l l

come to the top wil l be the integral of d8/2r between the l imits of x/6 and H.

The resul t i s 5/12. The coeff icient of d0 in the expression for P& i s ca l led o

probabil i ty density. In this spec:ial case, the probabil i ty density is l/:271.. In gen-

eral , for the continuous variablle 0, i f the probabil i ty of f inding 0 in the range

d0 is given by an expression of the form Pdo = p(H)do, thei? ~(19) i s c a l l e d

the probabil ity density. In our example, the probabil ity density, p(B),. was a

constant; but i f , for instance, there were more fr ict ion on one side of the axle

than the other, the wheel w(ould be more l ikely to stop in certain posit ions, and

p(0) would not be independent of 8.

Similarly, with a dif ferent plhysical si tuation described by a variable x, and

given the probabil i ty density p(x), the probabil i ty that x is to be found in the

range dx wil l be P& = p(x) dx. A probabil i ty density is thus the probabil i typer unit x, for the continuous variable x. Moreover, the probabil i ty that x wi l l be

Figure 2.9. Crosshatched area Jnder the probability density curve is the lpobobilitythat a measurement of x will yield a value between x1 and x2

3 2 Probability

s

2

found in the range between x1 and xz wi l l be given by p(x) dx. This is justx1

the area under the curve of p(x) , versus x between the l imits x1 and xz (seeFigure 2.9). Probability densities have the property that, when integrated over allposs ible values of x, the result must be the total probabi l i ty that some value of

x occurs, or/

p(x)c/x =: 1 .

2.18 EXPECTATION VALUES FROM PROBABILITY DENSITIES

We shal l next show how to compute an average us ing a probabi l i ty densi ty.You recal l that for a discrete probabi l i ty distr ibution function, P(x,), of the dis-crete variable x;, the rneon value of x is given by

@) = c xi Phi)

where P(x,) i s the probabi l i ty that xi occurs. I t i s unnecessary to div ide by

c,,rIX; P(x;) here, s ince the sum of the probabil i t ies is unity. Now consider theenti re range of the continuous var iable x to be broken up into smal l incrementsAx,. I f x, is a point in Ax,, then the probabi l i ty P(x,) that i t l ies in the range Ax,wi l l be given approximately by P(x;) = p(x;) Ax,, where p(x) is the probabi l i ty

density. Thus, (x) = c., x,p(xi) Ax;. Taking the l imit as Ax, 4 0, we get

(x) = /xp(x)dx. (2.51)

example 1. Consider the probabil i ty density defined for 0 5 x 5 1 by

A plot for p(x) is given in F igure 2.10. I f we wish to compute the average value

of some quantity us ing the given p(x), we should f i rst check to see that p(x) iscorrectly normalized. It will be correctly normalized if the integral

is equal to one. In this

2 , O % , p(x) is equal to zero. To find the expectation value (x), we shouldcompute the integral ,/xp(x:)dx. In this case also, the limits on the integral will befrom 0 to /2 The integration may then be performed as fol lows:

(x) = /- J2 2 x d x = ;0

2.78 Expecfofion voloes from probability densities 33

p(x)

.x1Figure 2.10.

2. Consider a part icle in a one ,dimensional box wi th ends at x == 0 and x = 2.

The probabi l i ty density for f inding the part icle outs ide the box is zero. In quan-

tum mechanics the probabi l i ty density is 1 G(x) 1 , where the wave funct ion #(x)sat i s f ies a wave equat ion callled the Schrtidinger equation. Suppose the proba-bi l i ty density, 1 G(x) / ?, i s g iven by

/ #(x) / 2 = p(x) =

for 0 5 x 5 2. Outside this range, p(x) is zero. A plot of this fulnction is, shownin Figure 2.1 1. This probability density p(x) is correctly normalized so that

p(x)

1 2

Figure 2.11.

3 4 Probabi/ity

s 20 p(x)dx = . w e can calculate the averuge value or expectation value of xas fol lows:

x p ( x ) d x = ; J,x(x2 - $ x4) d x

lb2 5--x24 ,=I)I

Let us also calculate the rms deviation of x from the mean. This is a measure of

the spread of the wave function q(x). The rms deviation is

((x - (x),) 2We know that the expectation value of x, +), is /,. Hence we wish to calculate

the expectation value

(i:x - :)) = (x) -- (x) = G) - gfrom Equation (2.31) It is

(x2)p(x)dx - ; = 5 12(x4 - ; x)dx - ;

Then the rms deviation from the mean is

j/qzs)3 = ~~5.1518 = 0 . 3 9 0

The same result can be obtained by straightforward calculation of

((x - ;)3 =12(x2 - 2+ y)p(x)dx

but the algebra is more tedious.

2.19 GAUSSIAN DISTRIBUTION

An interesting probabil i ty density, called the gaussian distr ibut ion, ar ises when a

fair coin is f l ipped an extremely large number of t imes. This same distr ibution

arises in the majority of physical measurements involving random errors. In

f-l lpping a coin N t imes, the discrete probabil i ty distr ibution function was, from

Equation (2.13),

PN(H) = (2Nqy - Q] (2.52)

2.22 Expectation values using CI gaussion distribufion 3 5

In the l imi t of very large N, th is d istr ibut ion funct ion is sharply peaked about

the average value of nH, % IN. Th is tendency may be seen by re fer r ing to

Figures 2.1 through 2.4.

We can obtain an approximate analyt ical expression for PN(nH), for large N,

by using St ir l ings approximation for the factor ia ls: For large n, In (n! ) E

% In (2x) + (n + Y2)ln (n) - n. This, together wi th the approx imat ion

In (1 + b) E h - /2 b2 for small b, leads to the following approximate resultfo r PN(nH):

p,(n,) 2If [

-$ exp I-.kN*1 (2.53)when N is large and nH is neor i t s averoge , % N. A graph of Equation (2.53)

is shown for N = 100 in F igure 2 .12. The corresponding discrete curve of Fig-

pm n,

0.1 r---

0.06 (

1L A?0 -

4 0 5 0 6 0 7 0

Figure 2.12. Comparison of the discrete probability distribution PI,, (n,,) with op-

proximate function, a Gaussian. The Gaussian curve is drawn with o dashed line.

ure 4 is shown on the some graph. It may be seen that for N = 100, the opproxi-

motion of Equation (2.53) is olready extremely good.

The exponent ia l curve of Equat ion (2.53), peaked symmetr ica l ly about Yz N, is

called a goussian or normal distribution. It occurs often in probability theory and

in classical statistical mechanics. Although nH is still a discrete variable ,toking onintegral volues, when N is sufficiently large we can lump many of these integral

values together and regard Pr.,(n) as a probability density.

20 EXPECTATION VALUES USING A GAUSSIAN DISTRIBUTION

In other chapters we will need o number of expectation values using the goussian

distribution. To illustrate the types of integrals which arise, let us find the root

3 6 Probability

mecrn squared deviation of x for the gaussian probability density p(x), given by

pf:x) d x = & e x p [y(x2izx1)?] d x

Here x ranges from -- 2 to LC, and the quant i t ies x1 and c are constonfs. First,we check that this probabil i ty density is normolized to unity. Referr ing to thetable of definite integrals, Table 2.4,

TABLE 2.4 Tobleof Integrals

and lett ing y = x - x1 w i th dx = dy, we f ind that

In calculat ing the rms deviat ion from the mean of x, we need f i rst to fincl themean. S ince the distr ibution is symmetr ic about x = xl, i t i s clear that (x) =: x1.I f this were not obvimous, the average value of x could be calculated by theequation

I - (x) = J_, XPb)dX (2.54)

In the case of the gaussian, this is

= ~ ( x - x , ) e x p [*2+] dx

+ & 1: e x p [3(x] d x (21.55)

The f irst integral on the r ight is zero because the integrand is odd in (x - xl).The second term is x1. Thus, for a goussian peaked about xl, the average valueof x is just the position of the center of the peak: (x) = x1.

Let us next calculate the rms deviat ion from the mean. This is Cl ,h!jo we first need to calculate the expectation value,

( ( x - x1)) = -----;; j m ( x - x1)2 e x p [mm2i2x)2 d x ] (1.56)--li

S u m m a r y 3 7

Again, with the subst i tut ion y = x - x1, this reduces to the second integral inTable 2.4. Therefore,((x - x1)) = r~. Hence, the rms deviat ion from the meanis equal to u. As a characteristic width of the gaussion curve, we might take thedistance from x1 to the polsnt where the curve is half i ts maximum. This i s atx - x, = ~5 ln2& = 1.180. Thus we see that the width at holf -maximumand the rms deviation are about the same.

CALCULATION OF PROBABILITY

The probability of an event is equal to the number of possible ways of getting thepart icular result , divided by the total number of ways of gett ing al l poss ible

results. I f A and 6 are two independent events, the total probabi l i ty of gett ingeither the result A or the result is is equal to the sum of the probabilities of gettingA and of gettiny 6 separately:

P ( e i t h e r A o r 8) = P ( A ) + P ( B )

The probability of getting both A and B is equal to the product of the probabili-ties of getting A and of getting B separately:

P(both A a n d 8) = P ( A ) P ( 6 )

The total probability of getting all possible results in a given situation is unity.

PROBABILITY DISTRIBUTION FUNCTIONS AND DENSITIES

A probabi l i ty P(n), which is a function of some discrete var iable n, and whic