Concepts of Modern Physics PDF

CD 0> CONCEPTS OF O z MODERN PHYSICS Second Edition n m H Arthur Beiser r o m TO < CO n (A CO ft o 3 m Q- 5: o CONCEPTS OF MODERN PHYSICS McGRAW-HILL SERIES IN FUNDAMENTALS OF PHYSICS: AN UNDERGRADUATE TEXTBOOK PROGRAM E. U. CONDON, EDITOR, UNIVERSITY OF CO...

CD 0> CONCEPTS OF O z MODERN PHYSICS Second Edition n m H Arthur Beiser r o m TO < CO n (A CO ft o 3 m Q- 5: o CONCEPTS OF MODERN PHYSICS McGRAW-HILL SERIES IN FUNDAMENTALS OF PHYSICS: AN UNDERGRADUATE TEXTBOOK PROGRAM E. U. CONDON, EDITOR, UNIVERSITY OF COLORADO MEMBERS OF THE ADVISORY BOARD CONCEPTS OF MODERN PHYSICS D. Allan Bromley, Ytilc Oiiirp.sili/ Arthur F. Kip, Hugh D. Young, ! nittmity of California, Berkeley Second Edition f?rmwfftf ftftiBrwn t 'niversity INTRODUCTORY TEXTS Beiscr Concepts of Modem Physics Arthur Beiser Kip luiHilmm-nliils of I'.teitriciti/ and Magnetism Young Fundamentals of Mechanics and Seal Young Fuiuliiuieiiliilfi of Optics tint! Modem I'liysics UPPER-DIVISION TEXTS INTERNATIONAL STUDENT EDITION Burger :mil ObsOfl. Ctas&oal Mechanics: A Modern Perspective Beiser. Pnsperiirr.s of Modem I'lit/xin Cohen CoRetpti of \tolror Physics Elmore and llculd. Physics of Waves kraut. frmilitiw nliit.i t>! Mathenuitical Physics Longn. fundamentals of tllementtiry Particle Physics Meyerhof i'.tcments (f Nuclear Physics - McGRAW-HILL KOGAKUSHA, LTD. Rcif Fundamentals of Statistical ami Thermal Physics Trail! and Potuilla Atomic Theory: An Introduction to Wave Mec.hnoii.-i Tokyo IXisseldorf Johannesburg London Mexico New Delhi Panama Rio de Janeiro Singapore Sydney  CONTENTS Preface ix Chapter 3 Wave Properties of List of Abbreviations Particles 73 3.1 De Broglie Waves 73 3.2 Wave Function 74 PART ONE BASIC CONCEPTS 3.3 De Broglie Wave Velocity 76 3.4 Phase and Croup Velocities 79 Chapter I Special Relativity 3 3.5 The Diffraction of Particles 82 1.1 The Michelson-Morlcy Experi- 3.6 The Uncertainty Principle ment 3 3.7 Applications of the Uncertainty Library of Congress Cataloging in Publication Data 1.2 The Special Theory of Principle 91 Relativity g 3.8 The Wave-particle Duality 93 Beiser, Arthur, 1.3 Time Dilation 12 Problems 96 Concepts of modern physics. 1.4 The Twin Paradox 16 1.5 length Contraction 17 (McGraw-Hill series in fundamentals of physics) 1.6 Meson Decay 20 PART TWO THE ATOM I. Matter-Constitution, 2. Quantum theory, 1.7 The Lorentz Transformation 22 "1.8 The Inverse Lorentj; Trans- Chapter 4 Atomic Structure 101 1. Title. formation 27 4.1 Atomic Models 101 QC173.B413 1973 530.1 72-7089 1.9 Velocity Addition 28 4.2 Alpha-particle Scattering 105 ISBN 0-07-004363-4 I.JO The Relativity of Mass 30 4.3 The Rutherford Scattering 1.11 Mass and Energy 35 Formula wa- CONCEPTS OF MODERN PHYSICS 1.12 Mass and Energy: Alternative 4.4 Nuclear Dimensions rn Derivation 37 4.5 Electron Orbits 113 INTERNATIONAL STUDENT EDITION Problems 39 4.6 Atomic Spectra in 4.7 The Bohr Atom 121 Exclusive rights by McGraw-Hill Kogakusha, Ltd., for manufacture and export. This Chapter 2 Particle Properties of 4.8 Energy levels and Spectra 125 book cannot be re-exported from the country to which it is consigned by McGraw-Hill. Waves 43 4.9 Nuclear Motion 129 2.1 The Photoelectric Effect 43 4.10 Atomic Excitation 131 2.2 The Quantum TTiaory of light 47 4.11 The Correspondence Principle 133 Copyright © 1963, 1967, 1973 by McGraw-Hill, Inc. All rights reserved. No part of 2.3 X Rays 51 Problems 135 this publication may be reproduced, stored in a retrieval system, or transmitted, 2.4 X-Ray Diffraction 56 in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, 2.5 The Compton Effect 60 Chapter 5 Quantum Mechanics 139 2.6 Pair Production 63 5.1 Introduction to Qiuui turn without the prior written permission of the publisher. 2.7 Gravitational Red Shift 68 Mechanics 139 Problems 70 5.2 The Wave Equation 140 PRINTED IN JAPAN  PART FOUR THE NUCLEUS 12.9 Cross Section 413 5.3 Schrodingrr"-, Equation: Time- PART THREE 12.10 The Compound Nucleus 117 df|H'tnlcnt Fortn 143 PROPERTIES OF MATTER 12.1 \lk [ear Fivsion 420 1.4 Expect sit ion Values 145 Chapter 1 1 The Atomic Nucleus 361 12.12 Trausuranic Elements 423 5.5 Sehrodinger'i Equation: Steady Chapter 8 The Physics 1 1.1 Atomic Masses 361 of Mole- 12.13 Thermonuclear Energy 424 statc Form 146 11.2 The Neutron 384 cules £43 Prohlems 127 5.8 Tin 1 Particle in » Box; Energy 11.3 Stable Nuclei 386 S. I Molecular Fun nation 243 Quantization 140 11.4 Nuclear Sizes and Shapes 370 8.2 Fleet ron Sharing 245 Chapter 13 Elementary Particles 431 5.7 The Particle in u Bus: Wave 8.3 The Molecular Ion 247 11.5 Binding Energy 372 6V 13.1 Antiparticlcs 431 Functions 153 8.4 The llj Molecule 252 "11.8 The Dcuteron 374 1 1.7 13.2 Meson Theory of Nuclear 5.8 The Particle in a Nan rigid Ground Stale of the Dculcrun 377 8.5 Molecular Orliitals 254 Forces 433 Box 156 Mi Hybrid 11.8 Triplet and Singlet States 379 Orliitals 261 13.3 Piraiv and Muons 5.0 milium 1.8 The Liquid-drop Model 436 '['In' 1 1. ()s t ill, ilni 158 Carbon-carbon Bonds 1 380 8.7 265 13.4 Kaons and Ilyperons 5.10 The Harmonic Oscillator: Solu- 11.10 The Shell Model 438 8.8 Rotational Energy Levels 269 383 13.5 Systematic* of Elementary tion of Schrddioger's Equation 163 Prohlems 397 8.9 Vibrational Energy Levels 272 Particles Problems 169 439 8.10 Electronic Spectra of Mole- 13.6 Strangeness Number +14 cules £81 Chapter 12 Nuclear Transformations 389 13.7 Isotoptc Spin 448 12.1 Radioactive Decay 389 13.8 Syn in let lies 10 id Conservation Chapter 6 Quantum Theory of the Chapter 9 Mechanics Radioactive Scries Statistical 287 12.2 393 Principles 447 Hydrogen Atom 173 9,1 Statistical DfetrflnitfOQ Uws 287 12.3 Alpha Decay 396 13.9 Theories of Elementary Particles 151 6.1 Schrodinger's Equation for the "12.4 8.2 Phase Space 288 Barrier Penetration 399 Problems Hydrogen Atom 173 M ax well-Bol nim n *12.5 Theory ft3 t I >isl ri 1 1 1 1 of Alpha Decay 404 °(j.2 Separation of Variables 176 lion 289 12.8 Reta Decay 408 Answers to Odd-numliercd Problems 457 "6.3 Quantum Numbers 178 "9.4 Evaluation of Constant^ 293 12.7 Inverse Reta Decay 411 6.4 Principal Quantum Number \m 9.5 Molecular Energies an Ideal 12.8 Camma Decay in 412 481 6.5 OrbitalQuantum Number ISO Gas 295 I'd Magnetic Quantuiii Number 184 9.6 Rotutional Spectra 298 6.7 1 !n- Normal Zeeman Effect 187 "9.7 Rose-Einstein Distribution 300 fl.8 Electron Probability Density 169 9.8 Rluck-hody RadUlimi 3tM 6.9 Radiative Transitions im 9.9 Fcnni-Dirac Distribution M) 6. t() Selection Rules 198 9.10 Comparison of Results 310 Problems 200 9.11 The Laser 311 Problems 314 Chapter 7 Many-electron Atoms 203 < ' i.i 1 1 1.- 1 10 The Solid State 317 7.1 Electron Spin 203 10.1 Crystalline and Amorphous 7.2 Spin -orbit Coupling 207 Solids 317 7.3 The Exclusion Principle 210 10.2 Ionic Crystals 318 7.1 El eel n )i i ( '. in i lii> 1 1 rat i i >i is 213 10.3 Covalent Crystals 325 7.5 The Periodic Table 215 111.4 Van Dcr Waals Forces 327 7.6 Hund's Rule 222 10.5 The Metallic Braid 331 °7.7 Total Angular Momentum 2-2-2 10.6 The Rand Theory of Solids 333 7.8 1.S Coupling 226 10.7 The Fermi Bong 1 339 7.9 ft Coupling 228 10.8 Electron-energy Distribution 342 *7.10 One-electron Spectra 229 10.9 Rrtllouin Zones 344 *7.1 1 Two-electron Spectra 232 "10.11) Origin y( Forhiddcn Bands 346 7.12 X-ray Spectra 235 10.11 Effect ive Mass 355 Problems 237 Problems 355 VI CONTENTS CONTENTS VII  PREFACE This book is intended for use with one-semester courses in modern physics that have elementary classical physics and calculus as prerequisites. Kelativity and quantum theory are considered first to provide a framework for under- standing the physics of atoms and nuclei. The theory of the atom is then de- veloped with emphasis on quantum-mechanical notions, and is followed by a discussion of the properties of aggregates of atoms. Finally atomic nuclei and elementary particles are examined. The balance here deliberately leans more toward ideas than toward experi- mental methods and practical applicatioas. because I believe that the beginning student is better served in his introduction to modern physics by a conceptual framework than by a mass of individual details. However, all physical theories live or die by the sword of experiment, and a nmnlier of extended derivations are included in order lo demonstrate exactly how an abstract concept can lie related to actual measurements. Many instructors will prefer not to hold their students responsible for the more complicated (though not necessarily mathe- matically difficult) discussions, and I have indicated with asterisks sections that can be passed over lightly without loss of continuity; problems based on the contents of these sections are also marked with asterisks. Other omis- sions are also passible, of course. Relativity, For example, may well have teen covered earlier, and Part 3 in its entirety may be skipped when its contents will lie the subject of later work. Thus there is scope for an instructor to fash- ion the type of course he wishes, whether a general survey or a deeper inquiry into selected subjects, and to choose the level of treatment appropriate to his audience. An expanded version of this book requiring no higher degree of mathematical preparation is my Perspectives of Modern Physics, an Upper Division Text in this series; other Upper Division Texts carry forward specific aspects of modem physics in detail.  In preparing this edition of Concepts of Modern Physics much of the original LIST OF ABBREVIATIONS text has been reorganized and rewritten, the coverage of a number of topics has been broadened, and some material of peripheral interest has been dropped. I am grateful to Y. Beers and T. Satoh for their helpful suggestions j n this regard. Arthur Beiser k angstrom amp ampere atm atmosphere b barn C coulomb Ci cmie d day eV electron volt F farad fm fermi g gram h hour Hz hertz joule ] K degree Kelvin m meter mi mile miri minute mol mole N newton s seeond T tesla u atomic mass unit V volt W watt yr year xi PREFACE  SPECIAL RELATIVITY i Our study of modern physics begins with a consideration of the special theory of relativity. This is a logical starting point, since all physics is ultimately concerned with measmeinent and relativity involves an analysis of how meas- urements depend upon the observer as weD as upon what is observed. From relativityemerges a new mechanics in which there are intimate relationships Iretween space and time, mass and energy. Without these relationships it would be impossible to understand the microscopic world within the atom whose elucidation is the central problem of modem physics. LI THE fJIICHELSONMORLEY EXPERIMENT The wave theory of light was devised and perfected several decades before the electromagnetic nature of the waves became known. It was reasonable for the pioneers in optics to regard light waves as undulations in an all-pervading elastic medium called the ether, and their successful explanation of diffraction and interference phenomena in terms Gf ether waves made the notion of the ether so familial- that its existence was accepted without question. Maxwell's develop- ment of the electromagnetic theory of light in 1864 and Hertz's experimental confirmation of it in 887 deprived the ether of most of its properties, but nobody 1 at the lime seemed willing to discard the hmdamentai idea represented by the ether: thai light propagates relative to some sort of universal frame of reference. Let us consider an example of what this idea implies with die help of a simple analogy. Figure 1-1 is D which Hows with the speed v. a sketch of a river of width Two boats start out from one bank of the river with the same speed V, Boal A crosses the river to a point on the other bank directly opposite the starting point Bud then returns, white boal B heads downstream for the distance D and then returns to the starling point. Let us calculate the time required for each round trip.  — component V as its net speed across the river. From Fig. 1-2 we see that these land speeds are related by the formula river V = 2 V' 2 + v2 so that the actual speed with which boat A crosses the river is V'= VV 2 - v2 rB CD- =3 4 Hence the time for the initial crossing is Lhe distance D divided by the CD- speed V, Since the reverse crossing involves exactly die same amount of time, the total round-trip time tA is twice D/V\ or FIGURE 1-1Boat A goes directly across the river and returns to starting point, while boat its B heads downstream for an identical distance and then returns. 2.D/V l.i \/l - v2 /V 2 We begin by considering Ixmt A. If A heads perpendicularly across Lhe river, The case of boat B is somewhat different. As it heads downstream, its speed the current will carry it downstream from its goal on the opposite bank relative to the shore is its own speed V plus the speed t; of the river (Fig. 1-3), (Fig. 1-2). It must therefore head somewhat upstream in order to compensate and it travels the distance D downstream in the time D/(V + c). On its return for the current. In order to accomplish this, its upstream component of velocity trip, however, B's speed relative to the shoTe own speed V minus the is its should be exactly —v in order to cancel out the river current v, leaving the speed u of the river. It therefore requires the longer time D/{V — c) to travel upstream the distance D to its starting point. The total round-trip time H t is FIGURE 1-2 Boat A must head upstream in order to compensate for the river current. the sum of these times, namely, = D D %.« V+ v V-v Using the common denominator (V + l)(V — v) for both terms, _ D(V ~ v) + D(V + v) ' B ~ (V + v)(V - o) 2DV V* -v" 2D/V 1.2 1 _ oVV a which is greater than f^, the corresponding round-trip time for the other boat. The ratio between the times f., and tB is 1.3 = VT - i;7 V 2 Jl 4 If we know the common speed Vof the two we can determine the speed v of the river. boats and measure the ratio t A /t 8 , BASIC CONCEPTS SPECIAL RELATIVITY L  — i mirror A V. B V B v v B X v parallel light from mirror B FIGURE 1-3 The speed of boat H downstream relative to the shore is increased single source by the speed of the river current while its speed upstream is reduced by the same amount. half -silvered mirror The reasoning used in this problem may be transferred to the analogous hypothetical problem of the passage of light waves through the ether. If there is an ether ether current viewing screen pervading space, we move through it with at least the 3 x |0* m/s (18.5 mi/s) speed of the earth's orbital motion about the sun; if the sun is also in motion, FIGURE 1-5 The MichetsonMortey experiment. our speed through the ether is even greater (Fig. 1-4). From the point of view of an observer on the earth, the ether is moving past the earth. To detect this motion, we can use the pair of light lieams formed by a half-silvered mirror mirror along a path perpendicular to the ether current, while the other goes instead of a pair of boats (Fig. 1-5). One of these light beams is directed to a to a mirror along a path parallel to the ether current. The optical arrangement is such that both beams return to the same viewing screen. The purpose of the clear glass plate is to ensure that both beams pass through the same thicknesses FIGURE 1-4 Motions of the earth through a hypothetical ether of air and glass. If the path lengths of the two beams are exactly the same, they will arrive at the screen hi phase and will interfere constructively to yield a bright Held of view. The presence of an ether current in the direction shown, however, would cause the beams to have different transit limes in going from the half-silvered mirror to the screen, so that they would no longer arrive at the screen in phase but would interfere destructively. In essence this is the famous experiment performed in 1887 by the American physicists Miehelson and Morley. In the actual experiment the two mirrors are not perfectly perpendicular, with the result that the viewing screen appears crossed with a series of bright and dark interference fringes due to differences in path length between adjacent light waves (Fig. 1-6). If either of the optical paths in the apparatus is varied in length, the fringes appear to move across the screen as reinforcement and cancellation of the waves succeed one another at each point. The stationary apparatus, then, ran tell us nothing about any time difference between die two paths. When die apparatus is rotated by 90", however, the two padis change their orientations relative to the hypothetical ether stream, so that the beam formerly requiring the time t A for the round tripnow requires tB and vice versa. If these times are different, the fringes will move across the screen during the rotation. BASIC CONCEPTS SPECIAL RELATIVITY  If d corresponds to the shifting of a fringes, d — nX where X is the wavelength of the light used. Equating these two formulas for FIGIWE 1-6 Fringe pat- d we t find that tern observed in Michel- son-Morley experiment. cAt _ Dv 2 Ac 2 In the actual experiment Michelson and Morley were able to make D about 10 m in effective length through the use of multiple reflections, and the wavelength of the light they used was about 5,000 A (1 A = 10~ 10 m). The expected fringe shift in each path when the apparatus is rotated by 90° is therefore Let us calculate the fringe shift expected on the basis of the ether theory. From Eqs. 1.1 and 1.2 the time difference between the two paths owing to the ether drift is M= t B - tA 10 m X (3 X L0 J m/s} 2 2D/V 2D/V "5x Mr- T mX (3 X 10 s m/s) 2 1 - v 2 /V 2 y/1 - v 2/V 2 = 0.2 fringe Here t> is the ether speed, which we shall take as the earth's orbital speed of Since both paths experience this fringe shift, the total shift should amount to 3 X 10 4 rn/s, and V is the speed of light c, where c = 3x lf* w m/s. Hence 2n or 0.4 fringe. A shift of this magnitude is readily observable, and therefore Michelson and Morley looked forward to establishing directly the existence of the ether. V2 whatever was found. When To everybody's surprise, no fringe shift the exper- io- fi iment was performed at different seasons of the year and in different locations, and when experiments of other kinds were tried for (he same purpose, the which is much smaller than 1. According to the binomial theorem, when x is conshisions were always identical: no motion through the ether was detected. extremely small compared with 1, The negative result of the Michelson-Morley experiment had two conse- (l±af « 1 ± nx quences. First, it rendered untenable the hypothesis of the ether by demon- We may therefore express Af to a good approximation as strating that the ether has no measurable properties — an ignominious end for what had once been a respected idea. Second, it suggested a new physical principle: the speed of light in free space is the same everywhere, regardless of any motion of source or ohserver. 1.2 THE SPECIAL THEORY OF RELATIVITY Here D is the distance between the half-silvered mirror and each of the other We mentioned earlier the role of the ether as a universal frame of reference mirrors. The path difference d corresponding to a time difference At is with respect to which light waves were supposed to propagate. Whenever we el = cA(. speak of "motion," of course, we really mean "motion relative to a frame of BASIC CONCEPTS SPECIAL RELATIVITY  reference." The frame of reference may be a road, the earth's surface, the sun, This postulate follows directly from the results of the Mfcbelson-Morley experi- the center of our galaxy; but in every case we must specify it. Stones dropped ment and many others. in Bermuda and in Perth, Australia, both fall "down," and yet the two move At first seem radical. Actually they subvert almost sight these postulates hardly in exactly opposite directions relative to the earth's center. Which is the correct and space we form on the basis of our daily all the intuitive concepts of lime location of the frame of reference in this situation, the earth's surface or experience. A simple example will illustrate this statement. In Fig. 1-7 we have its center? the two boats A and B once more, with boat A at rest in the water while boat B The answer is that all frames of reference are equally correct, aldiough one and so on drifts at the constant velocity v. There is a low-lying fog present, may be more convenient to use in a specific case. // there were an ether the moving one. At the neither boat does the observer have any idea which is pervading all space, we could refer all motion to it, and the inhabitants of The light from the flare travels instant that H is abreast of A, a flare is fired. Bermuda and Perth would escape from their quandary. The absence of an ether, uniformly in all directions, according to the second postulate of special relativity. then, implies that there is no universal frame of reference, since light (or, in An observer on either boat must find a sphere of light expanding with hlmsefy general, electromagnetic waves) is the only means whereby information can Ik; at its center, according to the first postulate of special relativity, even though transmitted through empty space. All motion exists solely relative to the person one of them is changing his position with respect to the point where the flare or instrument observing it. If we are in a free halloon above a uniform cloud undergoing such a change went off. The observers cannot detect which of them is bank and see another free balloon change its position relative to us, we have in position since the fog eliminates any frame of reference other than each boat no way of knowing which balloon is "really" moving. Should we be isolated itself, and so, since the speed of light is the same for both of (hem, they must in the universe, there would be no way in which we could determine whether hoth see the identical phenomenon. we were in motion or not, because without a frame of reference the concept Why is the situation of Fig. 1-7 unusual? Let us consider a more familiar of motion has no meaning. analog. The boats are at sea on a clear day and somel>ody on one of them drops The theory of relativity resulted from an analysis of the physical coii.se7c* J vx = L Vl- oVc 2 l > + c* which is the same as Eq. 1.10. Vl - dVc2 26 BASIC CONCEPTS SPECIAL RELATIVITY 27  so to him the duration of the interval / is df + sM. tr 1 = 1,-1, dt = Vl - v /c 2 2 ~ 'it'a and so VI - i'7r 2 *~ dt or 6. dx' + v (If / = v'l - H 1 + ff k- 0.5c + 0.9c (0.9c){0.5e) 1 + = 0.9655c which is less than c. We need go less than 1 percent faster than a space ship traveling at 0.9c in order to pass it at a relative speed of 0.5c. 1.10 THE RELATIVITY OF MASS collision as seen from frame S Until now we have been considering only the purely kinematical aspects of special relativity. The dynamical consequences of relativity are at least as remarkable, including as they do the variation of mass with velocity and the equivalence of mass and energy. We begin by considering an elastic collision (that is, a collision in which kinetic energy is conserved) between two particles A and B, as witnessed by observers in the reference frames S and S' which are in uniform relative motion. The properties of A and B are identical when determined in reference frames in which they are at rest. The frames S and $' are oriented as in Fig. 1-12, with S' moving in the +1 direction with respect to S at the velocity v. Before the collision, particle A had been at rest in frame S and particle B from frame collision as seen S': in frame S', Then, at the same instant, A was thrown in the +y direction at the speed VA while B was thrown in the — if direction at the speed V^, where 1.37 vA = vB Hence the behavior of A as seen from S is exactly the same as die behavior FIGURE 1-12 two different frames of reference, An elastic cplllslon is observed In of B as seen from §', When the two particles collide, A rebounds in the —y direction at the speed V A while B rebounds in the +y' direction at the speed , V'B. If the particles are thrown from positions Y apart, an observer in S finds that the collision occurs at y = /2 Y and one l in S' finds that it occurs at y' = %Y. SPECIAL RELATIVITY 31 30 BASIC CONCEPTS  In the above example both A and 8 are moving in S. In order to obtain a The round-trip time Tn for A as measured in frame S is therefore formula giving the mass m of a body measured while in motion in terms of its 1,38 *"*V mass m„ when measured at rest, we need only consider a similar example which V", and V^ are very small. In this case an observer in S will see B approach in and it is the same for H in S', A with Hie velocity v. make a glancing collision (since V'B < v), and then continue on. In S 'it- v , mA = irtf, and If momentum is conserved in the S frame, it must be true that mn = m and so 1.39 mA VA = mB VB 1.43 m= Relattvistic mass where mA and mB are the tnasses of A and B, and V^ and VB their velocities Vl - v 2 /c* a* measured in the S /ra»i«. In S the speed V' is found from w the speed v relative to an observer larger The mass of a body moving at is than its mass when at rest relative to the observer by the factor 1/ Vl - o'/c*. 1.40 V. =i This mass increase is reciprocal; to an observer in S' where T is the time required for H to make its round trip as measured in S. m, =m In S', however, B's trip requires the time '/;,, where and m„ = m„ 1.41 r= h rocket ship in night shorter than its twin still OB - Measured from the earth, is /I d»/c the ground and its mass is greater. To somebody on the rocket ship in Hight according to our previous results. Although observers in both frames see the the ship on the ground also appears shorter and to have a greater mass. (The same event, they disagree as to the length of time the particle thrown from the effect is, of course, unobservably small for actual rocket speeds.) Equation 1.43 other frame requires to make the collision and return. is plotted in Fig. 1-13. Replacing T in Eq. 1.40 with its equivalent in terms of T we , have Provided that momentum is defined as ryi - dVc2 B = ] 14 T \/l - v 2 /c* Prom Eq. 1.38 conservation of momentum is valid in special relativity just as in classical physics. However, Newton's second law of motion is correct only in the form A rQ Inserting these expressions for VA and VB in Eq. 1.39, we see that momentum conserved provided that _ tt T ma v 1 is us ~ rfiL \/l - dVc 2 -! 1.42 = m K Vl - »2 /c This is not equivalent to saying that Our was that A and B are identical when at rest with respect original hypothesis to an observer; the difference between m and m therefore means thai measure- F= ina A B ments of mass, like tho.se of space and time, depend upon the relative speed l>etween an observer and whatever he is observing. dt SPECIAL RELATIVITY 33 32 BASIC CONCEPTS  l.U MASS AND ENERGY The most famous relationship Einstein obtained from the postulates of special relativityconcerns mass and energy. This relationship can be derived directly from the definition of the kinetic energy T of a moving body as the work done in bringing it from rest to its state of motion. That is, -£** where F component of the applied force in the direction of the displace- is the ment ds and s is the distance over which the force acts. Using the relativistic form of the second law of motion rf(fflli) F= ~dT the expression for kinetic energy becomes d(mv) -[ dt da FIGURE 1-13 The relativity of mm. hib i; d{mc) J even with m given by Eq. 1.43, because { \y/l- v 2 /c 2 I d \ dv dm Integrating by parts («* < 1 since t: is much less than c, Tl*c equivalence of mass and energy can be demonstrated in a numlicr of different ways. An interesting derivation that is somewhat different from the = (1 + %^/c 2 )m v c 2 - m a c 2 T = %m v 2 * given above, but also suggested by Einstein, makes use of the basic notion owl the center of mass of an isolated system (one that does not Interact with 'Is surroundings) cannot be changed by any process occurring within the system, 1 terice at low speeds the relativistic expression for the kinetic energy of a moving particle reduces to the classical one. hi this derivation we imagine a closed box from one end of which a burst of The total energy of such a particle is "ferromagnetic radiation is emitted, as in Fig. 1-14. This radiation carries energy E= m c2 + %m v v2 aiMl momentum, and when the emission occurs, the lx>x recoils in order that the 36 BASIC CONCEPTS SPECIAL RELATIVITY 37  initial center of mass since m is much smaller than M. The time / during which the Ixjx moves is J- L 1 s 7burst of radiation — emitted ,'((iial is true to the time required lwx, a distance when in L away; < M). this by the radiation