Introduction to Modern Physics PDF

This page intentionally left blank Copyright © 2009, 2002, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected] ISBN (13) : 978-81-224-2922-0 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com PREFACE TO THE SECOND EDITION The standard undergraduate programme in physics of all Indian Universities includes courses on Special Theory of Relativity, Quantum Mechanics, Statistical Mechanics, Atomic and Molecular Spectroscopy, Solid State Physics, Semiconductor Physics and Nuclear Physics. To provide study material on such diverse topics is obviously a difficult task partly because of the huge amount of material and partly because of the different nature of concepts used in these branches of physics. This book comprises of self-contained study materials on Special Theory of Relativity, Quantum Mechanics, Statistical Mechanics, Atomic and Molecular Spectroscopy. In this book the author has made a modest attempt to provide standard material to undergraduate students at one place. The author realizes that the way he has presented and explained the subject matter is not the only way; possibilities of better presentation and the way of better explanation of intrigue concepts are always there. The author has been very careful in selecting the topics, laying their sequence and the style of presentation so that student may not be afraid of learning new concepts. Realizing the mental state of undergraduate students, every attempt has been made to present the material in most elementary and digestible form. The author feels that he cannot guess as to how far he has come up in his endeavour and to the expectations of esteemed readers. They have to judge his work critically and pass their constructive criticism either to him or to the publishers so that they can be incorporated in further editions. To err is human. The author will be glad to receive comments on conceptual mistakes and misinterpretation if any that have escaped his attention. A sufficiently large number of solved examples have been added at appropriate places to make the readers feel confident in applying the basic principles. I wish to express my thanks to Mr. Saumya Gupta (Managing Director), New Age International (P) Limited, Publishers, as well as the editorial department for their untiring effort to complete this project within a very short period. In the end I await the response this book draws from students and learned teachers. R.B. Singh This page intentionally left blank PREFACE TO THE FIRST EDITION This book is designed to meet the requirements of undergraduate students preparing for bachelor's degree in physical sciences of Indian universities. A decisive role in the development of the present work was played by constant active contact with students at lectures, exercises, consultations and examinations. The author is of the view that it is impossible to write a book without being in contact with whom it is intended for. The book presents in elementary form some of the most exciting concepts of modern physics that has been developed during the twentieth century. To emphasize the enormous significance of these concepts, we have first pointed out the shortcomings and insufficiencies of classical concepts derived from our everyday experience with macroscopic system and then indicated the situations that led to make drastic changes in our conceptions of how a microscopic system is to be described. The concepts of modern physics are quite foreign to general experience and hence for their better understanding, they have been presented against the background of classical physics. The author does not claim originality of the subject matter of the text. Books of Indian and foreign authors have been freely consulted during the preparation of the manuscript. The author is thankful to all authors and publishers whose books have been used. Although I have made my best effort while planning the lay-out of the text and the subject matter, I cannot guess as to how far I have come up to the expectations of esteemed readers. I request them to judge my work critically and pass their constructive criticisms to me so that any conceptual mistakes and typographical errors, which might have escaped my attention, may be eliminated in the next edition. I am thankful to my colleagues, family members and the publishers for their cooperation during the preparation of the text. In the end, I await the response, which this book draws from the learned scholars and students. R.B. Singh This page intentionally left blank CONTENTS UNIT I SPECIAL THEORY OF RELATIVITY CHAPTER 1 The Special Theory of Relativity.............................................................................. 3–46 1.1 Introduction............................................................................................................................... 3 1.2 Classical Principle of Relativity: Galilean Transformation Equations..................................... 4 1.3 Michelson-Morley Experiment (1881)..................................................................................... 7 1.4 Einstein’s Special Theory of Relativity..................................................................................... 9 1.5 Lorentz Transformations........................................................................................................ 10 1.6 Velocity Transformation.......................................................................................................... 13 1.7 Simultaneity............................................................................................................................. 15 1.8 Lorentz Contraction................................................................................................................. 15 1.9 Time Dilation........................................................................................................................... 16 1.10 Experimental Verification of Length Contraction and Time Dilation..................................... 17 1.11 Interval..................................................................................................................................... 18 1.12 Doppler’s Effect...................................................................................................................... 19 1.13 Relativistic Mechanics............................................................................................................. 22 1.14 Relativistic Expression for Momentum: Variation of Mass with Velocity............................. 22 1.15 The Fundamental Law of Relativistic Dynamics................................................................... 24 1.16 Mass-energy Equivalence........................................................................................................ 26 1.17 Relationship Between Energy and Momentum....................................................................... 27 1.18 Momentum of Photon............................................................................................................. 28 1.19 Transformation of Momentum and Energy........................................................................... 28 1.20 Verification of Mass-energy Equivalence Formula................................................................ 30 1.21 Nuclear Binding Energy.......................................................................................................... 31 Solved Examples..................................................................................................................... 31 Questions.................................................................................................................................. 44 Problems.................................................................................................................................. 45 x Contents UNIT II QUANTUM MECHANICS CHAPTER 1 Origin of Quantum Concepts................................................................................. 49–77 1.1 Introduction.......................................................................................................................... 49 1.2 Black Body Radiation............................................................................................................ 50 1.3 Spectral Distribution of Energy in Thermal Radiation........................................................ 51 1.4 Classical Theories of Black Body Radiation........................................................................ 52 1.5 Planck’s Radiation Law........................................................................................................ 54 1.6 Deduction of Stefan’s Law from Planck’s Law................................................................. 56 1.7 Deduction of Wien’s Displacement Law............................................................................. 57 Solved Examples................................................................................................................... 58 1.8 Photoelectric Effect.............................................................................................................. 60 Solved Examples................................................................................................................... 63 1.9 Compton’s Effect................................................................................................................. 65 Solved Examples................................................................................................................... 68 1.10 Bremsstrahlung..................................................................................................................... 70 1.11 Raman Effect........................................................................................................................ 72 Solved Examples................................................................................................................... 74 1.12 The Dual Nature of Radiation.............................................................................................. 75 Questions and Problems....................................................................................................... 76 CHAPTER 2 Wave Nature of Material Particles........................................................................ 78–96 2.1 Introduction.......................................................................................................................... 78 2.2 de Broglie Hypothesis........................................................................................................... 78 2.3 Experimental Verification of de Broglie Hypothesis............................................................. 80 2.4 Wave Behavior of Macroscopic Particles............................................................................ 82 2.5 Historical Perspective........................................................................................................... 82 2.6 The Wave Packet.................................................................................................................. 83 2.7 Particle Velocity and Group Velocity.................................................................................... 86 2.8 Heisenberg’s Uncertainty Principle or the Principle of Indeterminacy............................. 87 Solved Examples................................................................................................................... 89 Questions and Problems....................................................................................................... 96 CHAPTER 3 Schrödinger Equation............................................................................................. 97–146 3.1 Introduction.......................................................................................................................... 97 3.2 Schrödinger Equation........................................................................................................... 98 3.3 Physical Significance of Wave Function y....................................................................... 102 3.4 Interpretation of Wave Function y in terms of Probability Current Density................... 103 3.5 Schrödinger Equation in Spherical Polar Coordinates....................................................... 105 3.6 Operators in Quantum Mechanics..................................................................................... 106 Contents xi 3.7 Eigen Value Equation............................................................................................................112 3.8 Orthogonality of Eigen Functions....................................................................................... 113 3.9 Compatible and Incompatible Observables.........................................................................115 3.10 Commutator.........................................................................................................................116 3.11 Commutation Relations for Ladder Operators................................................................... 120 3.12 Expectation Value................................................................................................................ 121 3.13 Ehrenfest Theorem............................................................................................................. 122 3.14 Superposition of States (Expansion Theorem).................................................................. 125 3.15 Adjoint of an Operator........................................................................................................ 127 3.16 Self-adjoint or Hermitian Operator..................................................................................... 128 3.17 Eigen Functions of Hermitian Operator Belonging to Different Eigen Values are Mutually Orthogonal........................................................................................ 128 3.18 Eigen Value of a Self-adjoint (Hermitian Operator) is Real.............................................. 129 Solved Examples................................................................................................................. 129 Questions and Problems..................................................................................................... 144 CHAPTER 4 Potential Barrier Problems................................................................................. 147–168 4.1 Potential Step or Step Barrier............................................................................................. 147 4.2 Potential Barrier (Tunnel Effect)........................................................................................ 151 4.3 Particle in a One-dimensional Potential Well of Finite Depth........................................... 159 4.4 Theory of Alpha Decay...................................................................................................... 163 Questions............................................................................................................................. 167 CHAPTER 5 Eigen Values of Lˆ 2 and Lˆ z Axiomatic: Formulation of Quantum Mechanics............................................................................................... 169–188 5.1 Eigen Values and Eigen Functions of L̂2 And L̂z............................................................. 169 5.2 Axiomatic Formulation of Quantum Mechanics............................................................... 176 5.3 Dirac Formalism of Quantum Mechanics......................................................................... 178 5.4 General Definition of Angular Momentum........................................................................ 179 5.5 Parity................................................................................................................................... 186 Questions and Problems..................................................................................................... 187 CHAPTER 6 Particle in a Box.................................................................................................... 189–204 6.1 Particle in an Infinitely Deep Potential Well (Box)............................................................ 189 6.2 Particle in a Two Dimensional Potential Well.................................................................... 192 6.3 Particle in a Three Dimensional Potential Well.................................................................. 195 6.4 Degeneracy......................................................................................................................... 197 6.5 Density of States................................................................................................................. 198 6.6 Spherically Symmetric Potential Well................................................................................ 200 Solved Examples................................................................................................................. 202 Questions and Problems..................................................................................................... 204 xii Contents CHAPTER 7 Harmonic Oscillator............................................................................................. 205–217 7.1 Introduction........................................................................................................................ 205 Questions and Problems..................................................................................................... 215 CHAPTER 8 Rigid Rotator......................................................................................................... 218–224 8.1 Introduction........................................................................................................................ 218 Questions and Problems..................................................................................................... 224 CHAPTER 9 Particle in a Central Force Field........................................................................ 225–248 9.1 Reduction of Two-body Problem in Two Equivalent One-body Problem in a Central Force...................................................................................................................... 225 9.2 Hydrogen Atom................................................................................................................... 228 9.3 Most Probable Distance of Electron from Nucleus.......................................................... 238 9.4 Degeneracy of Hydrogen Energy Levels........................................................................... 241 9.5 Properties of Hydrogen Atom Wave Functions................................................................. 241 Solved Examples................................................................................................................. 243 Questions and Problems..................................................................................................... 245 UNIT III STATISTICAL MECHANICS CHAPTER 1 Preliminary Concepts.......................................................................................... 251–265 1.1 Introduction........................................................................................................................ 251 1.2 Maxwell-Boltzmann (M-B) Statistics................................................................................. 251 1.3 Bose-Einstein (B-E) Statistics............................................................................................ 252 1.4 Fermi-Dirac (F-D) Statistics.............................................................................................. 252 1.5 Specification of the State of a System............................................................................. 252 1.6 Density of States................................................................................................................. 254 1.7 N-particle System............................................................................................................... 256 1.8 Macroscopic (Macro) State............................................................................................... 256 1.9 Microscopic (Micro) State................................................................................................. 257 Solved Examples................................................................................................................. 258 CHAPTER 2 Phase Space........................................................................................................... 266–270 2.1 Introduction........................................................................................................................ 266 2.2 Density of States in Phase Space....................................................................................... 268 2.3 Number of Quantum States of an N-particle System....................................................... 270 CHAPTER 3 Ensemble Formulation of Statistical Mechanics............................................. 271–291 3.1 Ensemble............................................................................................................................. 271 Contents xiii 3.2 Density of Distribution (Phase Points) in g-space........................................................... 272 3.3 Principle of Equal a Priori Probability................................................................................ 272 3.4 Ergodic Hypothesis............................................................................................................. 273 3.5 Liouville’s Theorem............................................................................................................ 273 3.6 Statistical Equilibrium......................................................................................................... 277 Thermodynamic Functions 3.7 Entropy................................................................................................................................ 278 3.8 Free Energy......................................................................................................................... 279 3.9 Ensemble Formulation of Statistical Mechanics................................................................ 280 3.10 Microcanonical Ensemble................................................................................................... 281 3.11 Classical Ideal Gas in Microcanonical Ensemble Formulation.......................................... 282 3.12 Canonical Ensemble and Canonical Distribution............................................................... 284 3.13 The Equipartition Theorem................................................................................................. 288 3.14 Entropy in Terms of Probability......................................................................................... 290 3.15 Entropy in Terms of Single Particle Partition Function Z1............................................... 291 CHAPTER 4 Distribution Functions......................................................................................... 292–308 4.1 Maxwell-Boltzmann Distribution........................................................................................ 292 4.2 Heat Capacity of an Ideal Gas............................................................................................ 297 4.3 Maxwell’s Speed Distribution Function............................................................................. 298 4.4 Fermi-Dirac Statistics......................................................................................................... 302 4.5 Bose-Einstein Statistics....................................................................................................... 305 CHAPTER 5 Applications of Quantum Statistics................................................................... 309–333 Fermi-Dirac Statistics 5.1 Sommerfeld’s Free Electron Theory of Metals................................................................. 309 5.2 Electronic Heat Capacity.................................................................................................... 317 5.3 Thermionic Emission (Richardson-Dushmann Equation)................................................ 318 5.4 An Ideal Bose Gas............................................................................................................... 321 5.5 Degeneration of Ideal Bose Gas......................................................................................... 324 5.6 Black Body Radiation: Planck’s Radiation Law................................................................. 328 5.7 Validity Criterion for Classical Regime............................................................................... 329 5.8 Comparison of M-B, B-E and F-D Statistics..................................................................... 331 CHAPTER 6 Partition Function................................................................................................ 334–358 6.1 Canonical Partition Function.............................................................................................. 334 6.2 Classical Partition Function of a System Containing N Distinguishable Particles........... 335 6.3 Thermodynamic Functions of Monoatomic Gas.............................................................. 337 6.4 Gibbs Paradox..................................................................................................................... 338 xiv Contents 6.5 Indistinguishability of Particles and Symmetry of Wave Functions................................. 341 6.6 Partition Function for Indistinguishable Particles............................................................. 342 6.7 Molecular Partition Function.............................................................................................. 344 6.8 Partition Function and Thermodynamic Properties of Monoatomic Ideal Gas............... 344 6.9 Thermodynamic Functions in Terms of Partition Function............................................. 346 6.10 Rotational Partition Function.............................................................................................. 347 6.11 Vibrational Partition Function............................................................................................. 349 6.12 Grand Canonical Ensemble and Grand Partition Function................................................ 351 6.13 Statistical Properties of a Thermodynamic System in Terms of Grand Partition Function............................................................................................................... 354 6.14 Grand Potential F............................................................................................................... 354 6.15 Ideal Gas from Grand Partition Function.......................................................................... 355 6.16 Occupation Number of an Energy State from Grand Partition Function: Fermi-Dirac and Bose-Einstein Distribution...................................................................... 356 CHAPTER 7 Application of Partition Function...................................................................... 359–376 7.1 Specific Heat of Solids....................................................................................................... 359 7.1.1 Einstein Model.......................................................................................................... 359 7.1.2 Debye Model............................................................................................................ 362 7.2 Phonon Concept................................................................................................................. 365 7.3 Planck’s Radiation Law: Partition Function Method......................................................... 367 Questions and Problems..................................................................................................... 369 Appendix–A......................................................................................................................... 370 UNIT IV ATOMIC SPECTRA CHAPTER 1 Atomic Spectra–I.................................................................................................. 379–411 1.1 Introduction........................................................................................................................ 379 1.2 Thomson’s Model............................................................................................................... 379 1.3 Rutherford Atomic Model.................................................................................................. 381 1.4 Atomic (Line) Spectrum..................................................................................................... 382 1.5 Bohr’s Theory of Hydrogenic Atoms (H, He+, Li++)........................................................ 385 1.6 Origin of Spectral Series.................................................................................................... 389 1.7 Correction for Nuclear Motion.......................................................................................... 391 1.8 Determination of Electron-Proton Mass Ratio (m/MH)..................................................... 394 1.9 Isotopic Shift: Discovery of Deuterium............................................................................ 394 1.10 Atomic Excitation............................................................................................................... 395 1.11 Franck-Hertz Experiment................................................................................................... 396 1.12 Bohr’s Correspondence Principle...................................................................................... 397 Contents xv 1.13 Sommerfeld Theory of Hydrogen Atom............................................................................ 398 1.14 Sommerfeld’s Relativistic Theory of Hydrogen Atom...................................................... 403 Solved Examples................................................................................................................. 405 Questions and Problems..................................................................................................... 409 CHAPTER 2 Atomic Spectra–II................................................................................................. 412–470 2.1 Electron Spin....................................................................................................................... 412 2.2 Quantum Numbers and the State of an Electron in an Atom........................................... 412 2.3 Electronic Configuration of Atoms.................................................................................... 415 2.4 Magnetic Moment of Atom................................................................................................ 416 2.5 Larmor Theorem................................................................................................................. 417 2.6 The Magnetic Moment and Lande g-factor for One Valence Electron Atom.................. 418 2.7 Vector Model of Atom........................................................................................................ 420 2.8 Atomic State or Spectral Term Symbol............................................................................. 426 2.9 Ground State of Atoms with One Valence Electron (Hydrogen and Alkali Atoms)......... 426 2.10 Spectral Terms of Two Valence Electrons Systems (Helium and Alkaline-Earths)......... 427 2.11 Hund’s Rule for Determining the Ground State of an Atom............................................ 434 2.12 Lande g-factor in L-S Coupling......................................................................................... 435 2.13 Lande g-factor in J-J Coupling......................................................................................... 439 2.14 Energy of an Atom in Magnetic Field................................................................................ 440 2.15 Stern and Gerlach Experiment (Space Quantization): Experimental Confirmation for Electron Spin Concept........................................................................................................ 441 2.16 Spin Orbit Interaction Energy............................................................................................ 443 2.17 Fine Structure of Energy Levels in Hydrogen Atom......................................................... 446 2.18 Fine Structure of Hµ Line................................................................................................... 449 2.19 Fine Structure of Sodium D Lines..................................................................................... 450 2.20 Interaction Energy in L-S Coupling in Atom with Two Valence Electrons...................... 451 2.21 Interaction Energy In J-J Coupling in Atom with Two Valence Electrons...................... 455 2.22 Lande Interval Rule............................................................................................................. 458 Solved Examples................................................................................................................. 459 Questions and Problems..................................................................................................... 467 CHAPTER 3 Atomic Spectra-III............................................................................................... 471–498 3.1 Spectra of Alkali Metals...................................................................................................... 471 3.2 Energy Levels of Alkali Metals........................................................................................... 471 3.3 Spectral Series of Alkali Atoms......................................................................................... 474 3.4 Salient Features of Spectra of Alkali Atoms...................................................................... 477 3.5 Electron Spin and Fine Structure of Spectral Lines.......................................................... 477 3.6 Intensity of Spectral Lines.................................................................................................. 481 Solved Examples................................................................................................................. 484 xvi Contents 3.7 Spectra of Alkaline Earths.................................................................................................. 487 3.8 Transitions Between Triplet Energy States........................................................................ 493 3.9 Intensity Rules.................................................................................................................... 493 3.10 The Great Calcium Triads.................................................................................................. 493 3.11 Spectrum of Helium Atom.................................................................................................. 494 Questions and Problems..................................................................................................... 497 CHAPTER 4 Magneto-optic and Electro-optic Phenomena................................................... 499–519 4.1 Zeeman Effect..................................................................................................................... 499 4.2 Anomalous Zeeman Effect................................................................................................. 503 4.3 Paschen-back Effect.......................................................................................................... 506 4.4 Stark Effect......................................................................................................................... 512 Solved Examples................................................................................................................. 514 Questions and Problems..................................................................................................... 519 CHAPTER 5 X-Rays and X-Ray Spectra................................................................................. 520–538 5.1 Introduction........................................................................................................................ 520 5.2 Laue Photograph................................................................................................................. 520 5.3 Continuous and Characteristic X-rays............................................................................... 521 5.4 X-ray Energy Levels and Characteristic X-rays............................................................... 523 5.5 Moseley’s Law.................................................................................................................... 526 5.6 Spin-relativity Doublet or Regular Doublet........................................................................ 527 5.7 Screening (Irregular) Doublet............................................................................................ 528 5.8 Absorption of X-rays.......................................................................................................... 529 5.9 Bragg’s Law........................................................................................................................ 532 Solved Examples................................................................................................................. 535 Questions and Problems..................................................................................................... 538 UNIT V MOLECULAR SPECTRA OF DIATOMIC MOLECULES CHAPTER 1 Rotational Spectra of Diatomic Molecules....................................................... 541–548 1.1 Introduction........................................................................................................................ 541 1.2 Rotational Spectra—Molecule as Rigid Rotator................................................................ 543 1.3 Isotopic Shift...................................................................................................................... 547 1.4 Intensities of Spectral Lines............................................................................................... 548 CHAPTER 2 Vibrational Spectra of Diatomic Molecules...................................................... 549–554 2.1 Vibrational Spectra—Molecule as Harmonic Oscillator.................................................... 549 Contents xvii 2.2 Anharmonic Oscillator........................................................................................................ 550 2.3 Isotopic Shift of Vibrational Levels.................................................................................... 553 CHAPTER 3 Vibration-Rotation Spectra of Diatomic Molecules........................................ 555–561 3.1 Energy Levels of a Diatomic Molecule and Vibration-rotation Spectra........................... 555 3.2 Effect of Interaction (Coupling) of Vibrational and Rotational Energy on Vibration-rotation Spectra................................................................................................... 559 CHAPTER 4 Electronic Spectra of Diatomic Molecules........................................................ 562–581 4.1 Electronic Spectra of Diatomic Molecules........................................................................ 562 4.2 Franck-Condon Principle: Absorption............................................................................... 573 4.3 Molecular States................................................................................................................. 579 Examples............................................................................................................................. 581 CHAPTER 5 Raman Spectra...................................................................................................... 582–602 5.1 Introduction........................................................................................................................ 582 5.2 Classical Theory of Raman Effect..................................................................................... 584 5.3 Quantum Theory of Raman Effect.................................................................................... 586 Solved Examples................................................................................................................. 592 Questions and Problems..................................................................................................... 601 CHAPTER 6 Lasers and Masers................................................................................................ 603–612 6.1 Introduction........................................................................................................................ 603 6.2 Stimulated Emission............................................................................................................ 603 6.3 Population Inversion........................................................................................................... 606 6.4 Three Level Laser............................................................................................................... 608 6.5 The Ruby Laser.................................................................................................................. 609 6.6 Helium-Neon Laser............................................................................................................. 610 6.7 Ammonia Maser...................................................................................................................611 6.8 Characteristics of Laser.......................................................................................................611 Questions and Problems..................................................................................................... 612 Index........................................................................................................................... 613–618 This page intentionally left blank UNIT 1 SPECIAL THEORY OF RELATIVITY This page intentionally left blank CHAPTER THE SPECIAL THEORY OF RELATIVITY 1.1 INTRODUCTION All natural phenomena take place in the arena of space and time. A natural phenomenon consists of a sequence of events. By event we mean something that happens at some point of space and at some moment of time. Obviously the description of a phenomenon involves the space coordinates and time. The oldest and the most celebrated branch of science –mechanics- was developed on the concepts space and time that emerged from the observations of bodies moving with speeds very small compared with the speed of light in vacuum. Guided by intuitions and everyday experience Newton wrote about space and time: Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external and is otherwise called duration. In Newtonian (classical) mechanics, it assumed that the space has three dimensions and obeys Euclidean geometry. Unit of length is defined as the distance between two fixed points. Other distances are measured in terms of this standard length. To measure time, any periodic process may be used to construct a clock. Space and time are supposed to be independent of each other. This implies that the space interval between two points and the time interval between two specified events do not depend on the state of motion of the observers. Two events, which are simultaneous in one frame, are also simultaneous in all other frames. Thus the simultaneity is an absolute concept. In addition to this, the space and time are assumed to be homogeneous and isotropic. Homogeneity means that all points in space and all moments of time are identical. The space and time intervals between two given events do not depend on where and when these intervals are measured. Because of these properties of space and time, we are free to select the origin of coordinate system at any convenient point and conduct experiment at any moment of time. Isotropy of space means that all the directions of space are equivalent and this property allows us to orient the axes of coordinate system in any convenient direction. The description of a natural phenomenon requires a suitable frame of reference with respect to which the space and time coordinates are to be measured. Among all conceivable frames of reference, the most convenient ones are those in which the laws of physics appear simple. Inertial frames have this property. An inertial frame of reference is one in which Newton’s first law (the law of inertia) holds. In other words, an inertial frame is one in which a body moves uniformly and rectilinearly in 4 Introduction to Modern Physics absence of any forces. All frames of reference moving with constant velocity relative to an inertial frame are also inertial frames. A frame possessing acceleration relative to an inertial frame is called non-inertial frame. Newton’s first law is not valid in non-inertial frame. Reference frame with its origin fixed at the center of the sun and the three axes directed towards the stationary stars was supposed to be the fundamental inertial frame. In this frame, the motion of planets appear simple. Newton’s laws are valid this heliocentric frame. Let us see whether the earth is an inertial frame or not. The magnitude of acceleration associated with the orbital motion of earth around the sun is 0.006 m/s2 and that with the spin motion of earth at equator is 0.034 m/s2. For all practical purposes these accelerations are negligibly small and the earth may be regarded as an inertial frame but for precise work its acceleration must be taken into consideration. The entire classical mechanics was developed on these notions of space and time it worked efficiently. No deviations between the theoretical and experimental results were noticed till the end of the 19th century. By the end of 19th century particles (electrons) moving with speed comparable with the speed of light c were available; and the departures from classical mechanics were observed. For example, classical mechanics predicts that the radius r of the orbit of electron moving in a magnetic field of strength B is given by r = mv/qB, where m, v and q denote mass, velocity and charge of electron. The experiments carried out to measure the orbit radius of electron moving at low velocity give the predicted result; but the observed radius of electron moving at very high speed does not agree with the classical result. Many other experimental observations indicated that the laws of classical mechanics were no longer adequate for the description of motion of particles moving at high speeds. In 1905 Albert Einstein gave new ideas of space and time and laid the foundation of special theory of relativity. This new theory does not discard the classical mechanics as completely wrong but includes the results of old theory as a special case in the limit (v/c) ® 0. i.e., all the results of special theory of relativity reduce to the corresponding classical expressions in the limit of low speed. 1.2 CLASSICAL PRINCIPLE OF RELATIVITY: GALILEAN TRANSFORMATION EQUATIONS The Galilean transformation equations are a set of equations connecting the space-time coordinates of an event observed in two inertial frames, which are in relative motion. Consider two inertial frames S (unprimed) and S' (primed) with their corresponding axes parallel; the frame S' is moving along the common x-x' direction with velocity v relative to the frame S. Each frame has its own observer equipped with identical and compared measuring stick and clock. Assume that when the origin O of the frame S' passes over the origin O of frame S, both observers set their clocks at zero i.e., t = t' = 0. The event to be observed is the motion of a particle. At certain moment, the S-observer registers the space-time coordinates of the particle as (x, y, z, t) and S'- observer as (x', y', z', t'). It is evident that the primed coordinates are related to unprimed coordinates through the relationship x' = x – vt, y' = y, z' = z, t' = t...(1.2.1) The last equation t' = t has been written on the basis of the assumption that time flows at the same rate in all inertial frames. This notion of time comes from our everyday experiences with slowly moving objects and is confirmed in analyzing the motion of such objects. Equations (1.2.1) are called Galilean transformation equations. Relative to S', the frame S is moving with velocity v in negative The Special Theory of Relativity 5 direction of x-axis and therefore inverse transformation equations are obtained by interchanging the primed and unprimed coordinates and replacing v with –v. Thus x = x' + vt', y = y', z = z', t = t'...(1.2.2) Fig. 1.2.1 Galilean transformation Transformation of Length Let us see how the length of an object transforms on transition from S to S'. Consider a rod placed in frame S along its x-axis. The length of rod is equal to the difference of its end coordinates: l = x2 – x1. In frame S', the length of rod is defined by the difference of its end coordinates measured simultaneously. Thus: l' = x2′ − x1′ Making use of Galilean transformation equations we have l' = (x2 – vt) – (x1 – vt) = x2 – x1 = l Thus the distance between two points is invariant under Galilean transformation. Transformation of Velocity Differentiating the first equation of Galilean transformation, we have dx ′ dx = −v dt ′ dt...(1.2.3) ux′ = ux − v where ux and u'x are the x-components of velocity of the particle measured in frame S and S' respectively. Eqn. (1.2.3) is known as the classical law of velocity transformation. The inverse law is u x = u' + v...(1.2.4) These equations show that velocity is not invariant; it has different values in different inertial frames depending on their relative velocities. Transformation of Acceleration Differentiating equation (1.2.3) with respect to time, we have dux′ dux = ⇒ ax′ = ax...(1.2.5) dt dt 6 Introduction to Modern Physics where ax and a'x are the accelerations of the particle in S and S'. Thus we see that the acceleration is invariant with respect to Galilean transformation. Transformation of the Fundamental Law of Dynamics (Newton’s Law) The fundamental law of mechanics, which relates the force acting on a particle to its acceleration, is ma = F...(1.2.6) In classical mechanics, the mass of a particle is assumed to be independent of velocity of the moving particle. The well known position dependent forces–gravitational, electrostatic and elastic forces and velocity dependent forces- friction and viscous forces are also invariant with respect to Galilean transformation because of the invariance of length, relative velocity and time. Hence the fundamental law of mechanics is also invariant under Galilean transformation. Thus ma =F in frame S m' a' = F' in frame S' The invariance of the basic laws of mechanics ensures that all mechanical phenomena proceed identically in all inertial frames of reference consequently no mechanical experiment performed wholly within an inertial frame can tell us whether the given frame is at rest or moving uniformly in a straight line. In other words all inertial frames are absolutely equivalent, and none of them can be preferred to others. This statement is called the classical (Galilean) principle of relativity. The Galilean principle of relativity was successfully applied to the mechanical phenomena only because in Galileo’s time mechanics represented the whole physics. The classical notions of space, time and matter were regarded so fundamental that nobody ever felt necessity to raise any doubts about their truth. The Galilean principle of relativity did not worry physicists too much by the middle of the 19th century. By the middle of 19th century other branches of physics—electrodynamics, optics and thermodynamics—were developing and each of them required its own basic laws. A natural question arose: does the Galilean principle of relativity cover all physics as well? If the principle of relativity does not apply to other branches of physics then non-mechanical phenomena can be used to distinguish inertial frames thereby choosing a preferred frame. The basic laws of electrodynamics—Maxwell’s field equations—predicted that light was an electromagnetic phenomenon. The light propagates in vacuum with speed c = (m0e0) –½ = 3 × 108 m/s. The wave nature of light compelled the then physicists to assume a medium for the propagation of light and hypothetical medium luminiferous ether was postulated to meet this requirement. Ether was regarded absolutely at rest and light was supposed to travel with speed c relative to the ether. If a certain frame is moving with velocity v relative to the ether; the speed of light in that frame, according to Galilean transformation, is c ± v; the plus sign when c and v are oppositely directed and minus sign when c and v have the same direction. Making us of this result that the light has different speed in different frames; the famous Michelson-Morley experiment was set up to detect the motion of the earth with respect to the ether. When Galilean transformation equations were applied to the newly discovered laws of electrodynamics, the Maxwell’s equations, it was found that they change their shape on transition from one inertial frame to another. At first the validity of Maxwell’s equations was questioned and attempts were made to modify them in a way to make them consistent with the Galilean principle of The Special Theory of Relativity 7 relativity. But such attempts predicted new phenomenon, which could not be verified experimentally. It was then realized that Maxwell’s equations need no modifications. 1.3 MICHELSON-MORLEY EXPERIMENT (1881) The purpose of the experiment was to detect the motion of the earth relative to the hypothetical medium ether, which was supposed to be at rest. The instrument employed was the Michelson interferometer, which consists of two optically plane mirrors M1 and M2 fixed on two mutually perpendicular arms PM1 and PM2. At the point of intersection of the two arms, a glass plate P semi- silvered at its rear end is fixed. The glass plate P is inclined at 45° to each mirror. Monochromatic light from an extended source is allowed to fall on the plate P, which splits the incident beam into two beams—beam 1 that travels along the arm PM1 and beam 2 that travels along the arm PM2. The beam 1 is reflected back from mirror M1 and comes to the rear surface of the plate P where it suffers partial reflection and finally goes into the telescope T. The beam 2 also suffers reflection at the mirror M2 and is received into the telescope. These interfering beams produce interference fringes, which are observed in the telescope. Fig. 1.3.1 Michelson’s interferometer Now suppose that at the moment of the experiment the apparatus moves together with the earth with velocity v (= 3 × 104 m/s) in its orbit along the arm PM1. Relative to the apparatus the light traveling along the path PM1 has speed c – v and that traveling along the path M1P has speed c + v. If l is the length of the arm PM1 then the time spent by light to traverse the path PM1P is equal to −1 l l 2l 1 2l  v 2  2l  v 2  t|| = + = = 1 −  = 1 + ...(1.3.1) c − v c + v c 1 − v 2 / c 2 c  c 2  c  c 2  Since v/c c, the Lorentz transformations for x and t become imaginary; this means that motion with speed greater than that of speed of light is impossible. One of the thought-provoking features of the Lorentz transformations is that the time transformation equation contains spatial coordinate, which suggests that the space and time are inseparable. In other words, we should not speak separately of space and time but of unified space- time in which all phenomena take place. 1.6 VELOCITY TRANSFORMATION Consider an inertial frame S' moving relative to frame S with velocity v along the common x–x' direction. The space-time coordinates of a particle measured by S and S' observers are (x, y, z, t) 14 Introduction to Modern Physics and (x', y', z', t') respectively. Let the particle move through a distance dx in time dt in frame S; the corresponding quantities measured by S' observer are obtained by differentiating the Lorentz- transformation equations x' = g (x – vt), y' = y, z' = z t' = g (t – vx/c2) From these equations, we have dx' = g (dx – vdt), dy' = dy, dz' = dz...(1.6.1) 2 dt' = g (dt –vdx/c )...(1.6.2) Dividing Eqn. (1.6.1) by (1.6.2), we have dx ′ dx − vdt ( dx / dt) − v = = dt ′ dt − vdx / c2 1 − (v / c 2 ) (dx / dt ) ux − v ux′ =...(1.6.3) 1 − vux /c2 dy′ dy (dy / dt ) = = dt ′ γ (dt − vdx / c2 ) γ (1 − (v / c2 )dx / dt ) uy 1 − β2 uy′ =...(1.6.4) 1 − vux / c2 dz′ dz (dz / dt ) = = dt ′ γ (dt − vdx / c ) γ(1 − (v / c2 )dx / dt ) 2 uz 1 − β2 uz′ =...(1.6.5) 1 − vux / c2 Inverse velocity transformation equations are u′x + v u′y 1 − β2 uz′ 1 − β2 ux = , uy = , uz =...(1.6.6) 1 + vux′ / c2 1 + vux′ / c2 1 + vux′ / c2 Let us apply the transformation equation to the speed of light. If a photon moves with velocity ux = c in frame S, then its velocity in frame S' will be ux − v c−v u′x = = =c 1 − vux / c 2 1 − vc / c2 It can easily be seen that the relativistic formulae for transformation of velocity reduce to the Galilean transformation equations in the limit of low speed (v/c) ® 0. The Special Theory of Relativity 15 1.7 SIMULTANEITY In relativity the concept of simultaneity is not absolute. Two events occurring simultaneously in one inertial frame may not be simultaneous, in general, in other. Assume that the event 1 occurs at point x1, y1, z1 and at time t1 and event 2 occurs at point x2, y2, z2 and at time t2 in frame S. The space- time coordinates of these two events as measured in frame S', which is moving relative to S with velocity v in the common x-x' direction, can be obtained from Lorentz transformations x1′ = g (x1 –vt1), x2′ = g (x2 – vt2) t1′ = g (t1 – vx1/c2), t2′ = g (t2 – vx2/c2) The difference of space coordinates and time coordinates are in frame S' are x2′ – x1′ = g{(x2 – x1) – v(t2 – t1)}...(1.7.1) t2′ – t1′ = g{(t2 – t1) – (v/c2)(x2 – x1)}...(1.7.2) Eqn. (1.7.2) gives the time interval between the events as measured in frame S'. It is evident that if the two events are simultaneous (i.e., t2 – t 1 = 0) in frame S, they are not simultaneous (i.e., t2′ – t1′ ¹ 0) in frame S'. In fact t2′ – t1′ = – (gv/c2)(x2 – x1)...(1.7.3) The events are simultaneous in S' only if they occur at the same point in S (i.e., x2 – x1 = 0). Thus simultaneity is a relative concept. If t2′ – t1′ > 0, the events occur in frame S' in the same sequence as they occur in frame S. This always happens for events, which are related by cause and effect. That is, cause precedes the effect, which is known as the causality principle. If t2′ – t1′ < 0, the events occur in reverse sequence in S'. Such events cannot be related by cause and effect. It is important to point out that the relativity of simultaneity follows from the finiteness of the speed of light. In the limit c ® ¥ (classical assumption), simultaneity is an absolute concept i.e., t2′ – t1′ = t2 – t1. 1.8 LORENTZ CONTRACTION A moving body appears to be contracted in the direction of its motion. This phenomenon is called Lorentz (or Fitzgerald) contraction. Let us consider a rod arranged along the x'-axis and at rest relative to the frame S'. The length of the rod in frame S' is l0 = x2′ – x1′ where x1′ and x2′ are the coordinates of the rod ends. The length l0 is called the proper length of the rod. Now consider a frame S relative to which the frame S' is moving with velocity v along x–x' direction. To determine the length of rod in frame S, we must note the coordinates of the ends x1 and x2 at the same moment of time, say t0. The length of rod in frame S is l0 = x2 – x1. From Lorentz transformations, we have x1′ = γ ( x1 − vt0 ). x2′ = γ ( x2 − vt0 ) \ l0 = x2′ − x1′ = γ ( x2 − x1 ) = γ l 16 Introduction to Modern Physics l = (l0/g) = l0 1− β 2...(1.8.1) Evidently l < l0. Thus the moving rod appears to be contracted. (a) The rod is placed in frame S' (b) The rod is placed in frame S Fig. 1.8.1 Transformation of length If the rod is placed in frame S then its proper length is l0 = x2 – x1. Its length l in frame S' is equal to the difference of ends coordinates x1′ and x2′ measured at the same moment of time, say t0′. l0 = x2 − x1 = γ {( x2′ + vt0′ ) − ( x1′ + vt0′ )} = γ( x2′ − x1′ ) = γ l \ l = l0 / γ = l0 1 − β 2 Thus the length contraction is reciprocal. The rod in either frame appears to be shortened to the observer in the other frame. 1.9 TIME DILATION According to relativity there is no such thing as universal time. The rate of flow of time actually depends on the state of motion of the observer. Let us see how the time interval between two events measured in one inertial frame is related to that measured in another inertial frame, which is moving relative to the first one. Assume that an event 1 occurs at point x′0 at time t1′ in the frame S' and another event 2 also occurs at the same point but at time t2′. The interval between the two events is Dt' = t2′ – t1′. This time interval is measured on a single clock located at the point of occurrence of the events and is called the proper time interval and is usually denoted by Dt. The same two events are observed from a reference frame S relative to which the frame S' is moving with velocity v. Let t1 and t2 be the time of occurrence of the same events registered on the clocks of the frame S. Of course these times will be recorded on the clocks located at different points. The time interval The Special Theory of Relativity 17 Dt = t2 – t1 measured in the frame S is called non-proper or improper time interval. From Lorentz transformations t1 = γ (t1′ + vx0′ / c2 ), t2 = γ (t2′ + vx0′ / c 2 ) \ t2 − t1 = γ (t2′ − t1′ ) Dt = g Dt ∆τ Dt = , b = v/c...(1.9.1) 1− β 2 Fig. 1.9.1 Transformation of time interval Thus the time interval between two events has different values in different inertial frames, which are in relative motion. The time interval is least in the reference frame in which the events take place at the same point and hence registered on the same clock. Since the non-proper time is greater than the proper time, a moving clock appears to go slow. This phenomenon is called dilation of time. The variation of Dt with velocity v is shown in Fig 1.9.2. Fig. 1.9.2 Time dilation 1.10 EXPERIMENTAL VERIFICATION OF LENGTH CONTRACTION AND T TIME DILATION The conclusions of the special theory of relativity find direct experimental verification in many of the phenomena of particle physics. We shall illustrate this by an example. Muons are unstable sub- atomic particles, which decay into electron and neutrino. Their mean lifetime in a frame in which they are at rest is 2 µs. They are created in the upper atmosphere at a height 5 to 6 kms during the 18 Introduction to Modern Physics interaction of primary cosmic rays with the atmosphere. They are also found in considerable number at the sea level. The speed of muons is v = 0.998 c. Classical calculation shows that muons can travel in their lifetime a distance d = v t = (3 × 108m/s) (2 × 10–6 s) = 600 m This distance is much smaller than the height where the muons are born. Let us explain this paradox by relativistic calculation. The lifetime of muons is their proper life measured in their own frame. In laboratory frame their life is t = t /Ö(1 – b2) = 31.7 × 10–6s. In this time muons can travel a distance d = v t = (0.998 c) (31.7 × 10–6s) = 9.5 km. Thus muons can reach the sea level in their lifetime. We can arrive at the same conclusion by considering the length contraction formula. In muons frame the distance between the birthplace of muons and the sea level appears to be contracted to d = d0 Ö(1 – b2) = (6 ´ 103 m) Ö(1 – (0.998)2) = 379 m The time required to traverse this distance t = d/v = 379 m/(0.998 × 3 × 108 m/s) = 1.26 µs. This time is less than the proper lifetime of muons. 1.11 INTERVAL An event in a frame is characterized by space-time coordinates. Assume that an event 1 occurs at point x1, y1, z1 and at time t1. The corresponding coordinates for another event 2 are x2, y2, z2, t2. The quantity s12 defined by s212 = c2 ( t2 − t1 ) − ( x2 − x1 ) − ( y2 − y1 ) − ( z2 − z1 ) 2 2 2 2...(1.11.1) is called the interval between the events. If the events are infinitesimally close together, the interval is defined by ds2 = c2dt2 – dx2 – dy2 – dz2...(1.11.2) In frame S' the interval is defined by (ds′)2 = c 2 (dt ′)2 − (dx ′)2 − (dy′)2 − (dz′)2...(1.11.3) A remarkable property of interval is that it is invariant with respect to Lorentz transformations i.e., ds2 = ds' 2 From Lorentz transformations, we have dx − βcdt dx ′ = , dy′ = dy, dz′ = dz and 1− β 2 dt − (β / c ) dx dt ′ =...(1.11.4) 1 − β2 The Special Theory of Relativity 19 Substituting these values in Eqn. (1.11.3), we find {dt − (β / c)dx}2 (dx − β cdt )2 (ds′)2 = c2 − − dy2 − dz2 1− β 2 1− β 2 = c2 dt2 – dx2 – dy2 – dz2 = ds2 1.12 DOPPLER’S EFFECT The apparent change in frequency of a wave due to relative motion between the source of the wave and the observer receiving it, is called the Doppler’s effect. Let a monochromatic source placed at the origin of frame S' emit a plane wave in xy-plane in the direction making an angle q' with x'-axis. The equation of the wave in this frame is y' = a′ cos[ω′ t ′ − kx′ x′ − k ′y y′] 2π where kx′ = k ′ cos θ′, ky′ = k ′ sin θ′, k ′ = λ′ \ y' = a′ cos [ω′ t ′ − k ′x′ cos θ′ − k ′y′ sin θ′]...(1.12.1) The equation of the same wave in the frame S will be written as y = a cos [ω t − kx cos θ − ky sin θ]...(1.12.2) Fig. 1.12.1 Doppler’s effect The phase of a wave is invariant quantity i.e., j' = j. On transition from S' to S, the phase of the wave (1.12.1) becomes j = [ω′γ (t − vx / c2 ) − k ′γ ( x − vt )cos θ′ − k ′y sin θ′]   ω′v   =  γ(ω′ + k ′v cos θ′) t − γ  2 + k ′ cos θ′  x − −k ′y sin θ′...(1.12.3)   c   Comparing Eqn. (1.12.3) with the phase of the wave (1.12.2), we have w = γ (ω′ + k ′v cos θ′)...(1.12.4) 20 Introduction to Modern Physics  v  k cos q = γ  ω′ 2 + k ′ cos θ′   c  or, k cos q = γ (k ′β + k ′ cos θ′)...(1.12.5) k sin q = k' sin q'...(1.12.6) The first two equations give relativistic Doppler’s effect. Equation (1.12.4) can be transformed into a more convenient form as follows.  ω′  w = γ  ω′ + v cos θ′  since k' = w'/c  c  or w = γ (1 + β cos θ′ ) ω′...(1.12.7) Inverse transformation of Eqn. (1.12.7) is w' = γ (1 − β cos θ ) ω...(1.12.8) ω′ ω′ 1 − β2 \ w = = γ(1 − β cos θ) 1 − β cos θ 1 − β2 or v = ν′...(1.12.9) 1 − β cos θ Eqn. (1.12.9) gives relativistic Doppler’s shift. v' = proper frequency, v = observed frequency Fig. 1.12.2 Relativistic Doppler’s effect For q = 0 (velocity of source coincides with that of velocity of light) 1 − β2 1+ β v = ν′ = ν′...(1.12.10) 1− β 1−β Thus n > n'. Thus the observed frequency is greater than the emitted frequency For q = p (velocity of source is opposite to that of light) 1− β ν = ν′...(1.12.11) 1+ β The Special Theory of Relativity 21 In this case v < v'. Observed frequency is less than that emitted by source. For q = p/2, the relative velocity between the source and the observer is zero. However, even in this case there is a shift in frequency; the apparent frequency differs from the true frequency by a factor Ö(1 – b2). This is called transverse Doppler’s effect. In this case the observed frequency is always lower than the proper frequency. The transverse Doppler’s shift is a second order effect and does not exists in classical theory. Classical Doppler’s Effect Retaining the terms up to first order in b in relativistic expression for Doppler’s shift we get classical Doppler’s effect. Thus ν′ 1 − β 2 neglecting β2 ν′ ν= → = ν′ (1 + β cos θ)...(1.12.12) 1 − β cos θ 1 − β cos θ For q = 0 λ − λ′ υ n = n' (1 + b) or =− (violet shift)...(1.12.13) λ′ c and for q = p λ − λ′ υ n = n' ( 1 – b) or = (red shift)...(1.12.14) λ′ c  1  ν = ν′   = ν′(1 + β cos θ)  1 − β cos θ  Fig. 1.12.3 Classical Doppler’s effect Aberration of Light Dividing Eqn. (1.12.5) by (1.12.6), we obtain k ′ sin θ′ tan q = γ ( k β + k ′ cos θ′ ) ′ sin θ′ 1 − β 2 =...(1.12.15) cos θ′ +β 22 Introduction to Modern Physics The inverse transformation is sin θ 1 − β2 tan q' =...(1.12.16) cos θ − β Eqns. (1.12.15) and (1.12.16) connect the directions of light propagation q and q' as seen from two inertial frames S and S'. These are the relativistic equations for the aberration of light. 1.13 RELATIVISTIC MECHANICS In Newtonian mechanics the momentum of a particle is defined as the product of its mass and velocity. p = mν (classical) Here m is regarded as independent of velocity of particle. Newton’s laws are invariant with respect to the Galilean transformation but not with respect to the Lorentz transformation. If momentum is defined in a classical way then the law of conservation of momentum is found to be invariant under Galilean transformation but not under Lorentz transformation. The law of conservation of momentum is more fundamental than the Newton’s laws. To make this law invariant under Lorentz transformation, momentum must be redefined. 1.14 RELATIVISTIC EXPRESSION FOR MOMENTUM: VARIATION OF MASS WITH VELOCITY Let us consider inelastic collision of two identical balls. In frame S' the two balls approach each other with velocity u' along x-axis and after collision the stick together and the composite ball comes to rest. The same collision is observed from a frame S, which is fixed to one of the balls, say ball 2. Evidently the frame S' is moving with velocity v = u' relative to S. The second ball is at rest in the frame S. The velocity of the first ball in the frame S can be obtained from the relativistic law of addition of velocity u′ + u′ 2u′ u = =...(1.14.1) 1 + u′ / c 2 2 1 + u′2 / c2 This equation can be written as 2c2 u ′2 − u ′ + c2 = 0...(1.14.2) u 1  2  2 c 2  c 2  whence u' = ±   − c2  u  u     c2 c2 u2 or u' = ± 1− 2...(1.14.3) u u c The Special Theory of Relativity 23 Fig. 1.14.1 Collision of two identical particles as viewed from two inertial frames We must choose the negative sign before the radical because it gives the classical result (u = 2u') in the limit u/c ® 0. Hence c2 c2 u' = − 1 − u 2 / c2...(1.14.4) u u  c2 c2  and u – u' = u −  − 1 − u 2 / c2   u u  c2  u 2    2 − 1 + 1 − u / c  2 2 = u  c   c2 = 1 − u2 / c 2 1 − 1 − u2 / c2 ...(1.14.5) u   Now let us apply the law of conservation of mass and momentum in frame S. Let m be the mass of the ball 1 before collision. Since the ball 2 is at rest in this frame, we denote its mass by m0. After collision the composite ball comes to at rest in frame S' and hence it appears to move with velocity u'. In frame S, we have mu = Mu' m0 + m = M Eliminating M from these equations, we have m u′ = m0 u − u′ 24 Introduction to Modern Physics Making use of Eqns. (1.14.4) and (1.14.5), we get c2  1 − 1 − u 2 / c2  m u    1 = = m0 c2 1 − u 2 / c2 1 − u2 / c2 1 − 1 − u2 / c 2  u   m0 \ m= = g m0...(1.14.6) 1 − u2 / c 2 In general if particle with rest mass m0 moves with velocity v relative to an observer, its effective mass (or moving mass) is given by m0 m0 m= = = γ m0 2 2...(1.14.7) 1− v / c 1 − β2 The relativistic momentum is defined by m0 v p = mv = = γ m0 v (1.14.8) 1 − β2 The variation of Newtonian momentum and relativistic momentum of a particle with velocity v is shown in the Fig. (1.14.2 ). Fig. 1.14.2 Variation of mass and momentum with velocity 1.15 THE FUNDAMENTAL LAW OF RELATIVISTIC DYNAMICS The fundamental equation of classical mechanics (Newton’s laws) formulated in the form dv m =F dt is not invariant under Lorentz transformation. The correct law must, therefore, be formulated in The Special Theory of Relativity 25 such a way that it must be Lorentz invariant and should transform to the classical law in the limit v/c ® 0. If Newton’s law is formulated in the form d m0 v   =F dt  1 − v2 / c2  ...(1.15.1) it meets both the requirements. The formula F = ma a cannot be used in relativistic case because the acceleration vector a of a particle does not coincide in the general case with the direction of the force F. In the relativistic case, we have d (mv) = F dt Fig. 1.15.1 dm dv...(1.15.2) v+m =F dt dt This equation has been graphically illustrated in the Fig. (1.15.1). The acceleration vector a is not collinear with the force vector F in the general case. The direction of acceleration vector a coincides with that of F only in the two cases: (i) F is perpendicular to v. In this case |v| = constant and therefore the equation of motion becomes d m0 v    =F  dt  1 − v2 / c2   m0 dv =F 1 − v2 / c2 dt m0 a =F 1 − v2 / c2 F 1 − v2 / c2 a =...(1.15.3) m0 (ii) F is parallel to v. In this case the equation of motion may be written in the scalar form as d m0 v    =F  dt  1 − v2 / c2    dv v2 dv  m0  1 − v2 / c2 +   dt c2 1 − v2 / c2 dt  =F 1 − v2 / c2 26 Introduction to Modern Physics  1 v2 / c2  dv m0  + 2 2 3 / 2  =F  1 − v / c 2 2 (1 − v / c )  dt F(1 − v2 / c2 )3/2 \ a =...(1.15.4) m0 1.16 MASS-ENERGY EQUIVALENCE The work done by unbalanced force acting on a particle appears as increment in kinetic energy. The increment in kinetic energy dT due to the force F acting over the elementary path dr (= v dt) is given by d dT = F. dr = F. vdt = (mv). v dt = d (mv). v dt = dm v. v + m dv. v = v2 dm + m v. dv = v2 dm + mvdv...(1.16.1) The mass of the particle varies with velocity as m0 m = 1 − v2 / c2 whence m 2c 2 = m2v2 + m02c2 Taking differential of this equation we have 2mc2 dm = 2mv2 dm + 2m2 vdv Canceling the common factors we have c2dm = v2 dm + mvdv...(1.16.2) Making use of Eqn. (1.16.2), we can write the expression for increment in kinetic energy as dT = c2 dm...(1.16.3) The total kinetic energy of the particle at the instant it acquires velocity v is given by T m ∫ dT = c 2 ∫ dm 0 m0 T = ( m − m0 ) c2 = ( γ − 1) m0 c2...(1.16.4) The Special Theory of Relativity 27  1  T = m0 c2  − 1  ...(1.16.5)  1− v /c 2 2  This is the expression for the relativistic kinetic energy of a particle. For small velocities (v/c ® 0) Eqn. (1.16.5) reduces to the classical formula.   −1/ 2  v2 T = m0 c   1 − 2 2  − 1  c      1 v 2 3 v 4   = m0 c2  1 + + 4 +.......  − 1  2   2 c 8c   1 = m0 v2 (classical result) 2 The variation of relativistic and classical energy with velocity is shown in the figure. The expression (1.16.4) for kinetic energy can be written as mc 2 = T + m0c2...(1.16.6) 2 Einstein interpreted the term mc on the left hand side as the total energy E of the particle. Thus E = mc2 = T + m0c2...(1.16.7) If the particle is at rest, its kinetic energy T is zero but it still possesses energy equal to m0c2. This energy is called the rest energy. In relativistic physics total energy of a particle means the sum of kinetic energy and rest energy. The expression 2 m0 c2 E = mc = = γ m0 c2...(1.16.8) 2 2 1− v / c is one of the most fundamental laws of nature expressing the relationship between the total energy E of a particle and its mass. Equation (1.16.8) states that a change in total energy Fig. 1.16.1 Variation of kinetic energy of a particle by amount DE is equivalent to change in mass with velocity by amount Dm = DE/c2 and vice-versa. 1.17 RELATIONSHIP BETWEEN ENERGY AND MOMENTUM The total energy E and momentum p of a particle are E = γ m0 c2...(1.17.1) 28 Introduction to Modern Physics p = γ m0 v ∴ pc = γ m0 vc...(1.17.2) Now E2 − p2 c2 = γ 2 m02 (c4 − v2 c2 ) = γ 2 m02 c 4 (1 − v2 / c2 ) = ( m0 c 2 ) 2...(1.17.3) This equation gives relation between energy and momentum. The quantities on the right hand side are invariant. Therefore E2 – (pc)2 is also invariant. 1.18 MOMENTUM OF PHOTON The momentum of a particle moving with velocity v is given by E p = mv = v...(1.18.1) c2 Photon is a quantum of light with energy E = hv and velocity c. Its momentum is E hν h p = = =...(1.18.2) c c λ or E = pc...(1.18.3) From Eqn. (1.17.3), we see that the rest mass of photon is zero. In fact all particles moving with speed of light c have zero rest mass. Photon and neutrino are such particles. Converse is also true i.e., particles having zero rest mass always move with speed of light. 1.19 TRANSFORMATION OF MOMENTUM AND ENERGY The total energy E and momentum p of a particle are velocity dependent and hence are not invariant. In this section we shall obtain transformation equations for energy and momentum. Let a particle move from point (x, y, z) to point (x + dx, y + dy, z + dz). The distance dl covered in time dt is dl2 = (dx)2 + (dy)2 +(dz)2. The velocity of the particle is v = dl/dt. The elementary interval between the initial and the final point is 2 2 1/ 2 ds = [(cdt) − (dl) ] 1/ 2  v2  = cdt 1 − 2   c  cdt = γ The Special Theory of Relativity 29 whence γ ds dt =...(1.19.1) c Now the x-component of momentum of a particle is dx  m0 c  px = γ m0 vx = γ m0 = dx...(1.19.2) dt  ds  Similarly m c py =  0  dy  ds  m c...(1.19.3) pz =  0  dz  ds  The energy of the particle is given by

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