Elements of Modern Physics

Elements of Modern Physics

S. H. Patil

Elements of Modern Physics

123

S. H. PatilDepartment of PhysicsIndian Institute of Technology BombayMumbai, India

ISBN 978-3-030-70142-0 ISBN 978-3-030-70143-7 (eBook)https://doi.org/10.1007/978-3-030-70143-7

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© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer NatureSwitzerland AG 2021This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, andtransmission or information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publishers, the authors, and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publishers nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publishers remain neutral with regard tojurisdictional claims in published maps and institutional affiliations.

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Dedicatedto

My Parents

Preface

This book has been thoroughly revised and updated as per the requirement of the students. The book provides a perspective of the important concepts and applications in contemporary physics.

While modern physics developing so rapidly, there is a constant need to revise and update the presentation. The present book tries to do this. Starting with a discussion of special theory of relativity and quantum theory, it describes their applications to atoms, molecules, solids and nuclei. There are two special chapters on the modern description of elementary particles and on general theory of relativity and cosmology. The emphasis is on a logical development of ideas, and historical aspects are referred to mainly as an aid to this. An effort has been made to maintain rigour analytical discussions and precision in descriptions. It is hoped that the book will be useful to an advanced undergraduate student, and as a review to a graduate student.

I am grateful to my colleagues, Dr. S.M. Bharati. Dr. S.M. Chitre, Dr. P.P. Divakaran, Dr. Y.K. Gambhir, Dr. G.V. Dass, Dr. Dipan K. Ghosh, Dr. K.S. Kulkarni, Dr. R.C. Mehrotra, Dr. C.H. Mehta, Dr. G. Mukhopadhyay, Dr. R.S. Patil, Dr. G. Thyagarajan and Prof. Atul Mody, Dept. of Physics, VES, College of Arts, Science and Commerce, Mumbai who ungrudgingly gave me their valuable time in reading parts of the manuscript and made valuable suggestions. I also thank Mr. Sunil Somalwar for going through a part of the manuscript.

Mr. S.B Modak not only provided accurate typing but also executed the entire organization of the book with the help of Mr. D.S. Nakhawa, Mr. Kashipathy and Mr. C.A. Sarmalkar. I owe them gratitude. I acknowledge financial support from the curriculum development programme of IIT, Bombay.

S H Patil

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Fundamental Constants

c = 2.997925 × 108 m/s h = 6.6256 × 10–34 Js me = 9.109 × 10–31 kg = 0.511 MeV/c2

e = 1.60206 × 10–19 C k = 1.38044 × 10–23 J/K mp = 938.211 MeV/c2

mn = 939.505 MeV/c2

ε0 = 8.85434 × 10–12 F m or C2/Nm2

µ0 = 4π × 10–7 H/m or N/A2

N = 6.022 × 1026/kmol. number of atoms in 12 kg of 12C

viii

Contents

1. Special Theory of Relativity 1–30 1.1 Inertial Frames of Reference 2 1.2 Galilean Transformations 2 1.3 Velocity of Light 3 1.4 Postulates of Special Relativity 5 1.5 Lorentz Transformations 6 1.6 Simultaneity and Time Dilation 8 1.7 Length Contraction 11 1.8 Transformation of Velocities 12 1.9 Lorentz Four-Vectors 14 1.10 Energy-Momentum Four-Vector and Relativistic Dynamics 16 1.11 Electromagnetic Interaction 18 1.12 Zero-Mass Particles and Doppler Shift 21 1.13 Examples 23 Problems 28

2. Introduction to Quantum Ideas 31–64 2.1 Black-Body Radiation 32 2.2 Photoelectric Effect 37 2.3 Compton Effect 40 2.4 Wave Nature of Particles 43 2.5 Atomic Spectra 46 2.6 Nuclear Model of the Atom 49 2.7 Bohr Model 51 2.8 Examples 56 Problems 61

3. Elements of Quantum Theory 65–100 3.1 A Thought Experiment 66 3.2 The Wave Function 67 3.3 Postulates of Quantum Mechanics 70

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x Elements of Modern Physics

3.4 Some Properties of Observables and Wave Functions 72 3.5 Free Particle 74 3.6 Wave Packet and the Uncertainty Principle 76 3.7 Step Potential 78 3.8 Particle in a Box 83 3.9 Simple Harmonic Oscillator 87 3.10 Small Perturbations 89 3.11 Angular Momentum 90 3.12 Examples 94 Problems 98

4. The One-Electron Atom 101–130 4.1 Solutions of the Schrödinger Equation 102 4.2 Electron Spin 107 4.3 Total Angular Momentum 109 4.4 Fine Structure of One-Electron Atomic Spectra 110 4.5 Hyperfine Structure 115 4.6 Examples of One-Electron Atoms 118 4.7 Schrödinger Equation for Spin 1/2 Particles 120 4.8 Dirac Equation 122 4.9 Examples 125 Problems 129

5. Atoms and Molecules 131–172 5.1 Exchange Symmetry of Wave Functions 132 5.2 Shells and Subshells in Atoms 135 5.3 Periodic Table 137 5.4 Atomic Spectra 142 5.5 X-ray Spectra 151 5.6 Molecular Bonding 159 5.7 Molecular Spectra 162 5.8 Examples 166 Problems 171

6. Interaction with External Fields 173–208 6.1 The Hamiltonian 174 6.2 Atoms in a Magnetic Field 175 6.3 Interaction with Radiation 181 6.4 Spontaneous Transitions 184

xiContents

6.5 Lasers and Masers 188 6.6 Applications of Lasers 192 6.7 Some Experimental Methods 197 6.8 Examples 203 Problems 207

7. Quantum Statistics 209–253 7.1 Distinguishable Arrangements 210 7.2 Statistical Distributions 213 7.3 Applications of Maxwell-Boltzmann Distribution 218 7.4 Applications of Bose-Einstein Distribution 224 7.5 Applications of Fermi-Dirac Distribution 232 7.6 Superconductivity 238 7.7 Examples 246 Problems 251

8. Solid State Physics 255–316 8.1 Binding Forces in Solids 256 8.2 Crystal Structures 260 8.3 Band Theory of Solids 267 8.4 Semiconductors 274 8.5 Semiconductor Devices 283 8.6 Magnetic Properties 292 8.7 Dielectric Properties 302 8.8 Examples 308 Problems 313

9. The Nucleus 317–364 9.1 Properties of the Nucleus 318 9.2 Nuclear Forces 325 9.3 Models of the Nucleus 328 9.4 Weizsacker’s Mass Formula 336 9.5 Nuclear Stability 337 9.6 Nuclear Reactions 345 9.7 Fission Reactors 350 9.8 Thermonuclear Fusion 355 9.9 Examples 358 Problems 362

xii Elements of Modern Physics

10. Elementary Particles 365–390 10.1 Elementary Particles 366 10.2 Strong Interaction 368 10.3 Electromagnetic Interaction 372 10.4 Weak Interaction 373 10.5 Unified Approach 379 10.6 Production and Detection of Particles 380 10.7 Examples 387 Problems 390

11. General Relativity and Cosmology 391–418 11.1 Frames of Reference 392 11.2 Curved Space-Time 395 11.3 Schwarzschild Metric 401 11.4 Kinematics of the Universe 406 11.5 Dynamics of the Universe 409 11.6 The Early Universe 411 11.7 Examples 415 Problems 417 Reference 419–422Answers to Problems 423–428Index 429–432