Table Of Content

How to Be a Quantum Mechanic How to Be a Quantum Mechanic is an introduction to quantum mechanics at the upper-division level. It begins with wave-particle duality and ends with a brief introduction to the Dirac equation. Two attitudes went into its writing: examples are the best way to get into a subject, and numbers and equations do not always sum to understanding— results need exploration. Charles Wohl taught for 40 years at the University of California, Berkeley. He earned his PhD at Berkeley in experimental elementary-particle physics in the group led by Luis Alvarez. Taylor & Francis Taylor & Francis Group http://taylorandfrancis.com How to Be a Quantum Mechanic Charles G. Wohl First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Charles G. Wohl Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Wohl, C. G., author. Title: How to be a quantum mechanic / Charles G. Wohl. Description: First edition. | Boca Raton : CRC Press, 2022. | Series: Frontiers in physics | Includes bibliographical references and index. | Summary: “These lecture notes comprise an advanced undergraduate course in quantum mechanics as taught by Charles G. Wohl for over 30 years at the University of California, Berkeley. Each chapter covers a major subject in quantum mechanics, beginning with an accessible introduction and unfolding in subsections to signpost the reader’s progression through the topic. And, because examples are the best way to get into a subject, every chapter ends with a series of problems-over 175 total in the book-to provide plenty of hands-on practice in calculating. Targeted to upper-division physics students and lecturers, this textbook and its worked examples will teach students how to think like a quantum mechanic”-- Provided by publisher. Identifiers: LCCN 2021061434 (print) | LCCN 2021061435 (ebook) | ISBN 9781032256030 (hardback) | ISBN 9781032256023 (paperback) | ISBN 9781003284185 (ebook) Subjects: LCSH: Quantum theory. Classification: LCC QC174.12 .W633 2022 (print) | LCC QC174.12 (ebook) | DDC 530.12--dc23/eng20220412 LC record available at https://lccn.loc.gov/2021061434 LC ebook record available at https://lccn.loc.gov/2021061435 ISBN: 978-1-032-25603-0 (hbk) ISBN: 978-1-032-25602-3 (pbk) ISBN: 978-1-003-28418-5 (ebk) DOI: 10.1201/9781003284185 Typeset in CMR10 by KnowledgeWorks Global Ltd. Arwen—not for, but because of. v Taylor & Francis Taylor & Francis Group http://taylorandfrancis.com CONTENTS 1. Strangest Things 1 1.1. Planck, Einstein, Compton, de Broglie 2 1.2. Neutron Interference 4 1.3. Photon Interference 7 1.4. Bohr and Hydrogen 9 Problems 14 2. The Schro¨dinger Equation. Bound States 17 2.1. The Time-Dependent Schro¨dinger Equation 18 2.2. The Wave Function 21 2.3. The Time-Independent Equation. Energy Eigenstates 23 2.4. The Infinite Square Well 26 2.5. The Finite Square Well 28 2.6. The Delta-Function Well 34 2.7. Schro¨dinger in Three Dimensions 37 2.8. Two Important Energy Eigenstates 41 2.9. Qualitative Properties of Bound Energy Eigenstates 44 Problems 45 3. Simple Approximations for Bound States 54 3.1. Dimensions and Scaling 55 3.2. Fitting Wavelengths in a Well 58 3.3. Guessing the Ground-State Wave Function 62 3.4. Useful Integrals 67 Problems 68 4. Scattering in One Dimension 70 4.1. Particle Densities and Currents 71 4.2. Scattering by a Step 74 4.3. A General Rectangular Barrier 76 4.4. A Simple Rectangular Barrier 80 4.5. Designing with Rectangular Barriers 82 4.6. Thin Films and Light 86 4.7. Weak Tunneling 87 Problems 90 5. Mathematical Formalism 93 5.1. Vector Spaces. Dirac Notation 94 5.2. States as Vectors 98 5.3. Operators 100 5.4. Successive Operations. Commutators 102 5.5. Operators as Matrices 104 vii 5.6. Expectation Values 105 5.7. More Theorems 106 5.8. Revised Rules 108 Problems 109 6. The Harmonic Oscillator 111 6.1. The Classical Oscillator 112 6.2. The Quantum Oscillator: Series Solution 113 6.3. The Operator Solution 119 6.4. States as Vectors, Operators as Matrices 123 Problems 125 7. Uncertainty Relations. Simultaneous Eigenstates 130 7.1. Heisenberg Uncertainty Relations 131 7.2. The Schwarz Inequality 132 7.3. Proof of Uncertainty Relations 133 7.4. Fourier Transforms. Momentum Space 135 7.5. Time and Energy 138 7.6. When Operators Commute 142 Problems 144 8. Angular Momentum 147 8.1. Central Forces. Separation of Variables 148 8.2. Angular Momentum Commutation Relations 150 8.3. The Operator Solution 152 8.4. Certainty and Uncertainty 156 8.5. States as Vectors, Operators as Matrices (Again) 157 8.6. Differential Operators for Orbital Angular Momentum 159 8.7. Spherical Harmonics 161 8.8. Angular Momentum and the Oscillator 163 Problems 163 9. Hydrogen. The Isotropic Oscillator 165 9.1. The Effective Potential Energy 166 9.2. The Hydrogen Bound-State Energies 167 9.3. The Hydrogen Eigenfunctions 170 9.4. The Isotropic-Oscillator Energies 174 9.5. The Oscillator Eigenfunctions 177 9.6. Hydrogen and the Oscillator 177 Problems 180 10. Spin-1/2 Particles 183 10.1. Spinors. Eigenvalues and Eigenstates 184 10.2. The Polarization Vector 187 viii 10.3. Magnetic Interactions and Zeeman Splitting 189 10.4. Time Dependence and Larmor Precession 191 10.5. Time Dependence and Magnetic Resonance 192 10.6. Stern-Gerlach Experiments 196 10.7. Polarization and Light 199 Problems 201 11. Hyperfine Splitting. Two Angular Momenta. Isospin 204 11.1. Hyperfine Structure of the Hydrogen Ground State 205 11.2. The 21-cm Line and Astronomy 208 11.3. Coupling Two Spin-1/2 Particles 209 11.4. Coupling Any Two Angular Momenta 212 11.5. Clebsch-Gordan Coefficients 214 11.6. Particle Multiplets and Isospin 216 Problems 218 12. Cryptography. The EPR Argument. Bell’s Inequality 223 12.1. Quantum Cryptography 224 12.2. The EPR Argument 226 12.3. Bell’s Inequality 227 Problems 231 13. Time-Independent Perturbation Theory 233 13.1. The Nondegenerate Recipes 234 13.2. Examples of Nondegenerate Theory 236 13.3. The Nondegenerate Derivations 239 13.4. The Degenerate Recipe 241 13.5. Two Selection Rules. A Useful Relation 243 13.6. The Stark Effect in Hydrogen (Strong-Field Case) 245 13.7. Hydrogen Fine Structure: Experiment 248 13.8. Hydrogen Fine Structure: Theory 250 13.9. Atomic Magnetic Moments 255 13.10. The Zeeman Effect in Hydrogen (Weak-Field Case) 257 Problems 259 14. Identical Particles 263 14.1. Electrons in a Box 264 14.2. Electrons in an Atom: the Periodic Table 267 14.3. Electrons in an Atom: More Pauli 273 14.4. Two-Electron Symmetries 275 14.5. The Helium Ground State 276 14.6. The Electron-Electron Repulsion Integral 281 14.7. Helium Excited States. Exchange Degeneracy 283 ix