QUANTUM MECHANICS

1 QUANTUM MECHANICS . A Paradigms Approach This page intentionally left blank QUANTUM MECHANICS . A Paradigms Approach David H. McIntyre Oregon State University with contributions from Corinne A. Manogue, Janet Tate and the Paradigms in Physics group at Oregon State University Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montr al Toronto Delhi Mexico City S o Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Publisher: Jim Smith Cover Design: Mark Ong Editorial Manager: Laura Kenney Manufacturing Buyer: Kathy Sleys Senior Project Editor: Katie Conley Manager, Rights and Permissions: Zina Arabia Assistant Editors: Peter Alston and Steven Le Manager, Cover Visual Research &. Senior Marketing Manager: Kerry McGinnis Permissions: Karen Sanatar Managing Editor: Corinne Benson Printer and Binder: Courier, Westford Production Project Manager: Mary O'Connell Cover Printer: Courier, Westford Production Management and Composition: Cover Images: David H.

2 McIntyre Element LLC. Copyright 2012 Pearson Education, Inc., publishing as Pearson Addison-Wesley, 1301 Sansome St., San Francisco, CA 94111. All rights reserved. Manufactured in the United States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, 1900 E. Lake Ave., Glenview, IL 60025. For information regarding permissions, call (847) 486-2635. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks.

3 Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data McIntyre, David H. QUANTUM MECHANICS : a paradigms approach / David H. McIntyre ; with contributions from Corinne A. Manogue, Janet Tate, and the Paradigms in Physics group at Oregon State University. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-321-76579-6. ISBN-10: 0-321-76579-6. 1. QUANTUM theory. 2. MECHANICS . I. Manogue, Corinne A. II. Tate, Janet. III. Oregon State University. IV. Title. 2012. 2011039322. ISBN 10: 0-321-76579-6. ISBN 13: 978-0-321-76579-6. 1 2 3 4 5 6 7 8 9 10 CRW 16 15 14 13 12 11. Brief Contents 1 Stern-Gerlach Experiments 1.

4 2 Operators and Measurement 34. 3 Schr dinger Time Evolution 68. 4 QUANTUM Spookiness 97. 5 Quantized Energies: Particle in a Box 107. 6 Unbound States 161. 7 Angular Momentum 202. 8 Hydrogen Atom 250. 9 Harmonic Oscillator 275. 10 Perturbation Theory 312. 11 Hyper ne Structure and the Addition of Angular Momenta 355. 12 Perturbation of Hydrogen 382. 13 Identical Particles 410. 14 Time-Dependent Perturbation Theory 445. 15 Periodic Systems 469. 16 Modern Applications of QUANTUM MECHANICS 502. Appendices 529. Index 553. v This page intentionally left blank Contents Preface xiii Prologue xix 1 Stern-Gerlach Experiments 1. Stern-Gerlach Experiment 1. Experiment 1 5. Experiment 2 6. Experiment 3 7. Experiment 4 8. QUANTUM State Vectors 10. Analysis of Experiment 1 16. Analysis of Experiment 2 16.

5 Superposition States 19. Matrix Notation 22. General QUANTUM Systems 25. Postulates 27. Summary 28. Problems 29. Resources 32. Activities 32. Further Reading 33. 2 Operators and Measurement 34. Operators, Eigenvalues, and Eigenvectors 34. Matrix Representation of Operators 37. Diagonalization of Operators 38. New Operators 41. Spin Component in a General Direction 41. Hermitian Operators 44. Projection Operators 44. Analysis of Experiments 3 and 4 47. Measurement 50. Commuting Observables 54. Uncertainty Principle 56. S2 Operator 57. Spin-1 System 59. vii viii Contents General QUANTUM Systems 62. Summary 63. Problems 64. Resources 67. Activities 67. 3 Schr dinger Time Evolution 68. Schr dinger Equation 68. Spin Precession 72. Magnetic Field in the z-Direction 72. Magnetic Field in a General Direction 78.

6 Neutrino Oscillations 84. Time-Dependent Hamiltonians 87. Magnetic Resonance 87. Light-Matter Interactions 92. Summary 93. Problems 94. Resources 96. Activities 96. Further Reading 96. 4 QUANTUM Spookiness 97. Einstein-Podolsky-Rosen Paradox 97. Schr dinger Cat Paradox 102. Problems 105. Resources 106. Further Reading 106. 5 Quantized Energies: Particle in a Box 107. Spectroscopy 107. Energy Eigenvalue Equation 110. The Wave Function 112. In nite Square Well 119. Finite Square Well 128. Compare and Contrast 133. Wave Function Curvature 133. Nodes 135. Barrier Penetration 135. Inversion Symmetry and Parity 136. Orthonormality 136. Completeness 137. Superposition States and Time Dependence 137. Modern Application: QUANTUM Wells and Dots 146. Asymmetric Square Well: Sneak Peek at Perturbations 147.

7 Fitting Energy Eigenstates by Eye or by Computer 150. Qualitative (Eyeball) Solutions 150. Contents ix Numerical Solutions 151. General Potential Wells 154. Summary 154. Problems 156. Resources 159. Activities 159. Further Reading 160. 6 Unbound States 161. Free Particle Eigenstates 161. Energy Eigenstates 161. Momentum Eigenstates 163. Wave Packets 168. Discrete Superposition 168. Continuous Superposition 171. Uncertainty Principle 176. Energy Estimation 180. Unbound States and Scattering 181. Tunneling Through Barriers 188. Atom Interferometry 192. Summary 197. Problems 197. Resources 201. Activities 201. Further Reading 201. 7 Angular Momentum 202. Separating Center-of-Mass and Relative Motion 204. Energy Eigenvalue Equation in Spherical Coordinates 208. Angular Momentum 210.

8 Classical Angular Momentum 210. QUANTUM Mechanical Angular Momentum 210. Separation of Variables: Spherical Coordinates 215. Motion of a Particle on a Ring 218. Azimuthal Solution 220. QUANTUM Measurements on a Particle Con ned to a Ring 223. Superposition States 224. Motion on a Sphere 227. Series Solution of Legendre's Equation 228. Associated Legendre Functions 233. Energy Eigenvalues of a Rigid Rotor 236. Spherical Harmonics 237. Visualization of Spherical Harmonics 240. x Contents Summary 245. Problems 245. Resources 249. Activities 249. 8 Hydrogen Atom 250. The Radial Eigenvalue Equation 250. Solving the Radial Equation 252. Asymptotic Solutions to the Radial Equation 252. Series Solution to the Radial Equation 253. Hydrogen Energies and Spectrum 256. The Radial Wave Functions 261.

9 The Full Hydrogen Wave Functions 263. Superposition States 270. Summary 272. Problems 272. Resources 274. Activities 274. Further Reading 274. 9 Harmonic Oscillator 275. Classical Harmonic Oscillator 275. QUANTUM Mechanical Harmonic Oscillator 277. Wave Functions 284. Dirac Notation 289. Matrix Representations 293. Momentum Space Wave Function 296. The Uncertainty Principle 298. Time Dependence 300. Molecular Vibrations 305. Summary 307. Problems 308. Resources 311. Activities 311. Further Reading 311. 10 Perturbation Theory 312. Spin-1/2 Example 313. General Two-Level Example 317. Nondegenerate Perturbation Theory 319. First-Order Energy Correction 320. First-Order State Vector Correction 324. Second-Order Nondegenerate Perturbation Theory 329. Degenerate Perturbation Theory 336.

10 More Examples 343. Contents xi Harmonic Oscillator 343. Stark Effect in Hydrogen 346. Summary 351. Problems 352. 11 Hyper ne Structure and the Addition of Angular Momenta 355. Hyper ne Interaction 355. Angular Momentum Review 357. Angular Momentum Ladder Operators 359. Diagonalization of the Hyper ne Perturbation 361. The Coupled Basis 365. Addition of Generalized Angular Momenta 370. Angular Momentum in Atoms and Spectroscopic Notation 377. Summary 377. Problems 379. Resources 381. Activities 381. Further Reading 381. 12 Perturbation of Hydrogen 382. Hydrogen Energy Levels 382. Fine Structure of Hydrogen 386. Relativistic Correction 386. Spin-Orbit Coupling 388. Zeeman Effect 393. Zeeman Effect without Spin 394. Zeeman Effect with Spin 396. Weak magnetic eld 396. Strong magnetic eld 402.