Roger G Newton,Quantum Physics,A Text for Graduate Students. With 27 Figures,Roger G Newton,Department of Physics. Swain Hall West 117,Indiana University,Bloomington IN 47405. newton indiana edu,Series Editors,R Stephen Berry Joseph L Birman Mark P Silverman. Department of Chemistry Department of Physics Department of Physics. University of Chicago City College of CUNY Trinity College. Chicago IL 60637 New York NY 10031 Hartford CT 06106. USA USA USA,H Eugene Stanley Mikhail Voloshin, Center for Polymer Studies Theoretical Physics Institute. Physics Department Tate Laboratory of Physics,Boston University The University of Minnesota. Boston MA 02215 Minneapolis MN 55455, Library of Congress Cataloging in Publication Data. Newton Roger G, Quantum physics a text for graduate students Roger G Newton. p cm Graduate texts in contemporary physics,Includes bibliographical references and index. ISBN 0 387 95473 2 alk paper,1 Quantum theory I Title II Series. QC174 12 N49 2002,530 12 dc21 2002020945,ISBN 0 387 95473 2 Printed on acid free paper. 2002 Springer Verlag New York Inc, All rights reserved This work may not be translated or copied in whole or in part without the. written permission of the publisher Springer Verlag New York Inc 175 Fifth Avenue New York. NY 10010 USA except for brief excerpts in connection with reviews or scholarly analysis Use. in connection with any form of information storage and retrieval electronic adaptation computer. software or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names trademarks service marks and similar terms even if. they are not identified as such is not to be taken as an expression of opinion as to whether or not. they are subject to proprietary rights,Printed in the United States of America. 9 8 7 6 5 4 3 2 1 SPIN 10874003, Typesetting Pages created by the author using a Springer TEX macro package. www springer ny com,Springer Verlag New York Berlin Heidelberg. A member of BertelsmannSpringer Science Business Media GmbH. in the hope that he too will grow up to love science. The combination of quantum mechanics and quantum eld theory consti. tutes the most revolutionary and in uential physical theory of the twentieth. century Its impact is felt not only in almost all other sciences but the fruits. of its application are ubiquitous in everyday life This textbook is designed. to teach graduate students the underlying quantum physical ideas their. mathematical formulations and the basic problem solving techniques of. the discipline It assumes they have taken at least one introductory course. in quantum mechanics as undergraduates and are familiar with the history. of the subject and the basic experimental evidence that led to its adoption. as well as with many of its fundamental notions In contrast to most other. authors I am therefore not introducing the quantum theory via an histor. ical survey of its early successes Instead following the models of books. on classsical mechanics or electromagnetism I develop the theory from its. basic assumptions beginning with statics followed by the dynamics and. details of its speci c areas of use as well as the needed mathematical tech. Although this book inevitably deals largely with the behavior of point. particles under various conditions I do not regard particles as the funda. mental entities of the universe the most basic object is the quantum eld. with the observed particles arising from the eld as its quanta For this rea. son I introduce quantum elds right from the beginning and demonstrate. in the rst chapter how particles originate However this volume is not. intended to be a full edged text of quantum eld theory con ning itself to. the fundamental ideas of eld theory and their consequences glossing over. its mathematical di culties and pitfalls it does not deal with any of the. viii Preface, subtleties of that large subject For the development of the basic quantum. dynamics the book employs the Lagrangian technique with the principle. of stationary action The roots of this approach which includes generating. the canonical commutation rules go back to a course taught long ago by. Julian Schwinger ltered through and modi ed by many years of teach. ing the subject Similarly for the manner of introducing the matrices in. Chapter 10, The general physics preparation of the students for whom this book is. intended should comprise classical mechanics including its Lagrangian and. Hamiltonian formulations Maxwell s theory of electromagnetism and the. special theory of relativity as well as some understanding of statistical me. chanics As far as mathematics is concerned they are expected to have a. basic knowledge of linear algebra linear di erential equations both ordi. nary and partial and the theory of functions of a complex variable The. book s extensive mathematical appendices contain the needed elements of. vector spaces including Hilbert space the Dirac delta function linear. integral equations the required classical special functions and group rep. resentations, Rather than mixing this mathematical material with the explanation. of the physics though some details that the occasion demands are inter. spersed set o by horizontal lines and printed in a smaller font I collected. it at the end and the instructor will have to present or assign it when. needed These appendices are not meant to take the place of mathematics. courses appropriate for graduate students in physics or chemistry they can. serve at best as crutches or reminders However the inclusion of group rep. resentations allows me to apply this invaluable tool more extensively than. do most other comparable graduate texts A large number of exercises are. sprinkled as boldfaced footnotes throughout the text others collected at. the end of each chapter, A word about notation Many books and research articles employ the. very convenient Dirac notation in which a general state vector is denoted. by an eigenvector A of A with the eigenvalue A is written A and. the inner product becomes A A Although I shall not always. use this notation when it comes in handy I shall employ it. Since the subject of this book now pervades almost all of physics it is. hard to pin down my di use indebtedness I certainly owe much to the. four teachers from whom I learned quantum mechanics and quantum eld. theory as a student Norman Ramsey Walter Kohn John H Van Vleck. and above all Julian Schwinger Others to whom I owe a debt of gratitude. for their infusion of wisdom over the years are Res Jost Joseph Jauch. and my colleagues at Indiana University especially Emil Konopinski John. Challifour and Don Lichtenberg,Bloomington Indiana USA Roger G Newton. December 2001,Preface vii,I Physics xv,1 Quantum Basics Statics 1. 1 1 States and Dynamical Variables 1,1 1 1 Measurements 3. 1 1 2 Fields as dynamical variables 5,1 2 Probabilities 6. 1 2 1 Correlations 7,1 2 2 Interference 9,1 2 3 Expectation values and variance 11. 1 3 Mixed States 12,1 4 Representations 16,1 4 1 The con guration representation 19. 1 4 2 The momentum representation 22,1 4 3 The number representation 23. 1 5 Transformation Theory 27,1 6 Problems and Exercises 32. 2 Quantum Basics Dynamics 33,2 1 Time Dependence and Commutation Rules 33. 2 1 1 The principle of stationary action 34,2 1 2 The Lagrangian equations of motion 36. x Contents,2 1 3 The canonical commutation relations 37. 2 1 4 The Hamiltonian equations of motion 39,2 1 5 The Schro dinger picture 40. 2 2 Systems of Particles 43,2 2 1 Linear and angular momentum 43. 2 2 2 The equations of motion 44, 2 2 3 The time independent Schro dinger equation 46. 2 2 4 Example Gaussian wave packets 48,2 3 Fields 51. 2 3 1 The matter eld 51,2 3 2 In nitely many particles as a eld 56. 2 3 3 The electromagnetic eld 58,2 4 Canonical and Gauge Transformations 63. 2 5 Problems and Exercises 66,3 The Schro dinger Equation in One Dimension 67. 3 1 A Free Particle 67,3 2 A Particle With Potential Energy 69. 3 2 1 The square potential barrier 69,3 2 2 The potential well 74. 3 2 3 General transmission and re ection 77,3 2 4 The double well 78. 3 2 5 The Kronig Penney potential 80,3 2 6 The simple harmonic oscillator 82. 3 2 7 Coherent states 85,3 3 Problems and Exercises 89. 4 One and Two Particle Systems in Three Dimensions 91. 4 1 Free Particles 91,4 1 1 The propagator 96,4 1 2 Two particles 98. 4 2 Potentials with Spherical Symmerty 99,4 2 1 The three dimensional square well 100. 4 2 2 The scattering amplitude 103,4 2 3 Transformation to the laboratory system 110. 4 2 4 The Coulomb potential 112,4 2 5 The hydrogen atom 112. 4 2 6 Coulomb scattering 117,4 3 The Inverse Problem 121. 4 3 1 Solution procedure 122,4 3 2 The phase of the scattering amplitude 125. 4 4 Potentials without Spherical Symmetry 127,4 4 1 Green s functions 127. 4 4 2 The scattering amplitude 131,4 4 3 Solving the Lippmann Schwinger equation 134. Contents xi,4 5 Problems and Exercises 137,5 Symmetries 143. 5 1 The Angular Momentum Operator 143,5 1 1 Active and passive rotations 144. 5 1 2 An oscillator model 148,5 1 3 States of spin 1 and spin 1 2 150. 5 2 The Rotation Group 151,5 2 1 Angular momentum and the carrier space 155. 5 2 2 Polarization and the spin density matrix 160. 5 2 3 The magnetic moment 161,5 2 4 Addition of angular momenta 163. 5 2 5 Spherical tensors and selection rules 167,5 3 Time Reversal 172. 5 3 1 The time reversal operator 172,5 3 2 Time reversal invariance 174. 5 4 Problems and Exercises 177,6 Stationary Approximation Methods 181. 6 1 The WKB Approximation 181,6 1 1 The connecting formulas 184. 6 1 2 Examples 187,6 1 3 Scattering in the WKB approximation 190. 6 2 Bound State Perturbation Theory 195,6 2 1 The nondegenerate case 196. 6 2 2 Closely spaced levels 198,6 2 3 The case of degeneracy 199. 6 2 4 Second order shifts from rst order states 203. 6 2 5 E ects of symmetry 204, 6 2 6 Relativistic correction to hydrogenic levels 205. 6 2 7 The Stark e ect 206,6 3 Variational Methods 208. 6 3 1 Bound states 208,6 3 2 The helium atom 210,6 3 3 Scattering 212. 6 4 Problems and Exercises 214,7 Static Magnetic Fields 217. 7 1 The Larmor Precession 217,7 2 The Aharanov Bohm E ect 221. 7 3 Charged Particles with Spin 222, 7 3 1 Spin orbit coupling and the ne structure 223. 7 3 2 The Zeeman e ect 226,7 3 3 The hyper ne structure 229. 7 4 Problems and Exercises 233,xii Contents,8 Time Dependent Hamiltonians 235. 8 1 Perturbation Theory 235,8 1 1 Application to exponential decay 240. 8 1 2 Application to scattering 242,8 1 3 Second order perturbation theory 243. 8 1 4 Interaction with electromagnetic elds 244,8 2 Slowly Changing Hamiltonians 253. 8 3 Problems and Exercises 256,9 Multiparticle Systems 257. 9 1 Two Particles Interacting with a Center 257,9 2 Identical particles 264. 9 2 1 Fock space 269, 9 2 2 Bose Einstein and Fermi Dirac distributions 276. 9 3 Problems and Exercises 278,10 Relativistic Electron Theory 279. 10 1 Preliminaries 279,10 1 1 Relativistic spin 281. 10 1 2 The matrices 282,10 2 The Dirac Equation 292. 10 2 1 Electrons in an electromagnetic eld 299, 10 3 The Dirac Equation for a Central Potential 302. 10 3 1 The hydrogen atom 305,10 4 Problems and Exercises 308. II Mathematical Appendices 309,A The Dirac Delta Function 311. B Hilbert Space 315,B 1 Linear Vector Spaces 315,B 2 In nite Dimensional Linear Vector Spaces 318. B 3 Operators 319,B 3 1 Hermitian operators and their spectra 322. B 3 2 More on operators 328,B 4 Problems and Exercises 332. C Linear Integral Equations 333,C 1 Fredholm Methods 333. C 2 The Born Series 337, C 3 Application to the Lippmann Schwinger Equation 338. D Special Functions 341,D 1 Spherical Harmonics 341. Contents xiii,D 1 1 Legendre polynomials 342,D 1 2 Associated Legendre functions 345. D 1 3 Spherical harmonics 345,D 1 4 Zonal harmonics 346. D 2 Spherical Bessel Functions 347,D 3 Hermite Polynomials 348. D 4 The Hypergeometric Function 350,D 5 Laguerre Polynomials 351. D 6 Problems and Exercises 353,E Group Representations 355. E 1 Groups 355,E 2 Representations 357,E 2 1 Character 361. E 2 2 Real representations 364,E 2 3 Kronecker products 366. E 2 4 Carrier spaces 367,E 2 5 Clebsch Gordan coe cients 371. E 3 Lie Groups 373,E 3 1 Coordinate transformations 373. E 3 2 In nitesimal generators 374,E 3 3 The Lie algebra 379. E 4 Representations of Lie Groups 380,E 4 1 Multiple valuedness 385. E 4 2 The group O 2 386,E 4 3 The group SO 3 388,E 5 Problems and Exercises 397. Quantum Physics A Text for Graduate Students With 27 Figures IL 60637 New York Graduate texts in contemporary physics

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