Quantum Mechanics

Springer-Verlag Berlin Heidelberg GmbH

Physics and Astronomy ONLINE LIBRARY

http://www.springer.de/phys/

JULIAN SCHWINGER

Quantum Mechanics Symbolism of Atomic Measurements

Edited by Berthold-Georg Englert

, Springer

Julian Schwinger (1918-1994)

Dr. Berthold-Georg Englert Gleissenweg 23 85737 Ismaning, Germany

With 78 Drawings and Figures, and 351 Problems

Library of Congress Cataloging-in-Publication Data. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Schwinger, Julian Seymour:

Clarice Schwinger 10727 Stradella Court Los Angeles, CA 90077, USA

Quantum mechanics: symbolism of atomic measurements I Julian Schwinger. Ed. by Bertold-Georg Englert. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2001

(Physics and astronomy online library)

ISBN 978-3-642-07467-7 ISBN 978-3-662-04589-3 (eBook) DOI 10.1007/978-3-662-04589-3

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad­casting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 2001

Originally published by Springer-Verlag Berlin Heidelberg New York in 2001. Sof1cover reprint of the hardcover 1 st edition 2001

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant pro­tective laws and regulations and therefore free for general use.

Typesetting: Camera ready copy by the editor using a Springer TP<: macropackage Cover design: Erich Kirchner, Heidelberg

Printed on acid-free paper 55/3111 5 43 2 1 SPIN: 10980460

To Clarice and Ola

Julian SCHWINGER (1918-1994)

Preface

Julian Schwinger had plans to write a textbook on quantum mechanics since the 1950s when he was teaching the subject at Harvard University regularly.* Roger Newton remembers: t

[A] group of us (Stanley Deser, Dick Arnowitt, Chuck Zemach, Paul Martin and I forgot who else) wrote up lecture notes on his Quantum Mechanics course but he never wanted them published because he "had not yet found the perfect way to do quantum mechanics."

The only text of those days that got published eventually - following a sug­gestion by, and with the help of, Robert Kohler:!: - were the notes to the lectures that Schwinger presented at Les Houches in 1955. The book was reissued in 1991, with this Special Preface by Schwinger [3]:

The first two chapters of this book are devoted to Quantum Kine­matics. In 1985 I had the opportunity to review that development in connection with the celebration of the 100th anniversary of Hermann Weyl's birthday. [ ... ] In presenting my lecture [4] I felt the need to alter only one thing: the notation. Lest one think this rather triv­ial, recall that the ultimate abandonment, early in the 19th century, of Newton's method of fluxions in favor of the Leibnizian calculus, stemmed from the greater flexibility of the latter's notation.

Instead of the symbol of measurement: M(a',b'), I now write: I a' b' I, combining reference to what is selected and what is produced, with an indication that the act of measurement has a beginning and an end. Then, with the conceptual analysis of la'b'l into two stages, one of annihilation and one of creation, as symbolized by

la'b'l = la')(b'l, the fictitious null state, and the symbols 'I' and <1>, can be discarded.

As for Quantum Dynamics, I have long regretted that these chap­ters did not contain numerous examples of the practical use of the Quantum Action Principle in solving physical problems. Perhaps that can be remedied in another book, on Quantum Mechanics. [ ... ]

* See chapter 10 in the recent biography by Mehra and Milton [1]. t As quoted by Schweber in section 7.11 of [2]. tSee the preface to [3].

VIII Preface

The change in notation mentioned here was systematically incorporated in the set of lecture notes that Schwinger wrote up for the students of the three­quarter course on quantum mechanics that he taught twice in the mid-1980s at the University of California, Los Angeles (UCLA). I had the great luck to still be at UCLA during much of the first round, taking care, in fact, of office hours and problem sessions, and I continued to receive Schwinger's handwritten lecture notes after I had left.

Indeed, these notes were meant to be the basis of the intended book for which Schwinger had selected the natural title Quantum Mechanics and the less obvious - to others, that is, not to him - subtitle Symbolism of Atomic Measurements. This choice is the succinct pronouncement of his philosophy, which is spelt out in the Prologue. The quote from page 10:

[P]hysics is an experimental science; it is concerned only with those statements which in some sense can be verified by an experiment. The purpose of the theory is to provide a unification, a codification, or however you want to say it, of those results which can be tested by means of some experiment. Therefore, what is fundamental to any theory of a specific department of nature is the theory of measure­ment within that domain.

is to the point. Schwinger's continuing interest in frontier physics was a permanent and, in

hindsight, unfortunate distraction from the book-writing enterprise. Eventu­ally, his untimely death put an end to all plans, and so his quantum mechanics book is not to be.

Yet there are those UCLA lecture notes. Although they are certainly not identical to the book Schwinger would have written, they do represent a first draft and are the closest thing to the unwritten book that is available. From many conversations I had with him, I know that Schwinger was quite happy with the way he induces the general structure of quantum kinematics and establishes the dynamical principle, his quantum action principle. I think that he had finally "found the perfect way to do quantum mechanics."

I always thought that the notes should be put into a form that makes them accessible to the broad public, but it needed the encouragement of a few friends to actually go about it. Particularly decisive was the gentle push by Robert Finkelstein who, in response to my remark - during a lunch session at the UCLA faculty club (the Chatham had disappeared years earlier) -that somebody should put the notes into print, just said: "You should do it." Thank you, Bob.

And then, of course, there was the consistent support by Clarice Schwinger who gave me the feeling - very calmly and, I'm sure, very consciously - that she couldn't think of anyone else to do it. Thank you, Clarice.

My dear wife Ola had to be content with a much too small share of my attention while I was working on this project. Knowing well how much the book means to me, and why it does, she never complained. Dzi~kuj~ Ci, Olu.

Preface IX

The lecture notes of Schwinger's UCLA course consist of three parts cor­responding to the three quarters of teaching. Here is a brief summary of the contents.

Part A, the material of the fall quarter, begins with an analysis of experi­ments of the Stern-Gerlach type that accomplishes "a self-contained physical and mathematical development of the general structure of quantum kinemat­ics" [4]. ~luch technical material is delivered in passing. In particular, unitary transformations are studied from various angles, and the algebra of angular momentum is treated in depth. Then, an analysis of Galilean invariance yields the non-relativistic Hamilton operator.

The winter quarter, Part B, proceeds from there. The response to in­finitesimal time displacements establishes the equations of motion. Then the Quantum Action Principle is derived, and accepted as a fundamental prin­ciple. In a sense, the rest of Part B and all of Part C consist of instructive applications of the action principle - the "numerous examples" referred to above. Part B contains treatments of, among others, the (driven) harmonic oscillator, bound-state properties of hydrogenic atoms, and Rutherford scat­tering.

Part C (spring quarter) begins with the two-particle Coulomb problem, in­cluding the modifications for two identical particles. The treatment of systems with many identical particles follows, where the notion of second quantization eventually leads to the concept of the quantized field. As a first application, the Hartree-Fock and Thomas-Fermi approaches to many-electron atoms are presented, the latter in considerable detail. § The second and final application is the quantum theory of electromagnetic radiation, which is developed to the extent necessary for an understanding of (the non-relativistic aspects of) the Lamb shift.

During his oral lectures, and in the handed-out notes, Schwinger never took any credit for his own very substantial and highly original contributions. But, of course, he mentioned the names of others whenever appropriate. I de­cided to stick to this practice when preparing the notes for print.

Distributing notes to the students that attend your lectures is one thing, writing a book for the anonymous reader is quite another. So, some editing was unavoidable in the course of turning Schwinger's lecture notes into book form, but I tried to change as little as possible. In addition to the UCLA notes of the mid-1980s, Chapter 12 contains some material from lectures that Schwinger gave at the University of New Mexico, Albuquerque in 1987. The Prologue is based on the transcript of the audio record of a public lecture that he delivered in the early 1960s.'Il Most of the problems are as formulated by Schwinger; in addition to the ones that came with the lecture notes, I discovered many good problems in the Schwinger Papers [5] that are archived at the "CCLA Research Library. In fact, all the raw material that I used can

§This is an example that teaching and research were closely related activities for Schwinger. At the time of these lectures, he was working on refinements of the Thomas--Fermi method. ~ Section 7.10 in [2] comments on this lecture.

X Preface

be found there. Charlotte Brown, Curator of the VCLA Special Collections, has been very helpful in my search of the Schwinger Papers. I thank her sincerely.

I wish to thank Herbert Walther for the splendid hospitality extended to me over the years at the Max-Planck-Institut fUr Quantenoptik in Garching; the institute's infrastructure was of great help while I was working on this book. During the past year, the crucial stage of this undertaking, I was sup­ported by the Vniversitiit VIm; I thank Wolfgang Schleich for the generous invitation to join his Abteilung Quantenphysik temporarily.

I acknowledge with gratitude the support by the editorial staff of Springer­Verlag; in particular, the help of Wolf Beiglbock, Christian Caron, and Brigitte Reichel-Mayer was invaluable. And I thank Jens Schneider, who turned the handwritten notes into electronic files that I could then work on.

Ismaning, September 2000 BG Englert

References

1. Jagdish MEHRA and Kimball A. MILTON: Climbing the Mountain. The Scientific Biography of Julian Schwinger (Oxford University Press, Oxford and New York 2000)

2. Silvan S. SCHWEBER: QED and the Men Who Made It: Dyson, Feynman, Schwinger, and Tomonaga (Princeton University Press, Princeton 1994)

3. Julian SCHWINGER: Quantum Kinematics and Dynamics (W.A. Benjamin, New York 1970; reprinted by Addison~Wesley, Redwood City 1991)

4. Julian SCHWINGER: 'Hermann Weyl and Quantum Kinematics'. In: Exact Sci­ences and their Philosophical Foundations. Proceedings of the International Her­mann Weyl Congress, Kiel, Germany, 1985, edited by Wolfgang DEPPERT et al. (Verlag Peter Lang, Frankfurt/Main and New York 1988) pp. 107~129

5. Julian Schwinger Papers (Collection 371), Department of Special Collections, University Research Library, University of California, Los Angeles

Contents

Prologue...................................................... 1

Part A. Fall Quarter: Quantum Kinematics

1. Measurement Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. 29 1.1 Stern-Gerlach experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29 1.2 Measurement symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 1.3 State vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 1.4 Successive measurements. Probabilities. . . . . . . . . . . . . . . . . .. 38 1.5 Probability amplitudes. Interference ..................... 41 1.6 "Measurement disturbs the system" . . . . . . . . . . . . . . . . . . . . .. 46 1. 7 Observables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48 1.8 Algebra of Pauli's operators ............................ 50 1.9 Adjoint symbols, Hermitian symbols. . . . . . . . . . . . . . . . . . . .. 53 1.10 ~1atrix representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56 1.11 Traces............................................... 57 1.12 Unitary geometry ..................................... 59

1.12.1 Column and row vectors, wave functions. . . . . . . . . .. 59 1.12.2 Two arbitrary components of Pauli's vector operator. 63

1.13 Unitary operators..................................... 67 1.14 Unitary operator bases. Complementarity. . . . . . . . . . . . . . . .. 69 1.15 Quantum degrees of freedom. . . . . . . . . . . . .. . . . . . . . . . . . . . .. 76 1.16 The continuum limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82

1.16.1 Heisenberg's commutation relation. .. . . . . . . . . . . . . . .. 82 1.16.2 Schrodinger's differential-operator representation. . .. 86

Problems .................................................. 88

2. Continuous q, p Degree of Freedom ....................... 101 2.1 Wave functions ........................................ 101 2.2 Expectation values and their spreads ..................... 109 2.3 States of minimal uncertainty ........................... 111 2.4 States of stationary uncertainty ......................... 114 2.5 Hermite polynomials ................................... 118

XII Contents

2.6 Completeness of stationary-uncertainty states ............. 123 2.7 Eigenvectors of non-Hermitian operators ................. 125 2.8 Classical limit ......................................... 132 2.9 More about stationary-uncertainty states ................. 135 Problems .................................................. 136

3. Angular Momentum ...................................... 149 3.1 Infinitesimal unitary transformations ..................... 149 3.2 Infinitesimal rotations .................................. 150 3.3 Common eigenvectors of J2 and Jz ...................... 152 3.4 Decomposition into spins ............................... 155 3.5 Angular momentum of a composite system ............... 158 3.6 Finite rotations. Eulerian angles ......................... 162 3.7 Rotated angular-momentum eigenvectors ................. 168 Problems .................................................. 177

4. Galilean Invariance ....................................... 183 4.1 Generators of infinitesimal transformations ............... 183 4.2 Hamilton operator for a system of elementary particles ..... 190 Problems .................................................. 191

Part B. Winter Quarter: Quantum Dynamics

5. Quantum Action Principle . ............................... 195 5.1 Equations of motion ................................... 195 5.2 Conservation laws ..................................... 197 5.3 Sets of q, p pairs of variables . . . . . . . . . . . . . . . . . . . . . . . . . . .. 199 5.4 Wave functions for force-free motion ..................... 202 5.5 Quantum action principle .............................. 207 5.6 Principle of stationary action ........................... 210 5.7 Change of description .................................. 213 5.8 Permissible variations .................................. 214 Problems .................................................. 216

6. Elementary Applications . ................................. 223 6.1 Time transformation functions .......................... 223

6.1.1 Free particle .................................... 223 6.1.2 Constant force .................................. 224 6.1.3 Linear restoring force: Harmonic oscillator .......... 226

6.2 Short times ........................................... 227 6.3 Harmonic oscillator: Energy eigenvalues .................. 229 6.4 Free particle and constant force: State density ............. 231 6.5 Harmonic oscillator: Energy eigenstates .................. 234 6.6 Free particle and constant force: Energy eigenstates ........ 237

Contents XIII

6.7 Constant force: Asymptotic wave functions ............... 239 6.8 WKB approximation ................................... 243 6.9 Zeros and extrema of the Airy function ................... 248 6.10 Constant restoring force ................................ 252 6.11 Rayleigh-Ritz variational method ....................... 255 Problems .................................................. 257

7. Harmonic Oscillators ..................................... 269 7.1 Non-Hermitian operators ............................... 269 7.2 Driven oscillator ....................... , ............... 272

7.2.1 Time-independent drive .......... , ............... 274 7.2.2 Slowly varying drive ............................. 276 7.2.3 Temporary drive ................................ 278

7.3 Remarks on Laguerre polynomials ....................... 286 7.4 Two-dimensional oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.5 Three-dimensionaloscillator ............................ 295 Problems .................................................. 298

8. Hydrogenic Atoms ........................................ 303 8.1 Bound states .......................................... 303 8.2 Parameter dependence of energy eigenvalues .............. 307 8.3 Virial theorem ........................................ 309 8.4 Parabolic coordinates .................................. 313 8.5 Weak external electric field ............................. 316 8.6 Weak external magnetic field ............................ 319 8.7 Insertion: Charge in a homogeneous magnetic field ......... 324 8.8 Scattering states ...................................... 328 Problems .................................................. 333

Part C. Spring Quarter: Interacting Particles

9. Two-Particle Coulomb Problem .......................... 343 9.1 Internal and external motion ............................ 343 9.2 Rutherford scattering revisited .......................... 346 9.3 Additional short-range forces ............................ 352 9.4 Scattering of identical particles .......................... 355 9.5 Conserved axial vector ................................. 358 9.6 Weak external fields ................................... 365 Problems .................................................. 368

10. Identical Particles ........................................ 375 10.1 Modes. Creation and annihilation operators ............... 375 10.2 One-particle and two-particle operators .................. 381 10.3 ~Iulti-particle states ................................... 385

XIV Contents

10.4 Dynamical basics ...................................... 386 10.5 Example: General spin dynamics ........................ 387 10.6 General dynamics ..................................... 392 10.7 Operator fields ........................................ 395 10.8 Non-interacting particles ............................... 397 Problems .................................................. 403

11. Many-Electron Atoms . ................................... 405 11.1 Hartree-Fock method .................................. 405 11.2 Semiclassical treatment: Thomas-Fermi model ............ 410 11.3 Correction for strongly bound electrons .................. 420 11.4 Quantum corrections and exchange energy ................ 425 11.5 Energy oscillations ..................................... 428 Problems .................................................. 430

12. Electromagnetic Radiation . ............................... 437 12.1 Lagrangian, modes, equations of motion .................. 437 12.2 Effective action ....................................... 441 12.3 Consistency check ..................................... 444 12.4 Free-space photon mode functions ....................... 447 12.5 Physical mass ......................................... 449 12.6 Infrared photons ...................................... 452 12.7 Effective Hamiltonian .................................. 455 12.8 Energy shift .......................................... 459 12.9 Transition rates ....................................... 461 12.10 Thomson scattering .................................... 465 Problems .................................................. 467

Index ......................................................... 473