Documents on Modern Physics Edited by ELLIOTT W. MONTROLL, University of Rochester GEORGE H. VINEYA RD, Brookhaven National Laboratory MAURICE LEVY, Universite de Paris A. ABRAGAM L'Effet Mossbauer H. BACRY Lectures on Group Theory K. G. BUDDEN Lectures on Magnetoionic Theory J. w. CHAMBERLAIN Motion of Charged Particles in the Earth's Magnetic Field s. CHAPMAN Solar Plasma, Geomagnetism, and Aurora H.-Y. CHIU Neutrino Astrophysics A. H. COTTRELL Theory of Crystal Dislocations J. DANON Lectures on the Mossbauer Effect BRYCE s. DEWITT Dynamical Theory of Groups and Fields R. H. DICKE The Theoretical Significance of Experimental Relativity M. GOl..IRDIN Lagrangian Formalism and Symmetry Laws D. HESTENES Space-Time Algebra JOHN G. KIRKWOOD Selected Topics in Statistical Mechanics JOHN G. KIRKWOOD Macromolecules JOHN G. KIRKWOOD Theory of Liquids JOHN G. KIRKWOOD Theory of Solutions JOHN G. KIRKWOOD Proteins JOHN G. KIRKWOOD Quantum Statistics and Cooperative Phenomena JOHN G. KIRKWOOD Shock and Detonation Waves JOHN G. KIRKWOOD Dielectrics-Intermolecular Forces-Optical Rotation v. KOURGANOFF Introduction to the General Theory of Particle Transfer ROBERT LA TT ES Methods of Resolutions for Selected Boundary Problems in Mathematical Physics J. LEQUEUX Structure and Evolution of Galaxies F. E. LOW Symmetries and Elementary Particles P. H. E. MEDER Quantum Statistical Mechanics M. MOSHINSKY Group Theory and the Many-body Problem M. NIKOLIC Analysis of Scattering and Decay M. NIKOLIC Kinematics and Multiparticle Systems A. B. PIPPARD The Dynamics of Conduction Electrons H. REEVES Stellar Evolution and Nucleosynthesis L. SCHWARTZ Application of Distributions to the Theory of Elementary Particles in Quantum Mechanics J. SCHWINGER and D. s. SAXON Discontinuities in Waveguides M. TINKHAM Superconductivity Discontinuities in Waveguides Notes on lectures by Julian Schwinger JULIAN SCHWINGER Department ofP hysics University of Harvard Cambridge, Massachusetts DAY ID S. SAXON Department of Physics University of California Los Angeles, California WHOLESALE BOOK DEALERS Supplied by INTERNATIONAL SERVICE CO. 1909 BENSON AVENUE BROOKLYN, N.Y. 11214, U.S.A. Phone: CL 9..o403 Cable: INAJSERVE GORDON AND BREACH SCIENCE PUBLISHERS NEW YORK LONDON PARIS Copyright© 1968 by GORDON AND BREACH SCIENCE PUBLISHERS INC. 150 Fifth Avenue, New York, N.Y. 10011 Library of Congress Catalog Card Number: 68-28882 Editorial office for the United Kingdom: Gordon and Breach Science Publishers Ltd. 8 Bloomsbury Way London W.C.l Editorial office for France: Gordon & Breach 7-9 rue Emile Dubois Paris 14° Distributed in Canada by: The Ryerson Press 299 Queen Street West Toronto 2B, Ontario All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publishers. PRINTED TN GREAT BRITAIN BY ROBERT MACLEHOOE AND CO. LTD., THE UNIVERSITY PRESS, SCOTLAND Documents on Modem Physics Seventy years ago when the fraternity of physicists was smaller than the audience at a weekly physics colloquium in a major university, a J. Willard Gibbs could, after ten years of thought, summarize his ideas on a subject in a few monumental papers or in a classic treatise. His competition did not intimidate him into a muddled correspondence with his favorite editor nor did it occur to his colleagues that their own progress was retarded by his leisurely publication schedule. Today the dramatic phase of a new branch of physics spans less than a decade and subsides before the definitive treatise is published. Moreover modern physics is an extremely interconnected discipline and the busy practitioner of one of its branches must be kept aware of breakthroughs in other areas. An expository literature which is clear and timely is needed to relieve him of the burden of wading through tentative and hastily written papers scattered in many journals. To this end we have undertaken the editing of a new series, entitled Documents on Modern Physics, which will make available selected reviews, lecture notes, conference proceedings, and important collections of papers in branches of physics of special current interest. Complete coverage of a field will not be a primary aim. Rather, we will emphasize readability, speed of publication, and importance to students and research workers. The books will appear in low-cost paper-covered editions, as well as in cloth covers. The scope will be broad, the style informal. From time to time, the older branches of physics come alive again, and forgotten writings acquire relevance to recent developments. We expect to make a number of such works available by including them in this series along with new works. ELLIOTT M ONTROLL GEORGE H. VINEYARD MAURICE LEVY v Publishers' Foreword The history of these lecture notes is both curious and interesting. Originally the lectures were given by Professor Schwinger during the early years of World War II to a small group of colleagues at the MIT Radiation Laboratory. Notes were duplicated some time after each lecture and handed out to the participants. The projected series of lectures was, in fact, much more exten sive than published here-but the war ended before the series had finished, at which point all the people departed, leaving some lectures undelivered and some lecture notes unwritten. The knowledge of these sets of lecture notes was disseminated by the original group, so that, even by the end of 1945, the notes were duplicated again and sent out to an accumulated mailing list. A 'memorandum' and introduction were written by Professor Saxon at this stage to fill in some of the background for those who were now receiving the notes but had not attended the original lectures. Interest in these lectures has never waned; a steady stream of requests for a set of notes has reached Professors Schwinger and Saxon over the past twenty years. After the war, the interests of both Professor Saxon and Professor Schwinger veered away from the subject of the lectures (the latter's new direction led him to a Nobel Prize in 1965). And so we publish these lecture notes just as they stand. They are not particularly complete for the reasons already given (the end of the war cutting short the course; some delivered lectures not even being written). Nor have the inaccuracies been edited out in retrospect, this would ruin the flavour of the state of wartime study. The publishers do consider that this is a very important book-the survival of interest over 20 years proves this-and hence very worthy of publication in the present permanent form. GORDON AND BREACH vii Authors' Preface These notes are an interesting document of the fruitful interaction of different scientific disciplines. Attitudes and methods characteristic of quantum mechanics and nuclear physics were focused on the application of electro magnetic theory to practical microwave radar problems. And out of these rather special circumstances emerged a strategic lesson of wide impact. In seeking to apply a fundamental theory at the observational level, it is very advantageous to construct an intermediate theoretical structure, a pheno menological theory, which is capable on the one hand of organizing the body of experimental data into a relatively few numerical parameters, and on the other hand employs concepts that facilitate contact with the fundamental theory. The task of the latter thus becomes the explanation of the parameters of the phenomenological theory, rather than the direct confrontation with raw data. The effective range formulation of low energy nuclear physics was an early postwar application of this lesson. It was a substantial return on the initial investment, for now mathematical techniques developed for waveguide problems were applied to nuclear physics. There have been other applications of these methods, to neutron transport phenomena, to sound scattering problems. And it may be that there is still fertile ground for applying the basic lesson in high energy particle physics. J. SCHWINGER D. S. SAXON ix Contents Documents in Modern Physics v Publishers' Foreword vii Authors' Preface . ix Memorandum on Schwinger's Lectures xiii Introduction Preliminary Remarks: Equivalent Transmission Lines and Equi- valent Circuits 3 I Direct Application of the Variational Principle: Inductive Metallic Post in a Rectangular Waveguide 25 II Direct Application of the Variational Principle (continued): Dielectric Obstacles in Waveguides . 49 III Capacitive Discontinuities in Rectangular Waveguides: Metallic Obstacles of Zero Thickness 57 IV Capacitive Obstacles in Rectangular Waveguides: Metallic Obstacles of Zero Thickness (continued) 81 V Capacitive Discontinuities in Rectangular Waveguides: Change in Height ofa Guide 99 VI Application of Wiener-Hopf Techniques 125 xi