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The Feynman Lectures on Physics: Quantum Mechanics
The Feynman Lectures on Physics: Quantum Mechanics
Richard P. Feynman, Robert B. Leighton, Matthew Sands
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Edited for reading on tablets. Fully searchable text (edited in LaTeX). All figures and equations can be enlarged without losing any quality. Page numbering. Pdf table of contents. Clickable references to figures and equations.
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Vol. 3
Year:
2013
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New millenium edition for tablets
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Basic Books
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english
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688
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9780465072941
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probability^{376}
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equations^{233}
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frequency^{131}
atomic^{128}
proportional^{112}
beam^{112}
shown in fig^{109}
spin one^{107}
magnetic moment^{103}
scattering^{101}
photons^{100}
component^{95}
coefficients^{93}
protons^{92}
density^{88}
hydrogen atom^{86}
interference^{85}
conservation^{83}
detector^{81}
lattice^{81}
polarized^{78}
h11^{77}
stationary^{76}
ammonia^{76}
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2 comments
Wani Khawar
I can't describe how beautifully Richard Feynman describes the Quantum mechanical concepts in this book.
20 May 2018 (02:21)
Praveen aditya
My favourite book and awesome.
09 May 2019 (14:47)
You can write a book review and share your experiences. Other readers will always be interested in your opinion of the books you've read. Whether you've loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
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The Feynman LECTURES ON PHYSICS NEW MILLENNIUM EDITION FEYNMAN •LEIGHTON•SANDS VOLUME III Copyright © 1965, 2006, 2010 by California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer Published by Basic Books, A Member of the Perseus Books Group All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 250 West 57th Street, 15th Floor, New York, NY 10107. Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 8104145, ext. 5000, or email special.markets@perseusbooks.com. A CIP catalog record for the hardcover edition of this book is available from the Library of Congress. LCCN: 2010938208 Hardcover ISBN: 9780465024179 Ebook ISBN: 9780465072941 About Richard Feynman Born in 1918 in New York City, Richard P. Feynman received his Ph.D. from Princeton in 1942. Despite his youth, he played an important part in the Manhattan Project at Los Alamos during World War II. Subsequently, he taught at Cornell and at the California Institute of Technology. In 1965 he received the Nobel Prize in Physics, along with SinItiro Tomonaga and Julian Schwinger, for his work in quantum electrodynamics. Dr. Feynman won his Nobel Prize for successfully resolving problems with the theory of quantum electrodynamics. He also created a mathematical theory that accounts for the phenomenon of superfluidity in liquid helium. Thereafter, with Murray GellMann, he did fundamental work in the area of weak interactions such as beta decay. In later years Feynman played a key role in the development of quark theory by putting forwar; d his parton model of high energy proton collision processes. Beyond these achievements, Dr. Feynman introduced basic new computational techniques and notations into physics—above all, the ubiquitous Feynman diagrams that, perhaps more than any other formalism in recent scientific history, have changed the way in which basic physical processes are conceptualized and calculated. Feynman was a remarkably effective educator. Of all his numerous awards, he was especially proud of the Oersted Medal for Teaching, which he won in 1972. The Feynman Lectures on Physics, originally published in 1963, were described by a reviewer in Scientific American as “tough, but nourishing and full of flavor. After 25 years it is the guide for teachers and for the best of beginning students.” In order to increase the understanding of physics among the lay public, Dr. Feynman wrote The Character of Physical Law and QED: The Strange Theory of Light and Matter. He also authored a number of advanced publications that have become classic references and textbooks for researchers and students. Richard Feynman was a constructive public man. His work on the Challenger commission is well known, especially his famous demonstration of the susceptibility of the Orings to cold, an elegant experiment which required nothing more than a glass of ice water and a Cclamp. Less well known were Dr. Feynman’s efforts on the California State Curriculum Committee in the 1960s, where he protested the mediocrity of textbooks. iii A recital of Richard Feynman’s myriad scientific and educational accomplishments cannot adequately capture the essence of the man. As any reader of even his most technical publications knows, Feynman’s lively and multisided personality shines through all his work. Besides being a physicist, he was at various times a repairer of radios, a picker of locks, an artist, a dancer, a bongo player, and even a decipherer of Mayan Hieroglyphics. Perpetually curious about his world, he was an exemplary empiricist. Richard Feynman died on February 15, 1988, in Los Angeles. iv Preface to the New Millennium Edition Nearly fifty years have passed since Richard Feynman taught the introductory physics course at Caltech that gave rise to these three volumes, The Feynman Lectures on Physics. In those fifty years our understanding of the physical world has changed greatly, but The Feynman Lectures on Physics has endured. Feynman’s lectures are as powerful today as when first published, thanks to Feynman’s unique physics insights and pedagogy. They have been studied worldwide by novices and mature physicists alike; they have been translated into at least a dozen languages with more than 1.5 millions copies printed in the English language alone. Perhaps no other set of physics books has had such wide impact, for so long. This New Millennium Edition ushers in a new era for The Feynman Lectures on Physics (FLP): the twentyfirst century era of electronic publishing. FLP has been converted to eFLP, with the text and equations expressed in the LATEX electronic typesetting language, and all figures redone using modern drawing software. The consequences for the print version of this edition are not startling; it looks almost the same as the original red books that physics students have known and loved for decades. The main differences are an expanded and improved index, the correction of 885 errata found by readers over the five years since the first printing of the previous edition, and the ease of correcting errata that future readers may find. To this I shall return below. The eBook Version of this edition, and the Enhanced Electronic Version are electronic innovations. By contrast with most eBook versions of 20th century technical books, whose equations, figures and sometimes even text become pixellated when one tries to enlarge them, the LATEX manuscript of the New Millennium Edition makes it possible to create eBooks of the highest quality, in which all v features on the page (except photographs) can be enlarged without bound and retain their precise shapes and sharpness. And the Enhanced Electronic Version, with its audio and blackboard photos from Feynman’s original lectures, and its links to other resources, is an innovation that would have given Feynman great pleasure. Memories of Feynman's Lectures These three volumes are a selfcontained pedagogical treatise. They are also a historical record of Feynman’s 1961–64 undergraduate physics lectures, a course required of all Caltech freshmen and sophomores regardless of their majors. Readers may wonder, as I have, how Feynman’s lectures impacted the students who attended them. Feynman, in his Preface to these volumes, offered a somewhat negative view. “I don’t think I did very well by the students,” he wrote. Matthew Sands, in his memoir in Feynman’s Tips on Physics expressed a far more positive view. Out of curiosity, in spring 2005 I emailed or talked to a quasirandom set of 17 students (out of about 150) from Feynman’s 1961–63 class—some who had great difficulty with the class, and some who mastered it with ease; majors in biology, chemistry, engineering, geology, mathematics and astronomy, as well as in physics. The intervening years might have glazed their memories with a euphoric tint, but about 80 percent recall Feynman’s lectures as highlights of their college years. “It was like going to church.” The lectures were “a transformational experience,” “the experience of a lifetime, probably the most important thing I got from Caltech.” “I was a biology major but Feynman’s lectures stand out as a high point in my undergraduate experience . . . though I must admit I couldn’t do the homework at the time and I hardly turned any of it in.” “I was among the least promising of students in this course, and I never missed a lecture. . . . I remember and can still feel Feynman’s joy of discovery. . . . His lectures had an . . . emotional impact that was probably lost in the printed Lectures.” By contrast, several of the students have negative memories due largely to two issues: (i) “You couldn’t learn to work the homework problems by attending the lectures. Feynman was too slick—he knew tricks and what approximations could be made, and had intuition based on experience and genius that a beginning student does not possess.” Feynman and colleagues, aware of this flaw in the course, addressed it in part with materials that have been incorporated into Feynman’s Tips on Physics: three problemsolving lectures by Feynman, and vi a set of exercises and answers assembled by Robert B. Leighton and Rochus Vogt. (ii) “The insecurity of not knowing what was likely to be discussed in the next lecture, the lack of a text book or reference with any connection to the lecture material, and consequent inability for us to read ahead, were very frustrating. . . . I found the lectures exciting and understandable in the hall, but they were Sanskrit outside [when I tried to reconstruct the details].” This problem, of course, was solved by these three volumes, the printed version of The Feynman Lectures on Physics. They became the textbook from which Caltech students studied for many years thereafter, and they live on today as one of Feynman’s greatest legacies. A History of Errata The Feynman Lectures on Physics was produced very quickly by Feynman and his coauthors, Robert B. Leighton and Matthew Sands, working from and expanding on tape recordings and blackboard photos of Feynman’s course lectures† (both of which are incorporated into the Enhanced Electronic Version of this New Millennium Edition). Given the high speed at which Feynman, Leighton and Sands worked, it was inevitable that many errors crept into the first edition. Feynman accumulated long lists of claimed errata over the subsequent years—errata found by students and faculty at Caltech and by readers around the world. In the 1960’s and early 70’s, Feynman made time in his intense life to check most but not all of the claimed errata for Volumes I and II, and insert corrections into subsequent printings. But Feynman’s sense of duty never rose high enough above the excitement of discovering new things to make him deal with the errata in Volume III.‡ After his untimely death in 1988, lists of errata for all three volumes were deposited in the Caltech Archives, and there they lay forgotten. In 2002 Ralph Leighton (son of the late Robert Leighton and compatriot of Feynman) informed me of the old errata and a new long list compiled by Ralph’s † For descriptions of the genesis of Feynman’s lectures and of these volumes, see Feynman’s Preface and the Forewords to each of the three volumes, and also Matt Sands’ Memoir in Feynman’s Tips on Physics, and the Special Preface to the Commemorative Edition of FLP, written in 1989 by David Goodstein and Gerry Neugebauer, which also appears in the 2005 Definitive Edition. ‡ In 1975, he started checking errata for Volume III but got distracted by other things and never finished the task, so no corrections were made. vii friend Michael Gottlieb. Leighton proposed that Caltech produce a new edition of The Feynman Lectures with all errata corrected, and publish it alongside a new volume of auxiliary material, Feynman’s Tips on Physics, which he and Gottlieb were preparing. Feynman was my hero and a close personal friend. When I saw the lists of errata and the content of the proposed new volume, I quickly agreed to oversee this project on behalf of Caltech (Feynman’s longtime academic home, to which he, Leighton and Sands had entrusted all rights and responsibilities for The Feynman Lectures). After a year and a half of meticulous work by Gottlieb, and careful scrutiny by Dr. Michael Hartl (an outstanding Caltech postdoc who vetted all errata plus the new volume), the 2005 Definitive Edition of The Feynman Lectures on Physics was born, with about 200 errata corrected and accompanied by Feynman’s Tips on Physics by Feynman, Gottlieb and Leighton. I thought that edition was going to be “Definitive”. What I did not anticipate was the enthusiastic response of readers around the world to an appeal from Gottlieb to identify further errata, and submit them via a website that Gottlieb created and continues to maintain, The Feynman Lectures Website, www.feynmanlectures.info. In the five years since then, 965 new errata have been submitted and survived the meticulous scrutiny of Gottlieb, Hartl, and Nate Bode (an outstanding Caltech physics graduate student, who succeeded Hartl as Caltech’s vetter of errata). Of these, 965 vetted errata, 80 were corrected in the fourth printing of the Definitive Edition (August 2006) and the remaining 885 are corrected in the first printing of this New Millennium Edition (332 in volume I, 263 in volume II, and 200 in volume III). For details of the errata, see www.feynmanlectures.info. Clearly, making The Feynman Lectures on Physics errorfree has become a worldwide community enterprise. On behalf of Caltech I thank the 50 readers who have contributed since 2005 and the many more who may contribute over the coming years. The names of all contributors are posted at www.feynmanlectures. info/flp_errata.html. Almost all the errata have been of three types: (i) typographical errors in prose; (ii) typographical and mathematical errors in equations, tables and figures—sign errors, incorrect numbers (e.g., a 5 that should be a 4), and missing subscripts, summation signs, parentheses and terms in equations; (iii) incorrect cross references to chapters, tables and figures. These kinds of errors, though not terribly serious to a mature physicist, can be frustrating and confusing to Feynman’s primary audience: students. viii It is remarkable that among the 1165 errata corrected under my auspices, only several do I regard as true errors in physics. An example is Volume II, page 59, which now says “. . . no static distribution of charges inside a closed grounded conductor can produce any [electric] fields outside” (the word grounded was omitted in previous editions). This error was pointed out to Feynman by a number of readers, including Beulah Elizabeth Cox, a student at The College of William and Mary, who had relied on Feynman’s erroneous passage in an exam. To Ms. Cox, Feynman wrote in 1975,† “Your instructor was right not to give you any points, for your answer was wrong, as he demonstrated using Gauss’s law. You should, in science, believe logic and arguments, carefully drawn, and not authorities. You also read the book correctly and understood it. I made a mistake, so the book is wrong. I probably was thinking of a grounded conducting sphere, or else of the fact that moving the charges around in different places inside does not affect things on the outside. I am not sure how I did it, but I goofed. And you goofed, too, for believing me.” How this New Millennium Edition Came to Be Between November 2005 and July 2006, 340 errata were submitted to The Feynman Lectures Website www.feynmanlectures.info. Remarkably, the bulk of these came from one person: Dr. Rudolf Pfeiffer, then a physics postdoctoral fellow at the University of Vienna, Austria. The publisher, Addison Wesley, fixed 80 errata, but balked at fixing more because of cost: the books were being printed by a photooffset process, working from photographic images of the pages from the 1960s. Correcting an error involved retypesetting the entire page, and to ensure no new errors crept in, the page was retypeset twice by two different people, then compared and proofread by several other people—a very costly process indeed, when hundreds of errata are involved. Gottlieb, Pfeiffer and Ralph Leighton were very unhappy about this, so they formulated a plan aimed at facilitating the repair of all errata, and also aimed at producing eBook and enhanced electronic versions of The Feynman Lectures on Physics. They proposed their plan to me, as Caltech’s representative, in 2007. I was enthusiastic but cautious. After seeing further details, including a onechapter demonstration of the Enhanced Electronic Version, I recommended † Pages 288–289 of Perfectly Reasonable Deviations from the Beaten Track, The Letters of Richard P. Feynman, ed. Michelle Feynman (Basic Books, New York, 2005). ix that Caltech cooperate with Gottlieb, Pfeiffer and Leighton in the execution of their plan. The plan was approved by three successive chairs of Caltech’s Division of Physics, Mathematics and Astronomy—Tom Tombrello, Andrew Lange, and Tom Soifer—and the complex legal and contractual details were worked out by Caltech’s Intellectual Property Counsel, Adam Cochran. With the publication of this New Millennium Edition, the plan has been executed successfully, despite its complexity. Specifically: Pfeiffer and Gottlieb have converted into LATEX all three volumes of FLP (and also more than 1000 exercises from the Feynman course for incorporation into Feynman’s Tips on Physics). The FLP figures were redrawn in modern electronic form in India, under guidance of the FLP German translator, Henning Heinze, for use in the German edition. Gottlieb and Pfeiffer traded nonexclusive use of their LATEX equations in the German edition (published by Oldenbourg) for nonexclusive use of Heinze’s figures in this New Millennium English edition. Pfeiffer and Gottlieb have meticulously checked all the LATEX text and equations and all the redrawn figures, and made corrections as needed. Nate Bode and I, on behalf of Caltech, have done spot checks of text, equations, and figures; and remarkably, we have found no errors. Pfeiffer and Gottlieb are unbelievably meticulous and accurate. Gottlieb and Pfeiffer arranged for John Sullivan at the Huntington Library to digitize the photos of Feynman’s 1962–64 blackboards, and for George Blood Audio to digitize the lecture tapes—with financial support and encouragement from Caltech Professor Carver Mead, logistical support from Caltech Archivist Shelley Erwin, and legal support from Cochran. The legal issues were serious: In the 1960s, Caltech licensed to Addison Wesley rights to publish the print edition, and in the 1990s, rights to distribute the audio of Feynman’s lectures and a variant of an electronic edition. In the 2000s, through a sequence of acquisitions of those licenses, the print rights were transferred to the Pearson publishing group, while rights to the audio and the electronic version were transferred to the Perseus publishing group. Cochran, with the aid of Ike Williams, an attorney who specializes in publishing, succeeded in uniting all of these rights with Perseus (Basic Books), making possible this New Millennium Edition. Acknowledgments On behalf of Caltech, I thank the many people who have made this New Millennium Edition possible. Specifically, I thank the key people mentioned x above: Ralph Leighton, Michael Gottlieb, Tom Tombrello, Michael Hartl, Rudolf Pfeiffer, Henning Heinze, Adam Cochran, Carver Mead, Nate Bode, Shelley Erwin, Andrew Lange, Tom Soifer, Ike Williams, and the 50 people who submitted errata (listed at www.feynmanlectures.info). And I also thank Michelle Feynman (daughter of Richard Feynman) for her continuing support and advice, Alan Rice for behindthescenes assistance and advice at Caltech, Stephan Puchegger and Calvin Jackson for assistance and advice to Pfeiffer about conversion of FLP to LATEX, Michael Figl, Manfred Smolik, and Andreas Stangl for discussions about corrections of errata; and the Staff of Perseus/Basic Books, and (for previous editions) the staff of Addison Wesley. Kip S. Thorne The Feynman Professor of Theoretical Physics, Emeritus California Institute of Technology xi October 2010 LECTURES ON PHYSICS QUANTUM MECHANICS RICHARD P. FEYNMAN Richard Chace Tolman Professor of Theoretical Physics California Institute of Technology ROBERT B. LEIGHTON Professor of Physics California Institute of Technology MATTHEW SANDS Professor of Physics California Institute of Technology Copyright © 1965 CALIFORNIA INSTITUTE OF TECHNOLOGY ————————— Printed in the United States of America ALL RIGHTS RESERVED. THIS BOOK, OR PARTS THEREOF MAY NOT BE REPRODUCED IN ANY FORM WITHOUT WRITTEN PERMISSION OF THE COPYRIGHT HOLDER. Library of Congress Catalog Card No. 6320717 Third printing, July 1966 ISBN 0201021188P 0201021149H BBCCDDEEFFGGMU898 Feynman's Preface These are the lectures in physics that I gave last year and the year before to the freshman and sophomore classes at Caltech. The lectures are, of course, not verbatim—they have been edited, sometimes extensively and sometimes less so. The lectures form only part of the complete course. The whole group of 180 students gathered in a big lecture room twice a week to hear these lectures and then they broke up into small groups of 15 to 20 students in recitation sections under the guidance of a teaching assistant. In addition, there was a laboratory session once a week. The special problem we tried to get at with these lectures was to maintain the interest of the very enthusiastic and rather smart students coming out of the high schools and into Caltech. They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many would be very discouraged because there were really very few grand, new, modern ideas presented to them. They were made to study inclined planes, electrostatics, and so forth, and after two years it was quite stultifying. The problem was whether or not we could make a course which would save the more advanced and excited student by maintaining his enthusiasm. The lectures here are not in any way meant to be a survey course, but are very serious. I thought to address them to the most intelligent in the class and to make sure, if possible, that even the most intelligent student was unable to completely encompass everything that was in the lectures—by putting in suggestions of 3 applications of the ideas and concepts in various directions outside the main line of attack. For this reason, though, I tried very hard to make all the statements as accurate as possible, to point out in every case where the equations and ideas fitted into the body of physics, and how—when they learned more—things would be modified. I also felt that for such students it is important to indicate what it is that they should—if they are sufficiently clever—be able to understand by deduction from what has been said before, and what is being put in as something new. When new ideas came in, I would try either to deduce them if they were deducible, or to explain that it was a new idea which hadn’t any basis in terms of things they had already learned and which was not supposed to be provable—but was just added in. At the start of these lectures, I assumed that the students knew something when they came out of high school—such things as geometrical optics, simple chemistry ideas, and so on. I also didn’t see that there was any reason to make the lectures in a definite order, in the sense that I would not be allowed to mention something until I was ready to discuss it in detail. There was a great deal of mention of things to come, without complete discussions. These more complete discussions would come later when the preparation became more advanced. Examples are the discussions of inductance, and of energy levels, which are at first brought in in a very qualitative way and are later developed more completely. At the same time that I was aiming at the more active student, I also wanted to take care of the fellow for whom the extra fireworks and side applications are merely disquieting and who cannot be expected to learn most of the material in the lecture at all. For such students I wanted there to be at least a central core or backbone of material which he could get. Even if he didn’t understand everything in a lecture, I hoped he wouldn’t get nervous. I didn’t expect him to understand everything, but only the central and most direct features. It takes, of course, a certain intelligence on his part to see which are the central theorems and central ideas, and which are the more advanced side issues and applications which he may understand only in later years. In giving these lectures there was one serious difficulty: in the way the course was given, there wasn’t any feedback from the students to the lecturer to indicate how well the lectures were going over. This is indeed a very serious difficulty, and I don’t know how good the lectures really are. The whole thing was essentially an experiment. And if I did it again I wouldn’t do it the same way—I hope I don’t have to do it again! I think, though, that things worked out—so far as the physics is concerned—quite satisfactorily in the first year. 4 In the second year I was not so satisfied. In the first part of the course, dealing with electricity and magnetism, I couldn’t think of any really unique or different way of doing it—of any way that would be particularly more exciting than the usual way of presenting it. So I don’t think I did very much in the lectures on electricity and magnetism. At the end of the second year I had originally intended to go on, after the electricity and magnetism, by giving some more lectures on the properties of materials, but mainly to take up things like fundamental modes, solutions of the diffusion equation, vibrating systems, orthogonal functions, . . . developing the first stages of what are usually called “the mathematical methods of physics.” In retrospect, I think that if I were doing it again I would go back to that original idea. But since it was not planned that I would be giving these lectures again, it was suggested that it might be a good idea to try to give an introduction to the quantum mechanics—what you will find in Volume III. It is perfectly clear that students who will major in physics can wait until their third year for quantum mechanics. On the other hand, the argument was made that many of the students in our course study physics as a background for their primary interest in other fields. And the usual way of dealing with quantum mechanics makes that subject almost unavailable for the great majority of students because they have to take so long to learn it. Yet, in its real applications— especially in its more complex applications, such as in electrical engineering and chemistry—the full machinery of the differential equation approach is not actually used. So I tried to describe the principles of quantum mechanics in a way which wouldn’t require that one first know the mathematics of partial differential equations. Even for a physicist I think that is an interesting thing to try to do—to present quantum mechanics in this reverse fashion—for several reasons which may be apparent in the lectures themselves. However, I think that the experiment in the quantum mechanics part was not completely successful—in large part because I really did not have enough time at the end (I should, for instance, have had three or four more lectures in order to deal more completely with such matters as energy bands and the spatial dependence of amplitudes). Also, I had never presented the subject this way before, so the lack of feedback was particularly serious. I now believe the quantum mechanics should be given at a later time. Maybe I’ll have a chance to do it again someday. Then I’ll do it right. The reason there are no lectures on how to solve problems is because there were recitation sections. Although I did put in three lectures in the first year on how to solve problems, they are not included here. Also there was a lecture on inertial guidance which certainly belongs after the lecture on rotating systems, 5 but which was, unfortunately, omitted. The fifth and sixth lectures are actually due to Matthew Sands, as I was out of town. The question, of course, is how well this experiment has succeeded. My own point of view—which, however, does not seem to be shared by most of the people who worked with the students—is pessimistic. I don’t think I did very well by the students. When I look at the way the majority of the students handled the problems on the examinations, I think that the system is a failure. Of course, my friends point out to me that there were one or two dozen students who—very surprisingly—understood almost everything in all of the lectures, and who were quite active in working with the material and worrying about the many points in an excited and interested way. These people have now, I believe, a firstrate background in physics—and they are, after all, the ones I was trying to get at. But then, “The power of instruction is seldom of much efficacy except in those happy dispositions where it is almost superfluous.” (Gibbon) Still, I didn’t want to leave any student completely behind, as perhaps I did. I think one way we could help the students more would be by putting more hard work into developing a set of problems which would elucidate some of the ideas in the lectures. Problems give a good opportunity to fill out the material of the lectures and make more realistic, more complete, and more settled in the mind the ideas that have been exposed. I think, however, that there isn’t any solution to this problem of education other than to realize that the best teaching can be done only when there is a direct individual relationship between a student and a good teacher—a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned. But in our modern times we have so many students to teach that we have to try to find some substitute for the ideal. Perhaps my lectures can make some contribution. Perhaps in some small place where there are individual teachers and students, they may get some inspiration or some ideas from the lectures. Perhaps they will have fun thinking them through—or going on to develop some of the ideas further. Richard P. Feynman June, 1963 6 Foreword A great triumph of twentiethcentury physics, the theory of quantum mechanics, is now nearly 40 years old, yet we have generally been giving our students their introductory course in physics (for many students, their last) with hardly more than a casual allusion to this central part of our knowledge of the physical world. We should do better by them. These lectures are an attempt to present them with the basic and essential ideas of the quantum mechanics in a way that would, hopefully, be comprehensible. The approach you will find here is novel, particularly at the level of a sophomore course, and was considered very much an experiment. After seeing how easily some of the students take to it, however, I believe that the experiment was a success. There is, of course, room for improvement, and it will come with more experience in the classroom. What you will find here is a record of that first experiment. In the twoyear sequence of the Feynman Lectures on Physics which were given from September 1961 through May 1963 for the introductory physics course at Caltech, the concepts of quantum physics were brought in whenever they were necessary for an understanding of the phenomena being described. In addition, the last twelve lectures of the second year were given over to a more coherent introduction to some of the concepts of quantum mechanics. It became clear as the lectures drew to a close, however, that not enough time had been left for the quantum mechanics. As the material was prepared, it was continually discovered that other important and interesting topics could be treated with the elementary tools that had been developed. There was also a fear that the too brief treatment of the Schrödinger wave function which had been included in the twelfth lecture would not provide a sufficient bridge to the more conventional 7 treatments of many books the students might hope to read. It was therefore decided to extend the series with seven additional lectures; they were given to the sophomore class in May of 1964. These lectures rounded out and extended somewhat the material developed in the earlier lectures. In this volume we have put together the lectures from both years with some adjustment of the sequence. In addition, two lectures originally given to the freshman class as an introduction to quantum physics have been lifted bodily from Volume I (where they were Chapters 37 and 38) and placed as the first two chapters here—to make this volume a selfcontained unit, relatively independent of the first two. A few ideas about the quantization of angular momentum (including a discussion of the SternGerlach experiment) had been introduced in Chapters 34 and 35 of Volume II, and familiarity with them is assumed; for the convenience of those who will not have that volume at hand, those two chapters are reproduced here as an Appendix. This set of lectures tries to elucidate from the beginning those features of the quantum mechanics which are most basic and most general. The first lectures tackle head on the ideas of a probability amplitude, the interference of amplitudes, the abstract notion of a state, and the superposition and resolution of states— and the Dirac notation is used from the start. In each instance the ideas are introduced together with a detailed discussion of some specific examples—to try to make the physical ideas as real as possible. The time dependence of states including states of definite energy comes next, and the ideas are applied at once to the study of twostate systems. A detailed discussion of the ammonia maser provides the framework for the introduction to radiation absorption and induced transitions. The lectures then go on to consider more complex systems, leading to a discussion of the propagation of electrons in a crystal, and to a rather complete treatment of the quantum mechanics of angular momentum. Our introduction to quantum mechanics ends in Chapter 20 with a discussion of the Schrödinger wave function, its differential equation, and the solution for the hydrogen atom. The last chapter of this volume is not intended to be a part of the “course.” It is a “seminar” on superconductivity and was given in the spirit of some of the entertainment lectures of the first two volumes, with the intent of opening to the students a broader view of the relation of what they were learning to the general culture of physics. Feynman’s “epilogue” serves as the period to the threevolume series. As explained in the Foreword to Volume I, these lectures were but one aspect of a program for the development of a new introductory course carried out at the 8 California Institute of Technology under the supervision of the Physics Course Revision Committee (Robert Leighton, Victor Neher, and Matthew Sands). The program was made possible by a grant from the Ford Foundation. Many people helped with the technical details of the preparation of this volume: Marylou Clayton, Julie Curcio, James Hartle, Tom Harvey, Martin Israel, Patricia Preuss, Fanny Warren, and Barbara Zimmerman. Professors Gerry Neugebauer and Charles Wilts contributed greatly to the accuracy and clarity of the material by reviewing carefully much of the manuscript. But the story of quantum mechanics you will find here is Richard Feynman’s. Our labors will have been well spent if we have been able to bring to others even some of the intellectual excitement we experienced as we saw the ideas unfold in his reallife Lectures on Physics. Matthew Sands December, 1964 9 Contents Chapter 1. 11 12 13 14 15 16 17 18 Atomic mechanics . . . . . . . . . . . An experiment with bullets . . . . . . An experiment with waves . . . . . . . An experiment with electrons . . . . . The interference of electron waves . . Watching the electrons . . . . . . . . . First principles of quantum mechanics The uncertainty principle . . . . . . . Chapter 2. 21 22 23 24 25 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . 12 . 14 . 16 . 18 . 110 . 115 . 117 The Relation of Wave and Particle Viewpoints Probability wave amplitudes . . . . . . . . Measurement of position and momentum Crystal diffraction . . . . . . . . . . . . . The size of an atom . . . . . . . . . . . . Energy levels . . . . . . . . . . . . . . . . Philosophical implications . . . . . . . . . Chapter 3. 31 32 33 34 Quantum Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 . 23 . 28 . 210 . 213 . 215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . 38 . 312 . 316 Probability Amplitudes The laws for combining amplitudes . . The twoslit interference pattern . . . . Scattering from a crystal . . . . . . . . . Identical particles . . . . . . . . . . . . . 10 . . . . Chapter 4. 41 42 43 44 45 46 47 Bose particles and Fermi particles . States with two Bose particles . . . . States with n Bose particles . . . . Emission and absorption of photons The blackbody spectrum . . . . . . . Liquid helium . . . . . . . . . . . . . The exclusion principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Filtering atoms with a SternGerlach apparatus Experiments with filtered atoms . . . . . . . . SternGerlach filters in series . . . . . . . . . . Base states . . . . . . . . . . . . . . . . . . . . Interfering amplitudes . . . . . . . . . . . . . . The machinery of quantum mechanics . . . . . Transforming to a different base . . . . . . . . Other situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 . 59 . 511 . 513 . 516 . 521 . 524 . 527 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 . 75 . 79 . 716 . 718 Chapter 5. 51 52 53 54 55 56 57 58 Chapter 6. 61 62 63 64 65 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 46 410 413 415 422 423 Spin One Spin OneHalf Transforming amplitudes . . . . . . Transforming to a rotated coordinate Rotations about the zaxis . . . . . Rotations of 180◦ and 90◦ about y . Rotations about x . . . . . . . . . . Arbitrary rotations . . . . . . . . . . Chapter 7. 71 72 73 74 75 Identical Particles . . . . . system . . . . . . . . . . . . . . . . . . . . . . . . . . 61 64 610 615 620 622 The Dependence of Amplitudes on Time Atoms at rest; stationary states . . . . . . . Uniform motion . . . . . . . . . . . . . . . . Potential energy; energy conservation . . . Forces; the classical limit . . . . . . . . . . The “precession” of a spin onehalf particle 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 8. 81 82 83 84 85 86 Amplitudes and vectors . . . . Resolving state vectors . . . . . What are the base states of the How states change with time . The Hamiltonian matrix . . . . The ammonia molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 . 84 . 88 . 811 . 816 . 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 . 97 . 914 . 918 . 921 . 923 The hydrogen molecular ion . . . . . . . . . . Nuclear forces . . . . . . . . . . . . . . . . . . The hydrogen molecule . . . . . . . . . . . . The benzene molecule . . . . . . . . . . . . . Dyes . . . . . . . . . . . . . . . . . . . . . . . The Hamiltonian of a spin onehalf particle in The spinning electron in a magnetic field . . . . . . . a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 1010 1013 1017 1021 1022 1026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 119 1114 1115 1121 Chapter 9. 91 92 93 94 95 96 Chapter 11. 111 112 113 114 115 . . . . . . . . world? . . . . . . . . . . . . The The The The The . . . . . . . . . . . . . . . . . The Ammonia Maser The states of an ammonia molecule The molecule in a static electric field Transitions in a timedependent field Transitions at resonance . . . . . . . Transitions off resonance . . . . . . . The absorption of light . . . . . . . . Chapter 10. 101 102 103 104 105 106 107 The Hamiltonian Matrix . . . . . . . . . . . . . . . . . . . . . . Other TwoState Systems More TwoState Systems Pauli spin matrices . . . . . . . . . spin matrices as operators . . . . . solution of the twostate equations polarization states of the photon . neutral Kmeson . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 12. 121 122 123 124 125 126 Base states for a system with two spin onehalf particles The Hamiltonian for the ground state of hydrogen . . . The energy levels . . . . . . . . . . . . . . . . . . . . . . The Zeeman splitting . . . . . . . . . . . . . . . . . . . The states in a magnetic field . . . . . . . . . . . . . . . The projection matrix for spin one . . . . . . . . . . . . Chapter 13. 131 132 133 134 135 136 137 138 151 152 153 154 155 156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 124 1212 1215 1222 1226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 135 1310 1312 1314 1316 1320 1321 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 148 1412 1415 1419 1421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 157 159 1511 1518 1524 Semiconductors Electrons and holes in semiconductors . . Impure semiconductors . . . . . . . . . . The Hall effect . . . . . . . . . . . . . . . Semiconductor junctions . . . . . . . . . . Rectification at a semiconductor junction The transistor . . . . . . . . . . . . . . . . Chapter 15. . . . . . Propagation in a Crystal Lattice States for an electron in a onedimensional lattice States of definite energy . . . . . . . . . . . . . . Timedependent states . . . . . . . . . . . . . . . An electron in a threedimensional lattice . . . . Other states in a lattice . . . . . . . . . . . . . . Scattering from imperfections in the lattice . . . Trapping by a lattice imperfection . . . . . . . . Scattering amplitudes and bound states . . . . . Chapter 14. 141 142 143 144 145 146 The Hyperfine Splitting in Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . The Independent Particle Approximation Spin waves . . . . . . . . . . . . Two spin waves . . . . . . . . . . Independent particles . . . . . . The benzene molecule . . . . . . More organic chemistry . . . . . Other uses of the approximation . . . . . . . . . . . . . . . . . . 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 16. 161 162 163 164 165 166 Amplitudes on a line . . . . . The wave function . . . . . . States of definite momentum Normalization of the states in The Schrödinger equation . . Quantized energy levels . . . Chapter 17. 171 172 173 174 175 176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 167 1610 1614 1618 1623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 176 1713 1717 1721 1728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . photon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 186 189 1818 1823 1825 1839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 194 199 1917 1921 1925 Symmetry and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular Momentum Electric dipole radiation . . . . . . . . . Light scattering . . . . . . . . . . . . . . The annihilation of positronium . . . . . Rotation matrix for any spin . . . . . . Measuring a nuclear spin . . . . . . . . Composition of angular momentum . . Added Note 2: Conservation of parity in Chapter 19. 191 192 193 194 195 196 . . . x . . Symmetry . . . . . . . . . . . . . . Symmetry and conservation . . . . The conservation laws . . . . . . . Polarized light . . . . . . . . . . . The disintegration of the Λ0 . . . Summary of the rotation matrices Chapter 18. 181 182 183 184 185 186 188 The Dependence of Amplitudes on Position The Hydrogen Atom and The Periodic Table Schrödinger’s equation for the hydrogen atom Spherically symmetric solutions . . . . . . . . States with an angular dependence . . . . . The general solution for hydrogen . . . . . . The hydrogen wave functions . . . . . . . . . The periodic table . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 20. 201 202 203 204 205 206 207 Operations and operators . . . . Average energies . . . . . . . . . The average energy of an atom . The position operator . . . . . . The momentum operator . . . . Angular momentum . . . . . . . The change of averages with time Chapter 21. 211 212 213 214 215 216 217 218 219 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 205 209 2012 2015 2022 2025 The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity Schrödinger’s equation in a magnetic field The equation of continuity for probabilities Two kinds of momentum . . . . . . . . . . . The meaning of the wave function . . . . . Superconductivity . . . . . . . . . . . . . . The Meissner effect . . . . . . . . . . . . . . Flux quantization . . . . . . . . . . . . . . . The dynamics of superconductivity . . . . The Josephson junction . . . . . . . . . . . Feynman’s Epilogue Appendix Index Name Index List of Symbols 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 215 217 2110 2111 2114 2116 2122 2125 1 Quantum Behavior Note: This chapter is almost exactly the same as Chapter 37 of Volume I. 11 Atomic mechanics “Quantum mechanics” is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen. Newton thought that light was made up of particles, but then it was discovered that it behaves like a wave. Later, however (in the beginning of the twentieth century), it was found that light did indeed sometimes behave like a particle. Historically, the electron, for example, was thought to behave like a particle, and then it was found that in many respects it behaved like a wave. So it really behaves like neither. Now we have given up. We say: “It is like neither.” There is one lucky break, however—electrons behave just like light. The quantum behavior of atomic objects (electrons, protons, neutrons, photons, and so on) is the same for all, they are all “particle waves,” or whatever you want to call them. So what we learn about the properties of electrons (which we shall use for our examples) will apply also to all “particles,” including photons of light. The gradual accumulation of information about atomic and smallscale behavior during the first quarter of the 20th century, which gave some indications about how small things do behave, produced an increasing confusion which was finally resolved in 1926 and 1927 by Schrödinger, Heisenberg, and Born. They finally obtained a consistent description of the behavior of matter on a small scale. We take up the main features of that description in this chapter. 11 Because atomic behavior is so unlike ordinary experience, it is very difficult to get used to, and it appears peculiar and mysterious to everyone—both to the novice and to the experienced physicist. Even the experts do not understand it the way they would like to, and it is perfectly reasonable that they should not, because all of direct, human experience and of human intuition applies to large objects. We know how large objects will act, but things on a small scale just do not act that way. So we have to learn about them in a sort of abstract or imaginative fashion and not by connection with our direct experience. In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics. 12 An experiment with bullets To try to understand the quantum behavior of electrons, we shall compare and contrast their behavior, in a particular experimental setup, with the more familiar behavior of particles like bullets, and with the behavior of waves like water waves. We consider first the behavior of bullets in the experimental setup shown diagrammatically in Fig. 11. We have a machine gun that shoots a stream MOVABLE DETECTOR P1 1 GUN 2 WALL P12 x P2 BACKSTOP (a) P12 = P1 + P2 (b) Fig. 11. Interference experiment with bullets. 12 (c) of bullets. It is not a very good gun, in that it sprays the bullets (randomly) over a fairly large angular spread, as indicated in the figure. In front of the gun we have a wall (made of armor plate) that has in it two holes just about big enough to let a bullet through. Beyond the wall is a backstop (say a thick wall of wood) which will “absorb” the bullets when they hit it. In front of the wall we have an object which we shall call a “detector” of bullets. It might be a box containing sand. Any bullet that enters the detector will be stopped and accumulated. When we wish, we can empty the box and count the number of bullets that have been caught. The detector can be moved back and forth (in what we will call the xdirection). With this apparatus, we can find out experimentally the answer to the question: “What is the probability that a bullet which passes through the holes in the wall will arrive at the backstop at the distance x from the center?” First, you should realize that we should talk about probability, because we cannot say definitely where any particular bullet will go. A bullet which happens to hit one of the holes may bounce off the edges of the hole, and may end up anywhere at all. By “probability” we mean the chance that the bullet will arrive at the detector, which we can measure by counting the number which arrive at the detector in a certain time and then taking the ratio of this number to the total number that hit the backstop during that time. Or, if we assume that the gun always shoots at the same rate during the measurements, the probability we want is just proportional to the number that reach the detector in some standard time interval. For our present purposes we would like to imagine a somewhat idealized experiment in which the bullets are not real bullets, but are indestructible bullets— they cannot break in half. In our experiment we find that bullets always arrive in lumps, and when we find something in the detector, it is always one whole bullet. If the rate at which the machine gun fires is made very low, we find that at any given moment either nothing arrives, or one and only one—exactly one—bullet arrives at the backstop. Also, the size of the lump certainly does not depend on the rate of firing of the gun. We shall say: “Bullets always arrive in identical lumps.” What we measure with our detector is the probability of arrival of a lump. And we measure the probability as a function of x. The result of such measurements with this apparatus (we have not yet done the experiment, so we are really imagining the result) are plotted in the graph drawn in part (c) of Fig. 11. In the graph we plot the probability to the right and x vertically, so that the xscale fits the diagram of the apparatus. We call the probability P12 because the bullets may have come either through hole 1 or through hole 2. You will not be surprised that P12 is large near the middle of the graph but gets 13 small if x is very large. You may wonder, however, why P12 has its maximum value at x = 0. We can understand this fact if we do our experiment again after covering up hole 2, and once more while covering up hole 1. When hole 2 is covered, bullets can pass only through hole 1, and we get the curve marked P1 in part (b) of the figure. As you would expect, the maximum of P1 occurs at the value of x which is on a straight line with the gun and hole 1. When hole 1 is closed, we get the symmetric curve P2 drawn in the figure. P2 is the probability distribution for bullets that pass through hole 2. Comparing parts (b) and (c) of Fig. 11, we find the important result that P12 = P1 + P2 . (1.1) The probabilities just add together. The effect with both holes open is the sum of the effects with each hole open alone. We shall call this result an observation of “no interference,” for a reason that you will see later. So much for bullets. They come in lumps, and their probability of arrival shows no interference. 13 An experiment with waves Now we wish to consider an experiment with water waves. The apparatus is shown diagrammatically in Fig. 12. We have a shallow trough of water. A small object labeled the “wave source” is jiggled up and down by a motor and x x DETECTOR I1 I12 1 WAVE SOURCE I2 2 WALL ABSORBER I1 = h1 2 I2 = h2 2 (a) (b) I12 = h1 + h2 2 (c) Fig. 12. Interference experiment with water waves. 14 makes circular waves. To the right of the source we have again a wall with two holes, and beyond that is a second wall, which, to keep things simple, is an “absorber,” so that there is no reflection of the waves that arrive there. This can be done by building a gradual sand “beach.” In front of the beach we place a detector which can be moved back and forth in the xdirection, as before. The detector is now a device which measures the “intensity” of the wave motion. You can imagine a gadget which measures the height of the wave motion, but whose scale is calibrated in proportion to the square of the actual height, so that the reading is proportional to the intensity of the wave. Our detector reads, then, in proportion to the energy being carried by the wave—or rather, the rate at which energy is carried to the detector. With our wave apparatus, the first thing to notice is that the intensity can have any size. If the source just moves a very small amount, then there is just a little bit of wave motion at the detector. When there is more motion at the source, there is more intensity at the detector. The intensity of the wave can have any value at all. We would not say that there was any “lumpiness” in the wave intensity. Now let us measure the wave intensity for various values of x (keeping the wave source operating always in the same way). We get the interestinglooking curve marked I12 in part (c) of the figure. We have already worked out how such patterns can come about when we studied the interference of electric waves in Volume I. In this case we would observe that the original wave is diffracted at the holes, and new circular waves spread out from each hole. If we cover one hole at a time and measure the intensity distribution at the absorber we find the rather simple intensity curves shown in part (b) of the figure. I1 is the intensity of the wave from hole 1 (which we find by measuring when hole 2 is blocked off) and I2 is the intensity of the wave from hole 2 (seen when hole 1 is blocked). The intensity I12 observed when both holes are open is certainly not the sum of I1 and I2 . We say that there is “interference” of the two waves. At some places (where the curve I12 has its maxima) the waves are “in phase” and the wave peaks add together to give a large amplitude and, therefore, a large intensity. We say that the two waves are “interfering constructively” at such places. There will be such constructive interference wherever the distance from the detector to one hole is a whole number of wavelengths larger (or shorter) than the distance from the detector to the other hole. At those places where the two waves arrive at the detector with a phase difference of π (where they are “out of phase”) the resulting wave motion at 15 the detector will be the difference of the two amplitudes. The waves “interfere destructively,” and we get a low value for the wave intensity. We expect such low values wherever the distance between hole 1 and the detector is different from the distance between hole 2 and the detector by an odd number of halfwavelengths. The low values of I12 in Fig. 12 correspond to the places where the two waves interfere destructively. You will remember that the quantitative relationship between I1 , I2 , and I12 can be expressed in the following way: The instantaneous height of the water wave at the detector for the wave from hole 1 can be written as (the real part of) h1 eiωt , where the “amplitude” h1 is, in general, a complex number. The intensity is proportional to the mean squared height or, when we use the complex numbers, to the absolute value squared h1 2 . Similarly, for hole 2 the height is h2 eiωt and the intensity is proportional to h2 2 . When both holes are open, the wave heights add to give the height (h1 + h2 )eiωt and the intensity h1 + h2 2 . Omitting the constant of proportionality for our present purposes, the proper relations for interfering waves are I1 = h1 2 , I2 = h2 2 , I12 = h1 + h2 2 . (1.2) You will notice that the result is quite different from that obtained with bullets (Eq. 1.1). If we expand h1 + h2 2 we see that h1 + h2 2 = h1 2 + h2 2 + 2h1 h2  cos δ, (1.3) where δ is the phase difference between h1 and h2 . In terms of the intensities, we could write p (1.4) I12 = I1 + I2 + 2 I1 I2 cos δ. The last term in (1.4) is the “interference term.” So much for water waves. The intensity can have any value, and it shows interference. 14 An experiment with electrons Now we imagine a similar experiment with electrons. It is shown diagrammatically in Fig. 13. We make an electron gun which consists of a tungsten wire heated by an electric current and surrounded by a metal box with a hole in it. If the wire is at a negative voltage with respect to the box, electrons emitted by the wire will be accelerated toward the walls and some will pass through the hole. 16 x DETECTOR x P1 P12 1 ELECTRON GUN P2 2 WALL BACKSTOP P1 = φ1 2 P2 = φ2 2 (a) P12 = φ1 + φ2 2 (b) (c) Fig. 13. Interference experiment with electrons. All the electrons which come out of the gun will have (nearly) the same energy. In front of the gun is again a wall (just a thin metal plate) with two holes in it. Beyond the wall is another plate which will serve as a “backstop.” In front of the backstop we place a movable detector. The detector might be a geiger counter or, perhaps better, an electron multiplier, which is connected to a loudspeaker. We should say right away that you should not try to set up this experiment (as you could have done with the two we have already described). This experiment has never been done in just this way. The trouble is that the apparatus would have to be made on an impossibly small scale to show the effects we are interested in. We are doing a “thought experiment,” which we have chosen because it is easy to think about. We know the results that would be obtained because there are many experiments that have been done, in which the scale and the proportions have been chosen to show the effects we shall describe. The first thing we notice with our electron experiment is that we hear sharp “clicks” from the detector (that is, from the loudspeaker). And all “clicks” are the same. There are no “halfclicks.” We would also notice that the “clicks” come very erratically. Something like: click . . . . . clickclick . . . click . . . . . . . . click . . . . clickclick . . . . . . click . . . , etc., just as you have, no doubt, heard a geiger counter operating. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the two numbers are very 17 nearly the same. So we can speak of the average rate at which the clicks are heard (soandsomany clicks per minute on the average). As we move the detector around, the rate at which the clicks appear is faster or slower, but the size (loudness) of each click is always the same. If we lower the temperature of the wire in the gun, the rate of clicking slows down, but still each click sounds the same. We would notice also that if we put two separate detectors at the backstop, one or the other would click, but never both at once. (Except that once in a while, if there were two clicks very close together in time, our ear might not sense the separation.) We conclude, therefore, that whatever arrives at the backstop arrives in “lumps.” All the “lumps” are the same size: only whole “lumps” arrive, and they arrive one at a time at the backstop. We shall say: “Electrons always arrive in identical lumps.” Just as for our experiment with bullets, we can now proceed to find experimentally the answer to the question: “What is the relative probability that an electron ‘lump’ will arrive at the backstop at various distances x from the center?” As before, we obtain the relative probability by observing the rate of clicks, holding the operation of the gun constant. The probability that lumps will arrive at a particular x is proportional to the average rate of clicks at that x. The result of our experiment is the interesting curve marked P12 in part (c) of Fig. 13. Yes! That is the way electrons go. 15 The interference of electron waves Now let us try to analyze the curve of Fig. 13 to see whether we can understand the behavior of the electrons. The first thing we would say is that since they come in lumps, each lump, which we may as well call an electron, has come either through hole 1 or through hole 2. Let us write this in the form of a “Proposition”: Proposition A: Each electron either goes through hole 1 or it goes through hole 2. Assuming Proposition A, all electrons that arrive at the backstop can be divided into two classes: (1) those that come through hole 1, and (2) those that come through hole 2. So our observed curve must be the sum of the effects of the electrons which come through hole 1 and the electrons which come through hole 2. Let us check this idea by experiment. First, we will make a measurement for those electrons that come through hole 1. We block off hole 2 and make our counts of the clicks from the detector. From the clicking rate, we get P1 . 18 The result of the measurement is shown by the curve marked P1 in part (b) of Fig. 13. The result seems quite reasonable. In a similar way, we measure P2 , the probability distribution for the electrons that come through hole 2. The result of this measurement is also drawn in the figure. The result P12 obtained with both holes open is clearly not the sum of P1 and P2 , the probabilities for each hole alone. In analogy with our waterwave experiment, we say: “There is interference.” P12 6= P1 + P2 . For electrons: (1.5) How can such an interference come about? Perhaps we should say: “Well, that means, presumably, that it is not true that the lumps go either through hole 1 or hole 2, because if they did, the probabilities should add. Perhaps they go in a more complicated way. They split in half and . . . ” But no! They cannot, they always arrive in lumps . . . “Well, perhaps some of them go through 1, and then they go around through 2, and then around a few more times, or by some other complicated path . . . then by closing hole 2, we changed the chance that an electron that started out through hole 1 would finally get to the backstop . . . ” But notice! There are some points at which very few electrons arrive when both holes are open, but which receive many electrons if we close one hole, so closing one hole increased the number from the other. Notice, however, that at the center of the pattern, P12 is more than twice as large as P1 + P2 . It is as though closing one hole decreased the number of electrons which come through the other hole. It seems hard to explain both effects by proposing that the electrons travel in complicated paths. It is all quite mysterious. And the more you look at it the more mysterious it seems. Many ideas have been concocted to try to explain the curve for P12 in terms of individual electrons going around in complicated ways through the holes. None of them has succeeded. None of them can get the right curve for P12 in terms of P1 and P2 . Yet, surprisingly enough, the mathematics for relating P1 and P2 to P12 is extremely simple. For P12 is just like the curve I12 of Fig. 12, and that was simple. What is going on at the backstop can be described by two complex numbers that we can call φ1 and φ2 (they are functions of x, of course). The absolute square of φ1 gives the effect with only hole 1 open. That is, P1 = φ1 2 . The effect with only hole 2 open is given by φ2 in the same way. That is, P2 = φ2 2 . And the combined effect of the two holes is just P12 = φ1 + φ2 2 . The mathematics is the 19 same as that we had for the water waves! (It is hard to see how one could get such a simple result from a complicated game of electrons going back and forth through the plate on some strange trajectory.) We conclude the following: The electrons arrive in lumps, like particles, and the probability of arrival of these lumps is distributed like the distribution of intensity of a wave. It is in this sense that an electron behaves “sometimes like a particle and sometimes like a wave.” Incidentally, when we were dealing with classical waves we defined the intensity as the mean over time of the square of the wave amplitude, and we used complex numbers as a mathematical trick to simplify the analysis. But in quantum mechanics it turns out that the amplitudes must be represented by complex numbers. The real parts alone will not do. That is a technical point, for the moment, because the formulas look just the same. Since the probability of arrival through both holes is given so simply, although it is not equal to (P1 + P2 ), that is really all there is to say. But there are a large number of subtleties involved in the fact that nature does work this way. We would like to illustrate some of these subtleties for you now. First, since the number that arrives at a particular point is not equal to the number that arrives through 1 plus the number that arrives through 2, as we would have concluded from Proposition A, undoubtedly we should conclude that Proposition A is false. It is not true that the electrons go either through hole 1 or hole 2. But that conclusion can be tested by another experiment. 16 Watching the electrons We shall now try the following experiment. To our electron apparatus we add a very strong light source, placed behind the wall and between the two holes, as shown in Fig. 14. We know that electric charges scatter light. So when an electron passes, however it does pass, on its way to the detector, it will scatter some light to our eye, and we can see where the electron goes. If, for instance, an electron were to take the path via hole 2 that is sketched in Fig. 14, we should see a flash of light coming from the vicinity of the place marked A in the figure. If an electron passes through hole 1, we would expect to see a flash from the vicinity of the upper hole. If it should happen that we get light from both places at the same time, because the electron divides in half . . . Let us just do the experiment! Here is what we see: every time that we hear a “click” from our electron detector (at the backstop), we also see a flash of light either near hole 1 or near 110 x x P10 0 P12 1 LIGHT SOURCE ELECTRON GUN 2 A P20 0 = P0 + P0 P12 1 2 (a) (b) (c) Fig. 14. A different electron experiment. hole 2, but never both at once! And we observe the same result no matter where we put the detector. From this observation we conclude that when we look at the electrons we find that the electrons go either through one hole or the other. Experimentally, Proposition A is necessarily true. What, then, is wrong with our argument against Proposition A? Why isn’t P12 just equal to P1 + P2 ? Back to experiment! Let us keep track of the electrons and find out what they are doing. For each position (xlocation) of the detector we will count the electrons that arrive and also keep track of which hole they went through, by watching for the flashes. We can keep track of things this way: whenever we hear a “click” we will put a count in Column 1 if we see the flash near hole 1, and if we see the flash near hole 2, we will record a count in Column 2. Every electron which arrives is recorded in one of two classes: those which come through 1 and those which come through 2. From the number recorded in Column 1 we get the probability P10 that an electron will arrive at the detector via hole 1; and from the number recorded in Column 2 we get P20 , the probability that an electron will arrive at the detector via hole 2. If we now repeat such a measurement for many values of x, we get the curves for P10 and P20 shown in part (b) of Fig. 14. Well, that is not too surprising! We get for P10 something quite similar to what we got before for P1 by blocking off hole 2; and P20 is similar to what we got by blocking hole 1. So there is not any complicated business like going through both holes. When we watch them, the electrons come through just as we would 111 expect them to come through. Whether the holes are closed or open, those which we see come through hole 1 are distributed in the same way whether hole 2 is open or closed. But wait! What do we have now for the total probability, the probability that an electron will arrive at the detector by any route? We already have that information. We just pretend that we never looked at the light flashes, and we lump together the detector clicks which we have separated into the two columns. We must just add the numbers. For the probability that an electron will arrive 0 at the backstop by passing through either hole, we do find P12 = P10 + P20 . That is, although we succeeded in watching which hole our electrons come through, 0 we no longer get the old interference curve P12 , but a new one, P12 , showing no interference! If we turn out the light P12 is restored. We must conclude that when we look at the electrons the distribution of them on the screen is different than when we do not look. Perhaps it is turning on our light source that disturbs things? It must be that the electrons are very delicate, and the light, when it scatters off the electrons, gives them a jolt that changes their motion. We know that the electric field of the light acting on a charge will exert a force on it. So perhaps we should expect the motion to be changed. Anyway, the light exerts a big influence on the electrons. By trying to “watch” the electrons we have changed their motions. That is, the jolt given to the electron when the photon is scattered by it is such as to change the electron’s motion enough so that if it might have gone to where P12 was at a maximum it will instead land where P12 was a minimum; that is why we no longer see the wavy interference effects. You may be thinking: “Don’t use such a bright source! Turn the brightness down! The light waves will then be weaker and will not disturb the electrons so much. Surely, by making the light dimmer and dimmer, eventually the wave will be weak enough that it will have a negligible effect.” O.K. Let’s try it. The first thing we observe is that the flashes of light scattered from the electrons as they pass by does not get weaker. It is always the samesized flash. The only thing that happens as the light is made dimmer is that sometimes we hear a “click” from the detector but see no flash at all. The electron has gone by without being “seen.” What we are observing is that light also acts like electrons, we knew that it was “wavy,” but now we find that it is also “lumpy.” It always arrives—or is scattered—in lumps that we call “photons.” As we turn down the intensity of the light source we do not change the size of the photons, only the rate at which they are emitted. That explains why, when our source is dim, some electrons get 112 by without being seen. There did not happen to be a photon around at the time the electron went through. This is all a little discouraging. If it is true that whenever we “see” the electron we see the samesized flash, then those electrons we see are always the disturbed ones. Let us try the experiment with a dim light anyway. Now whenever we hear a click in the detector we will keep a count in three columns: in Column (1) those electrons seen by hole 1, in Column (2) those electrons seen by hole 2, and in Column (3) those electrons not seen at all. When we work up our data (computing the probabilities) we find these results: Those “seen by hole 1” have a distribution like P10 ; those “seen by hole 2” have a distribution like P20 (so that 0 those “seen by either hole 1 or 2” have a distribution like P12 ); and those “not seen at all” have a “wavy” distribution just like P12 of Fig. 13! If the electrons are not seen, we have interference! That is understandable. When we do not see the electron, no photon disturbs it, and when we do see it, a photon has disturbed it. There is always the same amount of disturbance because the light photons all produce the samesized effects and the effect of the photons being scattered is enough to smear out any interference effect. Is there not some way we can see the electrons without disturbing them? We learned in an earlier chapter that the momentum carried by a “photon” is inversely proportional to its wavelength (p = h/λ). Certainly the jolt given to the electron when the photon is scattered toward our eye depends on the momentum that photon carries. Aha! If we want to disturb the electrons only slightly we should not have lowered the intensity of the light, we should have lowered its frequency (the same as increasing its wavelength). Let us use light of a redder color. We could even use infrared light, or radiowaves (like radar), and “see” where the electron went with the help of some equipment that can “see” light of these longer wavelengths. If we use “gentler” light perhaps we can avoid disturbing the electrons so much. Let us try the experiment with longer waves. We shall keep repeating our experiment, each time with light of a longer wavelength. At first, nothing seems to change. The results are the same. Then a terrible thing happens. You remember that when we discussed the microscope we pointed out that, due to the wave nature of the light, there is a limitation on how close two spots can be and still be seen as two separate spots. This distance is of the order of the wavelength of light. So now, when we make the wavelength longer than the distance between our holes, we see a big fuzzy flash when the light is scattered by the electrons. 113 We can no longer tell which hole the electron went through! We just know it went somewhere! And it is just with light of this color that we find that the jolts given 0 to the electron are small enough so that P12 begins to look like P12 —that we begin to get some interference effect. And it is only for wavelengths much longer than the separation of the two holes (when we have no chance at all of telling where the electron went) that the disturbance due to the light gets sufficiently small that we again get the curve P12 shown in Fig. 13. In our experiment we find that it is impossible to arrange the light in such a way that one can tell which hole the electron went through, and at the same time not disturb the pattern. It was suggested by Heisenberg that the then new laws of nature could only be consistent if there were some basic limitation on our experimental capabilities not previously recognized. He proposed, as a general principle, his uncertainty principle, which we can state in terms of our experiment as follows: “It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern.” If an apparatus is capable of determining which hole the electron goes through, it cannot be so delicate that it does not disturb the pattern in an essential way. No one has ever found (or even thought of) a way around the uncertainty principle. So we must assume that it describes a basic characteristic of nature. The complete theory of quantum mechanics which we now use to describe atoms and, in fact, all matter, depends on the correctness of the uncertainty principle. Since quantum mechanics is such a successful theory, our belief in the uncertainty principle is reinforced. But if a way to “beat” the uncertainty principle were ever discovered, quantum mechanics would give inconsistent results and would have to be discarded as a valid theory of nature. “Well,” you say, “what about Proposition A? Is it true, or is it not true, that the electron either goes through hole 1 or it goes through hole 2?” The only answer that can be given is that we have found from experiment that there is a certain special way that we have to think in order that we do not get into inconsistencies. What we must say (to avoid making wrong predictions) is the following. If one looks at the holes or, more accurately, if one has a piece of apparatus which is capable of determining whether the electrons go through hole 1 or hole 2, then one can say that it goes either through hole 1 or hole 2. But, when one does not try to tell which way the electron goes, when there is nothing in the experiment to disturb the electrons, then one may not say that an electron goes either through hole 1 or hole 2. If one does say that, and starts to make any 114 deductions from the statement, he will make errors in the analysis. This is the logical tightrope on which we must walk if we wish to describe nature successfully. If the motion of all matter—as well as electrons—must be described in terms of waves, what about the bullets in our first experiment? Why didn’t we see an interference pattern there? It turns out that for the bullets the wavelengths were so tiny that the interference patterns became very fine. So fine, in fact, that with any detector of finite size one could not distinguish the separate maxima and minima. What we saw was only a kind of average, which is the classical curve. In Fig. 15 we have tried to indicate schematically what happens with largescale objects. Part (a) of the figure shows the probability distribution one might predict for bullets, using quantum mechanics. The rapid wiggles are supposed to represent the interference pattern one gets for waves of very short wavelength. Any physical detector, however, straddles several wiggles of the probability curve, so that the measurements show the smooth curve drawn in part (b) of the figure. x P12 (smoothed) P12 (a) (b) Fig. 15. Interference pattern with bullets: (a) actual (schematic), (b) observed. 17 First principles of quantum mechanics We will now write a summary of the main conclusions of our experiments. We will, however, put the results in a form which makes them true for a general class of such experiments. We can write our summary more simply if we first define 115 an “ideal experiment” as one in which there are no uncertain external influences, i.e., no jiggling or other things going on that we cannot take into account. We would be quite precise if we said: “An ideal experiment is one in which all of the initial and final conditions of the experiment are completely specified.” What we will call “an event” is, in general, just a specific set of initial and final conditions. (For example: “an electron leaves the gun, arrives at the detector, and nothing else happens.”) Now for our summary. Summary (1) The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number φ which is called the probability amplitude: P = probability, φ = probability amplitude, (1.6) P = φ . 2 (2) When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is interference: φ = φ1 + φ2 , P = φ1 + φ2 2 . (1.7) (3) If an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative. The interference is lost: P = P1 + P2 . (1.8) One might still like to ask: “How does it work? What is the machinery behind the law?” No one has found any machinery behind the law. No one can “explain” any more than we have just “explained.” No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced. We would like to emphasize a very important difference between classical and quantum mechanics. We have been talking about the probability that an electron 116 will arrive in a given circumstance. We have implied that in our experimental arrangement (or even in the best possible one) it would be impossible to predict exactly what would happen. We can only predict the odds! This would mean, if it were true, that physics has given up on the problem of trying to predict exactly what will happen in a definite circumstance. Yes! physics has given up. We do not know how to predict what would happen in a given circumstance, and we believe now that it is impossible—that the only thing that can be predicted is the probability of different events. It must be recognized that this is a retrenchment in our earlier ideal of understanding nature. It may be a backward step, but no one has seen a way to avoid it. We make now a few remarks on a suggestion that has sometimes been made to try to avoid the description we have given: “Perhaps the electron has some kind of internal works—some inner variables—that we do not yet know about. Perhaps that is why we cannot predict what will happen. If we could look more closely at the electron, we could be able to tell where it would end up.” So far as we know, that is impossible. We would still be in difficulty. Suppose we were to assume that inside the electron there is some kind of machinery that determines where it is going to end up. That machine must also determine which hole it is going to go through on its way. But we must not forget that what is inside the electron should not be dependent on what we do, and in particular upon whether we open or close one of the holes. So if an electron, before it starts, has already made up its mind (a) which hole it is going to use, and (b) where it is going to land, we should find P1 for those electrons that have chosen hole 1, P2 for those that have chosen hole 2, and necessarily the sum P1 + P2 for those that arrive through the two holes. There seems to be no way around this. But we have verified experimentally that that is not the case. And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say “at the present time,” but we suspect very strongly that it is something that will be with us forever—that it is impossible to beat that puzzle—that this is the way nature really is. 18 The uncertainty principle This is the way Heisenberg stated the uncertainty principle originally: If you make the measurement on any object, and you can determine the xcomponent of its momentum with an uncertainty ∆p, you cannot, at the same time, know its xposition more accurately than ∆x ≥ ~/2∆p, where ~ is a definite fixed number 117 given by nature. It is called the “reduced Planck constant,” and is approximately 1.05 × 10−34 jouleseconds. The uncertainties in the position and momentum of a particle at any instant must have their product greater than half the reduced Planck constant. This is a special case of the uncertainty principle that was stated above more generally. The more general statement was that one cannot design equipment in any way to determine which of two alternatives is taken, without, at the same time, destroying the pattern of interference. ROLLERS pa pb 1 ELECTRON GUN MOTION FREE ∆px DETECTOR 2 pa pb ∆px ROLLERS WALL BACKSTOP Fig. 16. An experiment in which the recoil of the wall is measured. Let us show for one particular case that the kind of relation given by Heisenberg must be true in order to keep from getting into trouble. We imagine a modification of the experiment of Fig. 13, in which the wall with the holes consists of a plate mounted on rollers so that it can move freely up and down (in the xdirection), as shown in Fig. 16. By watching the motion of the plate carefully we can try to tell which hole an electron goes through. Imagine what happens when the detector is placed at x = 0. We would expect that an electron which passes through hole 1 must be deflected downward by the plate to reach the detector. Since the vertical component of the electron momentum is changed, the plate must recoil with an equal momentum in the opposite direction. The plate will get an upward kick. If the electron goes through the lower hole, the plate should feel a downward kick. It is clear that for every position of the detector, the momentum received by the plate will have a different value for a traversal via hole 1 than for a traversal via hole 2. So! Without disturbing the electrons at all, but just by watching the plate, we can tell which path the electron used. 118 Now in order to do this it is necessary to know what the momentum of the screen is, before the electron goes through. So when we measure the momentum after the electron goes by, we can figure out how much the plate’s momentum has changed. But remember, according to the uncertainty principle we cannot at the same time know the position of the plate with an arbitrary accuracy. But if we do not know exactly where the plate is, we cannot say precisely where the two holes are. They will be in a different place for every electron that goes through. This means that the center of our interference pattern will have a different location for each electron. The wiggles of the interference pattern will be smeared out. We shall show quantitatively in the next chapter that if we determine the momentum of the plate sufficiently accurately to determine from the recoil measurement which hole was used, then the uncertainty in the xposition of the plate will, according to the uncertainty principle, be enough to shift the pattern observed at the detector up and down in the xdirection about the distance from a maximum to its nearest minimum. Such a random shift is just enough to smear out the pattern so that no interference is observed. The uncertainty principle “protects” quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, the quantum mechanics would collapse. So he proposed that it must be impossible. Then people sat down and tried to figure out ways of doing it, and nobody could figure out a way to measure the position and the momentum of anything—a screen, an electron, a billiard ball, anything—with any greater accuracy. Quantum mechanics maintains its perilous but still correct existence. 119 2 The Relation of Wave and Particle Viewpoints Note: This chapter is almost exactly the same as Chapter 38 of Volume I. 21 Probability wave amplitudes In this chapter we shall discuss the relationship of the wave and particle viewpoints. We already know, from the last chapter, that neither the wave viewpoint nor the particle viewpoint is correct. We would always like to present things accurately, or at least precisely enough that they will not have to be changed when we learn more—it may be extended, but it will not be changed! But when we try to talk about the wave picture or the particle picture, both are approximate, and both will change. Therefore what we learn in this chapter will not be accurate in a certain sense; we will deal with some halfintuitive arguments which will be made more precise later. But certain things will be changed a little bit when we interpret them correctly in quantum mechanics. We are doing this so that you can have some qualitative feeling for some quantum phenomena before we get into the mathematical details of quantum mechanics. Furthermore, all our experiences are with waves and with particles, and so it is rather handy to use the wave and particle ideas to get some understanding of what happens in given circumstances before we know the complete mathematics of the quantummechanical amplitudes. We shall try to indicate the weakest places as we go along, but most of it is very nearly correct—it is just a matter of interpretation. First of all, we know that the new way of representing the world in quantum mechanics—the new framework—is to give an amplitude for every event that can occur, and if the event involves the reception of one particle, then we can give the amplitude to find that one particle at different places and at different times. 21 The probability of finding the particle is then proportional to the absolute square of the amplitude. In general, the amplitude to find a particle in different places at different times varies with position and time. In some special case it can be that the amplitude varies sinusoidally in space and time like ei(ωt−k·r) , where r is the vector position from some origin. (Do not forget that these amplitudes are complex numbers, not real numbers.) Such an amplitude varies according to a definite frequency ω and wave number k. Then it turns out that this corresponds to a classical limiting situation where we would have believed that we have a particle whose energy E was known and is related to the frequency by E = ~ω, (2.1) and whose momentum p is also known and is related to the wave number by p = ~k. (2.2) (The symbol ~ represents the number h divided by 2π; ~ = h/2π.) This means that the idea of a particle is limited. The idea of a particle— its location, its momentum, etc.—which we use so much, is in certain ways unsatisfactory. For instance, if an amplitude to find a particle at different places is given by ei(ωt−k·r) , whose absolute square is a constant, that would mean that the probability of finding a particle is the same at all points. That means we do not know where it is—it can be anywhere—there is a great uncertainty in its location. On the other hand, if the position of a particle is more or less well known and we can predict it fairly accurately, then the probability of finding it in different places must be confined to a certain region, whose length we call ∆x. Outside this region, the probability is zero. Now this probability is the absolute square of an amplitude, and if the absolute square is zero, the amplitude is also zero, so that we have a wave train whose length is ∆x (Fig. 21), and the wavelength ∆x Fig. 21. A wave packet of length ∆x. 22 (the distance between nodes of the waves in the train) of that wave train is what corresponds to the particle momentum. Here we encounter a strange thing about waves; a very simple thing which has nothing to do with quantum mechanics strictly. It is something that anybody who works with waves, even if he knows no quantum mechanics, knows: namely, we cannot define a unique wavelength for a short wave train. Such a wave train does not have a definite wavelength; there is an indefiniteness in the wave number that is related to the finite length of the train, and thus there is an indefiniteness in the momentum. 22 Measurement of position and momentum Let us consider two examples of this idea—to see the reason that there is an uncertainty in the position and/or the momentum, if quantum mechanics is right. We have also seen before that if there were not such a thing—if it were possible to measure the position and the momentum of anything simultaneously—we would have a paradox; it is fortunate that we do not have such a paradox, and the fact that such an uncertainty comes naturally from the wave picture shows that everything is mutually consistent. Here is one example which shows the relationship between the position and the momentum in a circumstance that is easy to understand. Suppose we have a single slit, and particles are coming from very far away with a certain energy—so that they are all coming essentially horizontally (Fig. 22). We are going to concentrate on the vertical components of momentum. All of these particles have a certain horizontal momentum p0 , say, in a classical sense. So, in the classical sense, the vertical momentum py , before the particle goes through the hole, is C ∆θ B Fig. 22. Diffraction of particles passing through a slit. 23 definitely known. The particle is moving neither up nor down, because it came from a source that is far away—and so the vertical momentum is of course zero. But now let us suppose that it goes through a hole whose width is B. Then after it has come out through the hole, we know the position vertically—the yposition—with considerable accuracy—namely ±B.† That is, the uncertainty in position, ∆y, is of order B. Now we might also want to say, since we known the momentum is absolutely horizontal, that ∆py is zero; but that is wrong. We once knew the momentum was horizontal, but we do not know it any more. Before the particles passed through the hole, we did not know their vertical positions. Now that we have found the vertical position by having the particle come through the hole, we have lost our information on the vertical momentum! Why? According to the wave theory, there is a spreading out, or diffraction, of the waves after they go through the slit, just as for light. Therefore there is a certain probability that particles coming out of the slit are not coming exactly straight. The pattern is spread out by the diffraction effect, and the angle of spread, which we can define as the angle of the first minimum, is a measure of the uncertainty in the final angle. How does the pattern become spread? To say it is spread means that there is some chance for the particle to be moving up or down, that is, to have a component of momentum up or down. We say chance and particle because we can detect this diffraction pattern with a particle counter, and when the counter receives the particle, say at C in Fig. 22, it receives the entire particle, so that, in a classical sense, the particle has a vertical momentum, in order to get from the slit up to C. To get a rough idea of the spread of the momentum, the vertical momentum py has a spread which is equal to p0 ∆θ, where p0 is the horizontal momentum. And how big is ∆θ in the spreadout pattern? We know that the first minimum occurs at an angle ∆θ such that the waves from one edge of the slit have to travel one wavelength farther than the waves from the other side—we worked that out before (Chapter 30 of Vol. I). Therefore ∆θ is λ/B, and so ∆py in this experiment is p0 λ/B. Note that if we make B smaller and make a more accurate measurement of the position of the particle, the diffraction pattern gets wider. So the narrower we make the slit, the wider the pattern gets, and the more is the likelihood that we would find that the particle has sidewise momentum. Thus the † More precisely, the error in our knowledge of y is ±B/2. But we are now only interested in the general idea, so we won’t worry about factors of 2. 24 uncertainty in the vertical momentum is inversely proportional to the uncertainty of y. In fact, we see that the product of the two is equal to p0 λ. But λ is the wavelength and p0 is the momentum, and in accordance with quantum mechanics, the wavelength times the momentum is Planck’s constant h. So we obtain the rule that the uncertainties in the vertical momentum and in the vertical position have a product of the order h: ∆y ∆py ≥ ~/2. (2.3) We cannot prepare a system in which we know the vertical position of a particle and can predict how it will move vertically with greater certainty than given by (2.3). That is, the uncertainty in the vertical momentum must exceed ~/2∆y, where ∆y is the uncertainty in our knowledge of the position. Sometimes people say quantum mechanics is all wrong. When the particle arrived from the left, its vertical momentum was zero. And now that it has gone through the slit, its position is known. Both position and momentum seem to be known with arbitrary accuracy. It is quite true that we can receive a particle, and on reception determine what its position is and what its momentum would have had to have been to have gotten there. That is true, but that is not what the uncertainty relation (2.3) refers to. Equation (2.3) refers to the predictability of a situation, not remarks about the past. It does no good to say “I knew what the momentum was before it went through the slit, and now I know the position,” because now the momentum knowledge is lost. The fact that it went through the slit no longer permits us to predict the vertical momentum. We are talking about a predictive theory, not just measurements after the fact. So we must talk about what we can predict. Now let us take the thing the other way around. Let us take another example of the same phenomenon, a little more quantitatively. In the previous example we measured the momentum by a classical method. Namely, we considered the direction and the velocity and the angles, etc., so we got the momentum by classical analysis. But since momentum is related to wave number, there exists in nature still another way to measure the momentum of a particle—photon or otherwise—which has no classical analog, because it uses Eq. (2.2). We measure the wavelengths of the waves. Let us try to measure momentum in this way. Suppose we have a grating with a large number of lines (Fig. 23), and send a beam of particles at the grating. We have often discussed this problem: if the particles have a definite momentum, then we get a very sharp pattern in a 25 Nmλ = L Fig. 23. Determination of momentum by using a diffr