The feynman lectures on physics vol 3; quantum mechanics (1)

Preface to the New Millennium EditionNearly fifty years have passed since Richard Feynman taught the introductory physics course at Caltech that gave rise to these three volumes, The Fey

The Feynman

LECTURES ON PHYSICS

NEW MILLENNIUM EDITION

VOLUME III

Copyright © 1965, 2006, 2010 by California Institute of Technology,

Michael A Gottlieb, and Rudolf Pfeiffer Published by Basic Books,

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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 theManhattan Project at Los Alamos during World War II Subsequently, he taught

at Cornell and at the California Institute of Technology In 1965 he received theNobel Prize in Physics, along with Sin-Itiro Tomonaga and Julian Schwinger, forhis work in quantum electrodynamics

Dr Feynman won his Nobel Prize for successfully resolving problems with thetheory of quantum electrodynamics He also created a mathematical theory thataccounts for the phenomenon of superfluidity in liquid helium Thereafter, withMurray Gell-Mann, 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 ofquark theory by putting forward his parton model of high energy proton collisionprocesses

Beyond these achievements, Dr Feynman introduced basic new tional techniques and notations into physics—above all, the ubiquitous Feynmandiagrams that, perhaps more than any other formalism in recent scientific history,have changed the way in which basic physical processes are conceptualized andcalculated

computa-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 publicationsthat have become classic references and textbooks for researchers and students.Richard Feynman was a constructive public man His work on the Challengercommission is well known, especially his famous demonstration of the susceptibility

of the O-rings to cold, an elegant experiment which required nothing more than

a glass of ice water and a C-clamp Less well known were Dr Feynman’s efforts

on the California State Curriculum Committee in the 1960s, where he protestedthe mediocrity of textbooks

A recital of Richard Feynman’s myriad scientific and educational ments cannot adequately capture the essence of the man As any reader ofeven his most technical publications knows, Feynman’s lively and multi-sidedpersonality shines through all his work Besides being a physicist, he was atvarious times a repairer of radios, a picker of locks, an artist, a dancer, a bongoplayer, and even a decipherer of Mayan Hieroglyphics Perpetually curious abouthis world, he was an exemplary empiricist.

accomplish-Richard Feynman died on February 15, 1988, in Los Angeles

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 toFeynman’s unique physics insights and pedagogy They have been studiedworldwide by novices and mature physicists alike; they have been translatedinto at least a dozen languages with more than 1.5 millions copies printed in theEnglish language alone Perhaps no other set of physics books has had such wideimpact, for so long

This New Millennium Edition ushers in a new era for The Feynman Lectures

on Physics (FLP) : the twenty-first century era of electronic publishing FLP

electronic typesetting language, and all figures redone using modern drawingsoftware

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 knownand 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 firstprinting of the previous edition, and the ease of correcting errata that futurereaders 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 nical books, whose equations, figures and sometimes even text become pixellated

Edition makes it possible to create eBooks of the highest quality, in which all

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 itslinks to other resources, is an innovation that would have given Feynman greatpleasure

Memories of Feynman's Lectures

These three volumes are a self-contained pedagogical treatise They are also ahistorical record of Feynman’s 1961–64 undergraduate physics lectures, a courserequired of all Caltech freshmen and sophomores regardless of their majors.Readers may wonder, as I have, how Feynman’s lectures impacted the studentswho attended them Feynman, in his Preface to these volumes, offered a somewhatnegative 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 quasi-random set

of 17 students (out of about 150) from Feynman’s 1961–63 class—some who hadgreat difficulty with the class, and some who mastered it with ease; majors inbiology, 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 fromCaltech.” “I was a biology major but Feynman’s lectures stand out as a highpoint in my undergraduate experience though I must admit I couldn’t dothe homework at the time and I hardly turned any of it in.” “I was among theleast promising of students in this course, and I never missed a lecture Iremember 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 twoissues: (i) “You couldn’t learn to work the homework problems by attending thelectures Feynman was too slick—he knew tricks and what approximations could

be made, and had intuition based on experience and genius that a beginningstudent does not possess.” Feynman and colleagues, aware of this flaw in thecourse, addressed it in part with materials that have been incorporated into

Feynman’s Tips on Physics: three problem-solving lectures by Feynman, and

a set of exercises and answers assembled by Robert B Leighton and RochusVogt (ii) “The insecurity of not knowing what was likely to be discussed inthe next lecture, the lack of a text book or reference with any connection tothe lecture material, and consequent inability for us to read ahead, were veryfrustrating I found the lectures exciting and understandable in the hall, butthey 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 studentsstudied for many years thereafter, and they live on today as one of Feynman’sgreatest legacies

A History of Errata

The Feynman Lectures on Physicswas produced very quickly by Feynmanand his co-authors, Robert B Leighton and Matthew Sands, working fromand 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 firstedition Feynman accumulated long lists of claimed errata over the subsequentyears—errata found by students and faculty at Caltech and by readers aroundthe 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 insertcorrections into subsequent printings But Feynman’s sense of duty never rosehigh enough above the excitement of discovering new things to make him dealwith the errata in Volume III.‡ After his untimely death in 1988, lists of erratafor all three volumes were deposited in the Caltech Archives, and there they layforgotten

In 2002 Ralph Leighton (son of the late Robert Leighton and compatriot ofFeynman) 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.

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 oferrata and the content of the proposed new volume, I quickly agreed to overseethis project on behalf of Caltech (Feynman’s long-time 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, andcareful 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

antic-ipate was the enthusiastic response of readers around the world to an appealfrom 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 havebeen submitted and survived the meticulous scrutiny of Gottlieb, Hartl, and NateBode (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, seewww.feynmanlectures.info

Clearly, making The Feynman Lectures on Physics error-free has become a

world-wide community enterprise On behalf of Caltech I thank the 50 readerswho have contributed since 2005 and the many more who may contribute over the

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 andfigures—sign errors, incorrect numbers (e.g., a 5 that should be a 4), and missingsubscripts, summation signs, parentheses and terms in equations; (iii) incorrectcross references to chapters, tables and figures These kinds of errors, thoughnot terribly serious to a mature physicist, can be frustrating and confusing toFeynman’s primary audience: students

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 5-9, which now says “ no static distribution of charges inside a closed

grounded conductor can produce any [electric] fields outside” (the word groundedwas omitted in previous editions) This error was pointed out to Feynman by anumber of readers, including Beulah Elizabeth Cox, a student at The College ofWilliam 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 giveyou any points, for your answer was wrong, as he demonstrated using Gauss’slaw You should, in science, believe logic and arguments, carefully drawn, andnot authorities You also read the book correctly and understood it I made amistake, so the book is wrong I probably was thinking of a grounded conductingsphere, or else of the fact that moving the charges around in different placesinside does not affect things on the outside I am not sure how I did it, but Igoofed 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 Websitewww.feynmanlectures.info Remarkably, the bulk

of these came from one person: Dr Rudolf Pfeiffer, then a physics postdoctoralfellow 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 photo-offset process, working from photographic images of the pages fromthe 1960s Correcting an error involved re-typesetting the entire page, and toensure no new errors crept in, the page was re-typeset twice by two differentpeople, then compared and proofread by several other people—a very costlyprocess indeed, when hundreds of errata are involved

Gottlieb, Pfeiffer and Ralph Leighton were very unhappy about this, so theyformulated 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

one-chapter 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).

that Caltech cooperate with Gottlieb, Pfeiffer and Leighton in the execution oftheir plan The plan was approved by three successive chairs of Caltech’s Division

of Physics, Mathematics and Astronomy—Tom Tombrello, Andrew Lange, andTom Soifer—and the complex legal and contractual details were worked out byCaltech’s Intellectual Property Counsel, Adam Cochran With the publication of

this New Millennium Edition, the plan has been executed successfully, despite

its complexity Specifically:

(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 non-exclusive

for non-exclusive use of Heinze’s figures in this New Millennium English edition.

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 unbelievablymeticulous and accurate Gottlieb and Pfeiffer arranged for John Sullivan at theHuntington Library to digitize the photos of Feynman’s 1962–64 blackboards,and for George Blood Audio to digitize the lecture tapes—with financial supportand encouragement from Caltech Professor Carver Mead, logistical support fromCaltech Archivist Shelley Erwin, and legal support from Cochran

The legal issues were serious: In the 1960s, Caltech licensed to Addison Wesleyrights 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 tothe Pearson publishing group, while rights to the audio and the electronic versionwere transferred to the Perseus publishing group Cochran, with the aid of IkeWilliams, 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

above: Ralph Leighton, Michael Gottlieb, Tom Tombrello, Michael Hartl, RudolfPfeiffer, Henning Heinze, Adam Cochran, Carver Mead, Nate Bode, Shelley Erwin,Andrew Lange, Tom Soifer, Ike Williams, and the 50 people who submitted errata

(daughter of Richard Feynman) for her continuing support and advice, Alan Ricefor behind-the-scenes assistance and advice at Caltech, Stephan Puchegger and

Calvin Jackson for assistance and advice to Pfeiffer about conversion of FLP to

corrections of errata; and the Staff of Perseus/Basic Books, and (for previouseditions) the staff of Addison Wesley

Kip S Thorne

The Feynman Professor of Theoretical Physics, Emeritus

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 63-20717

Third printing, July 1966

ISBN 0-201-02118-8-P 0-201-02114-9-H BBCCDDEEFFGG-MU-898

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 180students gathered in a big lecture room twice a week to hear these lectures andthen they broke up into small groups of 15 to 20 students in recitation sectionsunder the guidance of a teaching assistant In addition, there was a laboratorysession once a week

The special problem we tried to get at with these lectures was to maintainthe interest of the very enthusiastic and rather smart students coming out ofthe high schools and into Caltech They have heard a lot about how interestingand exciting physics is—the theory of relativity, quantum mechanics, and othermodern ideas By the end of two years of our previous course, many would bevery discouraged because there were really very few grand, new, modern ideaspresented 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 excitedstudent by maintaining his enthusiasm

The lectures here are not in any way meant to be a survey course, but are veryserious I thought to address them to the most intelligent in the class and to makesure, if possible, that even the most intelligent student was unable to completelyencompass everything that was in the lectures—by putting in suggestions of

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 ideasfitted 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 bydeduction from what has been said before, and what is being put in as somethingnew 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—butwas just added in

At the start of these lectures, I assumed that the students knew somethingwhen they came out of high school—such things as geometrical optics, simplechemistry ideas, and so on I also didn’t see that there was any reason to make thelectures in a definite order, in the sense that I would not be allowed to mentionsomething until I was ready to discuss it in detail There was a great deal ofmention of things to come, without complete discussions These more completediscussions would come later when the preparation became more advanced.Examples are the discussions of inductance, and of energy levels, which are atfirst 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 aremerely 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 tounderstand 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 theoremsand central ideas, and which are the more advanced side issues and applicationswhich he may understand only in later years

In giving these lectures there was one serious difficulty: in the way the coursewas given, there wasn’t any feedback from the students to the lecturer to indicatehow 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 thephysics is concerned—quite satisfactorily in the first year

In the second year I was not so satisfied In the first part of the course, dealingwith electricity and magnetism, I couldn’t think of any really unique or differentway of doing it—of any way that would be particularly more exciting than theusual way of presenting it So I don’t think I did very much in the lectures onelectricity 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 onthe 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 theselectures again, it was suggested that it might be a good idea to try to give anintroduction to the quantum mechanics—what you will find in Volume III

It is perfectly clear that students who will major in physics can wait untiltheir third year for quantum mechanics On the other hand, the argument wasmade that many of the students in our course study physics as a background fortheir primary interest in other fields And the usual way of dealing with quantummechanics makes that subject almost unavailable for the great majority of studentsbecause they have to take so long to learn it Yet, in its real applications—especially in its more complex applications, such as in electrical engineeringand chemistry—the full machinery of the differential equation approach is notactually 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 partialdifferential 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 severalreasons which may be apparent in the lectures themselves However, I think thatthe experiment in the quantum mechanics part was not completely successful—inlarge part because I really did not have enough time at the end (I should, forinstance, have had three or four more lectures in order to deal more completelywith 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 wasparticularly serious I now believe the quantum mechanics should be given at alater 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 therewere recitation sections Although I did put in three lectures in the first year onhow to solve problems, they are not included here Also there was a lecture oninertial guidance which certainly belongs after the lecture on rotating systems,

but which was, unfortunately, omitted The fifth and sixth lectures are actuallydue to Matthew Sands, as I was out of town.

The question, of course, is how well this experiment has succeeded My ownpoint of view—which, however, does not seem to be shared by most of the peoplewho worked with the students—is pessimistic I don’t think I did very well bythe students When I look at the way the majority of the students handled theproblems 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—verysurprisingly—understood almost everything in all of the lectures, and who werequite 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 first-ratebackground 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 thosehappy 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 hardwork 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 thelectures and make more realistic, more complete, and more settled in the mindthe ideas that have been exposed

I think, however, that there isn’t any solution to this problem of educationother than to realize that the best teaching can be done only when there is adirect individual relationship between a student and a good teacher—a situation

in which the student discusses the ideas, thinks about the things, and talks aboutthe things It’s impossible to learn very much by simply sitting in a lecture, oreven by simply doing problems that are assigned But in our modern times wehave so many students to teach that we have to try to find some substitute forthe ideal Perhaps my lectures can make some contribution Perhaps in somesmall place where there are individual teachers and students, they may get someinspiration or some ideas from the lectures Perhaps they will have fun thinkingthem through—or going on to develop some of the ideas further

Richard P Feynman

June, 1963

A great triumph of twentieth-century physics, the theory of quantum ics, is now nearly 40 years old, yet we have generally been giving our studentstheir introductory course in physics (for many students, their last) with hardlymore than a casual allusion to this central part of our knowledge of the physicalworld We should do better by them These lectures are an attempt to presentthem with the basic and essential ideas of the quantum mechanics in a waythat would, hopefully, be comprehensible The approach you will find here isnovel, particularly at the level of a sophomore course, and was considered verymuch 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, roomfor improvement, and it will come with more experience in the classroom Whatyou will find here is a record of that first experiment

mechan-In the two-year sequence of the Feynman Lectures on Physics which weregiven from September 1961 through May 1963 for the introductory physics course

at Caltech, the concepts of quantum physics were brought in whenever they werenecessary for an understanding of the phenomena being described In addition,the last twelve lectures of the second year were given over to a more coherentintroduction 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 leftfor the quantum mechanics As the material was prepared, it was continuallydiscovered that other important and interesting topics could be treated with theelementary tools that had been developed There was also a fear that the toobrief treatment of the Schrödinger wave function which had been included in thetwelfth lecture would not provide a sufficient bridge to the more conventional

treatments of many books the students might hope to read It was thereforedecided to extend the series with seven additional lectures; they were given tothe sophomore class in May of 1964 These lectures rounded out and extendedsomewhat the material developed in the earlier lectures.

In this volume we have put together the lectures from both years with someadjustment of the sequence In addition, two lectures originally given to thefreshman class as an introduction to quantum physics have been lifted bodily

chapters here—to make this volume a self-contained unit, relatively independent

of the first two A few ideas about the quantization of angular momentum(including a discussion of the Stern-Gerlach experiment) had been introduced in

convenience of those who will not have that volume at hand, those two chaptersare reproduced here as an Appendix

This set of lectures tries to elucidate from the beginning those features of thequantum mechanics which are most basic and most general The first lecturestackle 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 areintroduced together with a detailed discussion of some specific examples—to try

to make the physical ideas as real as possible The time dependence of statesincluding states of definite energy comes next, and the ideas are applied at once

to the study of two-state systems A detailed discussion of the ammonia maserprovides the frame-work for the introduction to radiation absorption and inducedtransitions 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 completetreatment of the quantum mechanics of angular momentum Our introduction

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 theentertainment lectures of the first two volumes, with the intent of opening to thestudents a broader view of the relation of what they were learning to the generalculture of physics Feynman’s “epilogue” serves as the period to the three-volumeseries

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

California Institute of Technology under the supervision of the Physics CourseRevision Committee (Robert Leighton, Victor Neher, and Matthew Sands) Theprogram was made possible by a grant from the Ford Foundation Many peoplehelped with the technical details of the preparation of this volume: MarylouClayton, Julie Curcio, James Hartle, Tom Harvey, Martin Israel, Patricia Preuss,Fanny Warren, and Barbara Zimmerman Professors Gerry Neugebauer andCharles Wilts contributed greatly to the accuracy and clarity of the material byreviewing 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 evensome of the intellectual excitement we experienced as we saw the ideas unfold inhis real-life Lectures on Physics

Matthew Sands

December, 1964

Chapter 1 Quantum Behavior

1-1 Atomic mechanics 1-1

1-2 An experiment with bullets 1-2

1-3 An experiment with waves 1-4

1-4 An experiment with electrons 1-6

1-5 The interference of electron waves 1-8

1-6 Watching the electrons 1-10

1-7 First principles of quantum mechanics 1-15

1-8 The uncertainty principle 1-17

Chapter 2 The Relation of Wave and Particle Viewpoints

2-1 Probability wave amplitudes 2-1

2-2 Measurement of position and momentum 2-3

2-3 Crystal diffraction 2-8

2-4 The size of an atom 2-10

2-5 Energy levels 2-13

2-6 Philosophical implications 2-15

Chapter 3 Probability Amplitudes

3-1 The laws for combining amplitudes 3-1

3-2 The two-slit interference pattern 3-8

3-3 Scattering from a crystal 3-12

3-4 Identical particles 3-16

Chapter 4 Identical Particles

4-1 Bose particles and Fermi particles 4-1

4-2 States with two Bose particles 4-6

4-3 States with n Bose particles 4-10

4-4 Emission and absorption of photons 4-13

4-5 The blackbody spectrum 4-15

4-6 Liquid helium 4-22

4-7 The exclusion principle 4-23

Chapter 5 Spin One

5-1 Filtering atoms with a Stern-Gerlach apparatus 5-1

5-2 Experiments with filtered atoms 5-9

5-3 Stern-Gerlach filters in series 5-11

5-4 Base states 5-13

5-5 Interfering amplitudes 5-16

5-6 The machinery of quantum mechanics 5-21

5-7 Transforming to a different base 5-24

5-8 Other situations 5-27

Chapter 6 Spin One-Half

6-1 Transforming amplitudes 6-1

6-2 Transforming to a rotated coordinate system 6-4

6-3 Rotations about the z-axis 6-10

6-4 Rotations of 180◦and 90◦ about y 6-15

6-5 Rotations about x 6-20

6-6 Arbitrary rotations 6-22

Chapter 7 The Dependence of Amplitudes on Time

7-1 Atoms at rest; stationary states 7-1

7-2 Uniform motion 7-5

7-3 Potential energy; energy conservation 7-9

7-4 Forces; the classical limit 7-16

7-5 The “precession” of a spin one-half particle 7-18

Chapter 8 The Hamiltonian Matrix

8-1 Amplitudes and vectors 8-1

8-2 Resolving state vectors 8-4

8-3 What are the base states of the world? 8-8

8-4 How states change with time 8-11

8-5 The Hamiltonian matrix 8-16

8-6 The ammonia molecule 8-17

Chapter 9 The Ammonia Maser

9-1 The states of an ammonia molecule 9-1

9-2 The molecule in a static electric field 9-7

9-3 Transitions in a time-dependent field 9-14

9-4 Transitions at resonance 9-18

9-5 Transitions off resonance 9-21

9-6 The absorption of light 9-23

Chapter 10 Other Two-State Systems

10-1 The hydrogen molecular ion 10-1

10-2 Nuclear forces 10-10

10-3 The hydrogen molecule 10-13

10-4 The benzene molecule 10-17

10-5 Dyes 10-21

10-6 The Hamiltonian of a spin one-half particle in a magnetic field 10-22

10-7 The spinning electron in a magnetic field 10-26

Chapter 11 More Two-State Systems

11-1 The Pauli spin matrices 11-1

11-2 The spin matrices as operators 11-9

11-3 The solution of the two-state equations 11-14

11-4 The polarization states of the photon 11-15

11-5 The neutral K-meson 11-21

Chapter 12 The Hyperfine Splitting in Hydrogen

12-1 Base states for a system with two spin one-half particles 12-1

12-2 The Hamiltonian for the ground state of hydrogen 12-4

12-3 The energy levels 12-12

12-4 The Zeeman splitting 12-15

12-5 The states in a magnetic field 12-22

12-6 The projection matrix for spin one 12-26

Chapter 13 Propagation in a Crystal Lattice

13-1 States for an electron in a one-dimensional lattice 13-1

13-2 States of definite energy 13-5

13-3 Time-dependent states 13-10

13-4 An electron in a three-dimensional lattice 13-12

13-5 Other states in a lattice 13-14

13-6 Scattering from imperfections in the lattice 13-16

13-7 Trapping by a lattice imperfection 13-20

13-8 Scattering amplitudes and bound states 13-21

15-4 The benzene molecule 15-11

15-5 More organic chemistry 15-18

15-6 Other uses of the approximation 15-24

Chapter 16 The Dependence of Amplitudes on Position

16-1 Amplitudes on a line 16-1

16-2 The wave function 16-7

16-3 States of definite momentum 16-10

16-4 Normalization of the states in x 16-14

16-5 The Schrödinger equation 16-18

16-6 Quantized energy levels 16-23

Chapter 17 Symmetry and Conservation Laws

17-1 Symmetry 17-1

17-2 Symmetry and conservation 17-6

17-3 The conservation laws 17-13

17-4 Polarized light 17-17

17-5 The disintegration of the Λ0 17-21

17-6 Summary of the rotation matrices 17-28

Chapter 18 Angular Momentum

18-1 Electric dipole radiation 18-1

18-2 Light scattering 18-6

18-3 The annihilation of positronium 18-9

18-4 Rotation matrix for any spin 18-18

18-5 Measuring a nuclear spin 18-23

18-6 Composition of angular momentum 18-25

18-8 Added Note 2: Conservation of parity in photon emission 18-39

Chapter 19 The Hydrogen Atom and The Periodic Table

19-1 Schrödinger’s equation for the hydrogen atom 19-1

19-2 Spherically symmetric solutions 19-4

19-3 States with an angular dependence 19-9

19-4 The general solution for hydrogen 19-17

19-5 The hydrogen wave functions 19-21

19-6 The periodic table 19-25

Chapter 20 Operators

20-1 Operations and operators 20-1

20-2 Average energies 20-5

20-3 The average energy of an atom 20-9

20-4 The position operator 20-12

20-5 The momentum operator 20-15

20-6 Angular momentum 20-22

20-7 The change of averages with time 20-25

Chapter 21 The Schrödinger Equation in a Classical Context: A Seminar

on Superconductivity

21-1 Schrödinger’s equation in a magnetic field 21-1

21-2 The equation of continuity for probabilities 21-5

21-3 Two kinds of momentum 21-7

21-4 The meaning of the wave function 21-10

21-5 Superconductivity 21-11

21-6 The Meissner effect 21-14

21-7 Flux quantization 21-16

21-8 The dynamics of superconductivity 21-22

21-9 The Josephson junction 21-25

Quantum Behavior

Note: This chapter is almost exactly the same as Chapter37of Volume I

1-1 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 experienceabout They do not behave like waves, they do not behave like particles, they donot behave like clouds, or billiard balls, or weights on springs, or like anythingthat you have ever seen

Newton thought that light was made up of particles, but then it was discoveredthat it behaves like a wave Later, however (in the beginning of the twentiethcentury), it was found that light did indeed sometimes behave like a particle.Historically, the electron, for example, was thought to behave like a particle, andthen 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 Thequantum 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 tocall them So what we learn about the properties of electrons (which we shall usefor our examples) will apply also to all “particles,” including photons of light.The gradual accumulation of information about atomic and small-scale be-havior during the first quarter of the 20th century, which gave some indicationsabout how small things do behave, produced an increasing confusion which wasfinally resolved in 1926 and 1927 by Schrödinger, Heisenberg, and Born Theyfinally obtained a consistent description of the behavior of matter on a smallscale We take up the main features of that description in this chapter

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 thenovice and to the experienced physicist Even the experts do not understand itthe 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 largeobjects 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 orimaginative fashion and not by connection with our direct experience

In this chapter we shall tackle immediately the basic element of the mysteriousbehavior 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

the basic peculiarities of all quantum mechanics

1-2 An experiment with bullets

To try to understand the quantum behavior of electrons, we shall compareand contrast their behavior, in a particular experimental setup, with the morefamiliar behavior of particles like bullets, and with the behavior of waves likewater waves We consider first the behavior of bullets in the experimental setup

GUN

(a) WALL

1

2

x

MOVABLE DETECTOR

of bullets It is not a very good gun, in that it sprays the bullets (randomly) over afairly 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) whichwill “absorb” the bullets when they hit it In front of the wall we have an objectwhich we shall call a “detector” of bullets It might be a box containing sand Anybullet 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 x-direction).

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 definitelywhere any particular bullet will go A bullet which happens to hit one of theholes 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 thesame 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 inlumps, 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 anygiven moment either nothing arrives, or one and only one—exactly one—bulletarrives 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 weare really imagining the result) are plotted in the graph drawn in part (c) of

because the bullets may have come either through hole 1 or through hole 2 You

small if x is very large You may wonder, however, why P12has 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

value of x which is on a straight line with the gun and hole 1 When hole 1 is

distribution for bullets that pass through hole 2 Comparing parts (b) and (c) of

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.1-3 An experiment with waves

Now we wish to consider an experiment with water waves The apparatus

small object labeled the “wave source” is jiggled up and down by a motor and

WAVE

SOURCE

(a) WALL

1

2 DETECTOR

makes circular waves To the right of the source we have again a wall with twoholes, 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 x-direction, as before The

detector is now a device which measures the “intensity” of the wave motion Youcan 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 interesting-looking

curve marked I12in part (c) of the figure

We have already worked out how such patterns can come about when westudied the interference of electric waves in Volume I In this case we wouldobserve that the original wave is diffracted at the holes, and new circular wavesspread out from each hole If we cover one hole at a time and measure theintensity 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

wave from hole 2 (seen when hole 1 is blocked)

of I1and I2 We say that there is “interference” of the two waves At some places

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 Therewill be such constructive interference wherever the distance from the detector toone hole is a whole number of wavelengths larger (or shorter) than the distancefrom 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

the detector will be the difference of the two amplitudes The waves “interferedestructively,” and we get a low value for the wave intensity We expect such lowvalues wherever the distance between hole 1 and the detector is different from thedistance between hole 2 and the detector by an odd number of half-wavelengths.

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 waterwave at the detector for the wave from hole 1 can be written as (the real part

intensity is proportional to the mean squared height or, when we use the complex

is h2e iωt and the intensity is proportional to |h2|2 When both holes are open,

the wave heights add to give the height (h1+ h2)e iωt and the intensity |h1+ h2|2.Omitting the constant of proportionality for our present purposes, the proper

relations for interfering waves are

intensity can have any value, and it shows interference

1-4 An experiment with electrons

Now we imagine a similar experiment with electrons It is shown

heated by an electric current and surrounded by a metal box with a hole in it Ifthe wire is at a negative voltage with respect to the box, electrons emitted bythe wire will be accelerated toward the walls and some will pass through the hole

GUN

(a) WALL

1

2 DETECTOR

Fig 1-3 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 thebackstop 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 (asyou could have done with the two we have already described) This experimenthas never been done in just this way The trouble is that the apparatus wouldhave 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 proportionshave 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 “half-clicks.”

We would also notice that the “clicks” come very erratically Something like:click click-click click click click-click click , etc.,just as you have, no doubt, heard a geiger counter operating If we count theclicks which arrive in a sufficiently long time—say for many minutes—and thencount again for another equal period, we find that the two numbers are very

nearly the same So we can speak of the average rate at which the clicks are

heard (so-and-so-many 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 lowerthe temperature of the wire in the gun, the rate of clicking slows down, but stilleach 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 whateverarrives 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 Weshall say: “Electrons always arrive in identical lumps.”

Just as for our experiment with bullets, we can now proceed to find imentally the answer to the question: “What is the relative probability that

exper-an electron ‘lump’ will arrive at the backstop at various distexper-ances x from the

center?” As before, we obtain the relative probability by observing the rate ofclicks, 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.

1-5 The interference of electron waves

Now let us try to analyze the curve of Fig.1-3to see whether we can understandthe behavior of the electrons The first thing we would say is that since theycome in lumps, each lump, which we may as well call an electron, has come eitherthrough 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 bedivided into two classes: (1) those that come through hole 1, and (2) those thatcome through hole 2 So our observed curve must be the sum of the effects ofthe electrons which come through hole 1 and the electrons which come throughhole 2 Let us check this idea by experiment First, we will make a measurementfor those electrons that come through hole 1 We block off hole 2 and make

The result of the measurement is shown by the curve marked P1in part (b) ofFig.1-3 The result seems quite reasonable In a similar way, we measure P2, theprobability distribution for the electrons that come through hole 2 The result ofthis measurement is also drawn in the figure.

experiment, we say: “There is interference.”

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, andthen they go around through 2, and then around a few more times, or by someother 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

in terms of individual electrons going around in complicated ways through the

in terms of P1 and P2

extremely simple For P12is just like the curve I12of Fig.1-2, and that was simple.

What is going on at the backstop can be described by two complex numbers that

of φ1 gives the effect with only hole 1 open That is, P1= |φ1|2 The effect with

combined effect of the two holes is just P12= |φ1+ φ2|2 The mathematics is the

same as that we had for the water waves! (It is hard to see how one could getsuch a simple result from a complicated game of electrons going back and forththrough the plate on some strange trajectory.)

We conclude the following: The electrons arrive in lumps, like particles, andthe probability of arrival of these lumps is distributed like the distribution ofintensity of a wave It is in this sense that an electron behaves “sometimes like aparticle 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 complexnumbers 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 themoment, 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 alarge 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

1-6 Watching the electrons

We shall now try the following experiment To our electron apparatus weadd a very strong light source, placed behind the wall and between the two holes,

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

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 thesame 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

(b)

P 0 1

P 0 2

(c)

P 0 12

P 0

12 = P 0

1 + P 0 2

Fig 1-4 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 atthe electrons we find that the electrons go either through one hole or the other.Experimentally, Proposition A is necessarily true

and find out what they are doing For each position (x-location) 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 thisway: whenever we hear a “click” we will put a count in Column 1 if we see theflash 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

2,the probability that an electron will arrive at the detector via hole 2 If we now

1and P0 2

shown in part (b) of Fig.1-4

what we got before for P1by blocking off hole 2; and P0

2is 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

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 isopen 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 thatinformation We just pretend that we never looked at the light flashes, and welump 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

12= P0

1+ P0

2 That

is, although we succeeded in watching which hole our electrons come through,

we no longer get the old interference curve P12, but a new one, P0

12, 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 onour light source that disturbs things? It must be that the electrons are verydelicate, and the light, when it scatters off the electrons, gives them a jolt thatchanges 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 tothe electron when the photon is scattered by it is such as to change the electron’s

wavy interference effects

You may be thinking: “Don’t use such a bright source! Turn the brightnessdown! The light waves will then be weaker and will not disturb the electrons somuch 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 firstthing we observe is that the flashes of light scattered from the electrons as they

pass by does not get weaker It is always the same-sized 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

by without being seen There did not happen to be a photon around at the timethe electron went through.

This is all a little discouraging If it is true that whenever we “see” the electron

we see the same-sized flash, then those electrons we see are always the disturbed

ones Let us try the experiment with a dim light anyway Now whenever wehear 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 P0

1; those “seen by hole 2” have a distribution like P0

2(so that

seen at all” have a “wavy” distribution just like P12 of Fig.1-3! 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 sameamount of disturbance because the light photons all produce the same-sizedeffects and the effect of the photons being scattered is enough to smear out anyinterference 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 themomentum 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 avoiddisturbing the electrons so much

Let us try the experiment with longer waves We shall keep repeating ourexperiment, each time with light of a longer wavelength At first, nothing seems tochange 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 natureof 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 oflight 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.