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THE THEORY of QUANTUM MECHANICS: Sflvtin LI.,Lc'[URI.]S F OR S'l'ui)L.Ln'l.Sol..- TN'froi)UC'i'o RY UNI V I.]RSI'i'y I' H YSICS

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THE THEORY of QUANTUM MECHANICS: Sflvtin LI.,Lc'[URI.]S F OR S'l'ui)L.Ln'l.Sol..- TN'froi)UC'i'o RY UNI V I.]RSI'i'y I' H YSICS THE THEORY OF QUANTUM MECHANICS: SFlVtiN LI.,lC'[URI.]S F OR S'l'UI)l.lN'l.SOl..- TN'fROI)UC'I'O RY UNI V I.]RSI'I'Y I' H YSICS by I)aniel'l'.Gillespie ResearchDepartment; Naual WeaponsCenter China Loke, California gSSSs Approvedfor publicrelease; distribution is unlimited" Preparedfor the Final Reportof the Carleton College Conference (August lg88) of the lntroductory University Physics Project I'REI.'ACE I came to write these lectures in the course of participating in the [ntroductory University Physics Project (IUPP), which was organized in 1987 by the American Ph-vsicalSociet-v and the American Association of Physics Teachers under the sponsorship of the Nationarl Science Foundation. The ultimate goarlof IUPP is to develop some new models for the introductory, calculus-based, university physics course. As envisioned by IUPP chairman John lligden, the new course models should have a "leaner story line," and should if at all possible contain something meaningful about quantum mechanics. My own involvement in tUPP is as a membcr of the working group on quanturn mechanics,which is chilired by llugen Merzbacher and co-chaired by Thomas Moore. I musl emphusize tlwt lhe set of'lecture,spresente.d here. is rutt patterned after the syllabus 'l'he recommended hy the IUPP quctntum mechanics worhin14!:roup. approach taken in these lectures is frankly too abstract and too formal, too long on theory and too short on applications, for mttst of thc students rvho tuke the introductory carlculus-basedphysics course. But.sorne of those students, those with a more mathemartical and philosophical bent, might possibly prolit flrom thcse lccturcs. And pcrhaps also some physics tcachcrs might find onc or more fleaturesof thesc lectures worth incorporating into their own lecturesand rvritings. Wh-v should a univcrsity student bother to learn quantum mcchanics? For students intent on becoming prolessional physicists, the motive harsalways been clear and compelling: [f -vou don't learn quantum mcchanics, then you can't join the club. Obviously, a different motive must impel students on a broadcr spectrum, and pcrhaps should impcl physics majors as wcll. The motive that I have tried to implicitly establish in these lectures is roughly as [ollows. Nature has been found to behave, on the microscale, in ways thart seem to utterly dely comrnon sense. Mankind has thus been presented with an immense chtrllenge: N{akc scnse of Nature on thc microscarle!Quantum mcchanics, within carefully prescribed limits, has successfullymet that challenge. So the theory of quantum mechanicsrepresents an intellectuarlachievement of truly heroic proportions, arnd may rightly be regarded as "a principal jewel in the cultural crown of civilization." To appreciate the beauty and mystery of that cultural jewel is alone sufficient reason for any student of an.y university to undcrtake r.rscrious study of quantum mechanics. It seems that there are two arpproachcsto tcaching physics in general and quantum rnechanics in parrticular: In the ax,ioma,ticapproerch, one first sets forth a minimal number of assumptions or axioms, and one then rigorously deduces their various consequences. Ily contrast, in the organic approach one pays little attention to logical structure, and simply allows the various ftrctsof the theory to arrange themselves in whatever way seems convenient. While I have respect for the organic approach, and in fact suspect th.rt physicists of that camp are more likely than their opposites to make significant advances in physics, I havc delibcratcly writtcn these lectures in the axiomatic vein. I bclievc that students with an aptitude for mathematics and logic.rl reasoning strongly prefer t<t learn by thc axiornatic 'l'he approach. "axioms" of querntum mechanics are presented here in the torm of fiue rules; Lhe first four rulcs are stated in Lecturc 3, and the fifth rule is stated in [,ccture 6. The lang,uulleof those rules, which provides thc allowed mechanisms of inference, is l,he mathemartical langunge 'l'h.rt of lincar algcbra, which I have prcfcrred hcre to call "gencralizcd vcctor thcnry." language is developed, in I think a rather novel way, in Lecture 2. Lecture 2 is therefore a makc-or-break lccture for thc whole scries. Although each of thc scvcn lectures can bc rearl in tlne hour, l think nrostwill takc lruo50-nrinutc lccturc sessionslo successfullydetiuer. Lecttrrcs 5 and 7 are lhe most challenging of the scrics, and could be omittcd for a lcss demanding fivc- lecture sequence;however, u glance at the table of contents will show that omission of Lecturcs 5 and 7 will cut out several major topics. u I feel obliged to forcwarn teachers who are contemplating using these lectures that I have taken what some might consider to be an unnecessarily doctrinaire position, narmely, that obseruablesfor microscale systems do not ulways haue ualue,s. [n [act, I make thert premise to be a principal motive for devising quantum mechanics. A teacher who feels that lhis premise is unwarranted, or even false, will nol like these lectures. []ut bcfore such a teacher larysthese pages aside for that reason, I would respectfully entreat a reading of the entertaining article by N. David Mermin in the April 1985 issue of Physics Today. In that article, Mermin discusscs the implictrtions of Bell's theorem in terms of trn idealized cxperiment thut is equivalent to an electron-pair spin-correlation experiment, but which is happily dcvoid of the esotericjargon oI quantum physics. Mcrmin shows with rernarkablc clarit.v th.rt there is simply no way of expleriningthe results of his experiment without doing great violence to one's sense of "reasonableness." I think that a defensible interpretartion of the specific conclusions o[ the 'lhere Mermin experiment is this: is no way to assign fixed values (red or green) to each observable (1, 2 and 3) o[ both particles in such a way that the experimental data can be quantitatively acounted for, hence, we cannot generall-v ascribe simultaneous values, euen unh,nown ualues, to non-commuting observables. This is onc of several non-common-sensical features of quantum mccharnicsthat, have bccn exposed by.Iohn []cll's work in 1964, and more rccently confirmcd experimcntally by Alain Aspect and coworkers. I believe it is wrong not to bc up-front with studcnts about such issucs" I belicve it is wrong to pretend to students, on the prctcxt of "doing physics rather than philosophy," that although quantum mechanicsmay seem strangc and unusual there is nothing about it that should disturb our view of Reality. Indeed, I feeI th.rt the strong claim which quanlum mcchanics hars to prolound cultural significatnce stems largely lrom the fact that il hus raised serious and as yet unanswered questions about Rculity. We nccd not necessarily rnuke a philosophical study of such issucs in a physics course, arndwe do not do that in these lectures, but we should not deny the existence of these issues. As evidence in these lectures for the "fact" thut ohservablcs for microscale systems do not alwarys have values, I havc invoked thc doublc-slit experiment. Thc cvidencc provided by that experiment is strong, but not unassailnble. Evidence of a more compelling nature would have been providcd by an elcctron-pair spin-corrclation experimcnt, but unfortunatel-v etsattisfactory 'Ihe quantum analysis of such an experiment lics beyond the reach of thcse lectures" double-slit experiment on the o[her harndcunbe analyzed within the framework of thcse lcctures, arndin fact it provides us with a thcmatic bridge from Lecture I to l,ecture 7. As to the treatment givcn in thcsc lectures of the quantum theory itself, I havc indccd organized and presented the ingredients of the theory in a very different way than is usually done. Ilut the basic view of quantum mechanics taken in these lectures is quite orthodox, arndis fully in line with what one will find in such standard textbooks as l)irac, lllessiah and Merzbacher" I have tricd to prescnt in these lcctures a highly simplified and hcnce unconventional rendering of conventional quantum theory. I would like to th:rnk John lligdcn and Ougen Merzbacher for their encouragement in preparing these lectures. I am happy to acknowledge the benelits of conversations with other physicists who participated in the 1988 IUPP conferences at llarvey Mudd College and Carlcton College, cspecially those in the quantum mechanics working group. And I want to thank Ron [)crr, [lead of thc RcscarrchDcpartmcnt of thc Naval Wcapons Center, for allowing me to perrticip.rte in IUPP and develop these lectures within the frermework of the Cen[er's lndependent Rcsearch [)rogram" China l,ilkc, Cal iflornra I)."1. Gillcspic May, 1989 Ilr CONTENTS paSe Prelace Lecture l. Introduction and Motivation I l.l 'lhe fundamentalproblem of mechantcs. I l.2 The failure of classicarlmechanics on the microscale I 1.3 The aim and planof theselcctures 3 1.4Complcx numbcrs 4 Lecture 2. The Mathematics of Generalized Vectors 6 2"I Vectors,scalars, components, basis expansions 6 2.2 Operators,lincarity, eigenvectorsand eigenvalues,eigenbarses . l0 2.3Summarv ... t2 Lecture 3. Quantum Mechanics: Statics I l3 3.1 Statestrndobservables,vectorsandoperators ....... 13 3.2 ltesultsand effectsof measuremcnts.. 15 3.3AnswertotheF'POMforl=0.... ....... 17 Lecture 4. Quantum Mechanics: Statics Il l8 4.1 Recapitulation r8 4.2 Compatible and incompatible observables 20 4.3 Intcrference. The "value" of an observarble 2T f LectureS. Quantum Mechanics: Statics III 23 5.1 The position and momentum opcrators. Wave lunctions 23 5.2 The wave-particleduality 25 5.3 The lleiscnbergUnccrtainty Principlc ,.7 Lectuie 6. Quantum Mechanics: l)ynamics I 29 'l'he 6.1 Hamiltonian operator and time-evolution 29 6.2 Conserved quantities. Stationary states 31 6.3 Answerto the FPOM lor any I > 0 32 6.4 '['ransitionamplitudes arnd virtual processes 33 t LectureT.
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