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Lecture Notes in Physics EditorialBoard R.Beig,Wien,Austria W.Beiglbo¨ck,Heidelberg,Germany W.Domcke,Garching,Germany B.-G.Englert,Singapore U.Frisch,Nice,France P.Ha¨nggi,Augsburg,Germany G.Hasinger,Garching,Germany K.Hepp,Zu¨rich,Switzerland W.Hillebrandt,Garching,Germany D.Imboden,Zu¨rich,Switzerland R.L.Jaffe,Cambridge,MA,USA R.Lipowsky,Potsdam,Germany H.v.Lo¨hneysen,Karlsruhe,Germany I.Ojima,Kyoto,Japan D.Sornette,Nice,France,andZu¨rich,Switzerland S.Theisen,Potsdam,Germany W.Weise,Garching,Germany J.Wess,Mu¨nchen,Germany J.Zittartz,Ko¨ln,Germany TheLectureNotesin Physics TheseriesLectureNotesinPhysics(LNP),foundedin1969,reportsnewdevelopments in physics research and teaching – quickly and informally, but with a high quality and theexplicitaimtosummarizeandcommunicatecurrentknowledgeinanaccessibleway. Bookspublishedinthisseriesareconceivedasbridgingmaterialbetweenadvancedgrad- uatetextbooksandtheforefrontofresearchandtoservethreepurposes: (cid:129) tobeacompactandmodernup-to-datesourceofreferenceonawell-definedtopic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) tobeasourceofadvancedteachingmaterialforspecializedseminars,coursesand schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Pro- ceedingswillnotbeconsideredforLNP. VolumespublishedinLNParedisseminatedbothinprintandinelectronicformats,the electronicarchivebeingavailableatspringerlink.com.Theseriescontentisindexed,ab- stractedandreferencedbymanyabstractingandinformationservices,bibliographicnet- works,subscriptionagencies,librarynetworks,andconsortia. ProposalsshouldbesenttoamemberoftheEditorialBoard,ordirectlytothemanaging editoratSpringer: ChristianCaron SpringerHeidelberg PhysicsEditorialDepartmentI Tiergartenstrasse17 69121Heidelberg/Germany [email protected] J.G. Muga R. Sala Mayato ´ I.L. Egusquiza (Eds.) Time in Quantum Mechanics Second Edition Editors J.G.Muga Í.L.Egusquiza DepartamentodeQu´ımica-F´ısica F´ısicaTeorikoarenSalla FacultaddeCienciayTecnologia ZientziaetaTeknologia UniversidaddelPa´ısVasco EuskalHerrikoUnibertsitatea Apdo644 644PK. 48080Bilbao,Spain Bilbao,Spain [email protected] [email protected] R.SalaMayato UniversidaddeLaLaguna DepartamentodeF´ısicaFundamentalII 38203LaLaguna(S/CdeTenerife),Spain [email protected] J.G.Muga,R.SalaMayatoandÍ.L.Egusquiza(Eds.),TimeinQuantumMechanics,Lect. NotesPhys.734(Springer,BerlinHeidelberg2008),DOI10.1007/978-3-540-73473-4 ThisfirsteditionofthisbookwasoriginallypublishedasVol.m72intheseriesLecture NotesinPhysics LibraryofCongressControlNumber:2007937501 ISSN0075-8450 ISBN978-3-540-73472-7 SpringerBerlinHeidelbergNewYork ISBN978-3-540-43294-41stEditionSpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved, whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember 9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagBerlinHeidelberg2002,2008 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandIntegrausingaSpringerLATEXmacropackage Coverdesign:eStudioCalamarS.L.,F.Steinen-Broo,Pau/Girona,Spain Printedonacid-freepaper SPIN:12083183 543210 Preface to the Second Edition Time and quantum mechanics are, by and unto themselves, words of such force and attraction that the first edition of this book quite rapidly went out of print. The idea of bringing out a second edition became, thus, more compellingastime(indeed)flewon.Amongthedifferentpossiblities(awholly new text, a mere reprinting) we finally settled on a middle way, namely that each of the contributors has decided to what extent their respective chapters needed updating. One of the reasons for this decision is that the field has indeed been evolving, and the increase in editions of the TQM workshops heldinLaLagunaandBilbaohasprovideduswithquiteanumberofpossible contributorsforpreparing,inadditiontothissecondedition,asecondvolume with entirely new chapters. But more important is that the results and texts of the authors have withstood the passage of time and have weathered very wellits ravages.All chaptershavebeen updated, however,in orderto include new results and bibliography. We hope that this new edition meets with as much success as its prede- cessor and, as we write this, that in the future we can provide the interested reader with more contributions of other participants in our TQM workshops. We are grateful to R. F. Snider and G. N. Flemming for pointing out several typos and errors of the first edition. We also acknowledge additional support from Ministerio de Educacioń y Ciencia (Grants BFM2003-01003, FIS2006-10268-C03-01andFIS2004/05687),GobiernoAutońomodeCanarias (PI2004/025),and UPV-EHU (Grant 00039.310-15968/2004). Bilbao - La Laguna, J. Gonzalo Muga May 2007 Rafael Sala Mayato In˜igo L. Egusquiza Preface to the First Edition Time and quantum mechanics have, each of them separately, captivated sci- entists and laymen alike, as shown by the abundance of popular publications on“time”oronthemanyquantummysteriesorparadoxes.Wetoohavebeen seduced by these two topics, and in particular by their combination. Indeed, the treatment of time in quantum mechanics is one of the important and challenging open questions in the foundations of quantum theory. This book describes the problems, and the attempts and achievements in defining, formalizing and measuring different time quantities in quantum theory, such as the parametric (clock) time, tunneling times, decay times, dwelltimes,delaytimes,arrivaltimes,orjumptimes.Thetheoreticalanalysis of several of these quantities has been controversial and is still subject to debate. For example, there are literally hundreds of research papers on the tunneling time. In fact, the standard recipe to link the observables and the formalism does not seem to apply, at least in an obvious manner, to time observables.Thishasposedthechallengeofextendingthedomainofordinary quantum mechanics. Thedifficultiesindealingwithtimeinquantumtheoryweremadeexplicit veryearlyon,mostclearlybyPauliinhisfamous“theorem,”whichseemedto impose a serious limitation on the possibility of formulating time as a quantum observable and which has hindered the investigationof time in quantum mechanics for many years. Another disturbing historical landmark is the dis- covery of quantum Zeno’s effect, a paradox that arises when attempting to find an algorithm for computing time probabilities by means of frequently repeated measurements: in the continuum limit these measurements “freeze” theoccurrenceofevents.Morerecently,however,inthelast15years,therehas been much interest in overcomingthese difficulties. Researchers from atomic, molecular, and optical sciences, or from mesoscopic, high-energy,and mathe- maticalphysics,haveconvergedtostudydifferenttimequantitiesinquantum mechanics. In addition, modern laser technology and the ability to manipu- lateatomicandmolecularmotionsandthe internalstateofquantumsystems allownowadaysthe experimentalrealizationofsome ofthe questions ontime VIII Preface to theFirst Edition in quantum mechanics.As a matter of fact, a number oftime observablesare alreadyroutinely measuredin laboratories,for example arrivaltimes in time- of-flightexperiments,butthe theoreticalfoundationofthesemeasurementsis still being discussed. Thebookreflectsagoodnumberoftheserecenttrends,butitisnotanen- cyclopedicattempttocoverallthe manydifferentopenquestions,evenentire fields,wheretimeplaysanimportantrole,frequentlyapuzzlingone,inquan- tum theory. It is not possible to cover all these subjects in a comprehensive manner unless the treatment becomes very superficial, so we abandonedthat objective. While many of these topics are strongly related to the ones pre- sentedhere,othersarebasicallydecoupled.Wehavethuspreferredto makea more compact selection of topics based on the workshops on “Time in Quan- tum Mechanics” (TQM) in La Laguna and Bilbao.Our aim is to edit further volumes containing new aspects of quantum time not treated in the present one with additional material from old and new TQM workshops. Most authors are or have been involved in the definition or study of “characteristictimes”or“timeobservables”inquantummechanics,andmore specifically on tunneling and/or arrival times, which have been central subjects in the TQM workshop series. Even so, this book goes clearly beyond these two seed subjects. Thorny issues such as the relations between quantum mechanics and the world of classical events, the theory of measurement, hiddenvariabletheories,time–energyuncertaintyprinciples,superluminalef- fects,orextensionsofthestandardformalismarefrequentlyingredientsofthis research, as demonstrated in several chapters. It became evident to us that these topics, surrounding the central ones, had to be addressed too, in order to get a better handle on the originalproblems. Understanding the mysteries of time in quantum mechanics is thus inextricably linked to understanding quantum mechanics itself. The chapters that follow are reviews that may serve both as an introduc- tory guide for the non-initiated and as a useful tool for the expert. We have essentially allowed full freedom to each contributing author in choosing their presentation and emphasis. This has the major advantage of freshness and, in this manner, the actual state of the issue is much better presented: in this field,thereisahostofdiverseapproaches,tools,languages,notations,nomenclature ...and results.Nonetheless,there has been progress,andconsensusin some topics has been achieved. This consensus is also well reflected in the following chapters, as the reader will soon discover. The main disadvantage is of course that the presentation is not fully unified, and that notation and nomenclature are occasionally divergent. In particular, terms such as “tunneling,” “traversal,” “delay,” or “arrival” times are used in different ways in different chapters. Finally,wewouldliketoacknowledgeourcoworkersD.Alonso,A.D.Baute, S. Brouard, M. Buttiker, J.A. Damborenea, V. Delgado, C.R. Leavens, D.Mac´ıas,J.P.Palao,A.Pérez,D.Sokolovski,R.F.Snider,andG.W.Weifor all their support in timing quantum mechanics and for so many good times Preface to theFirst Edition IX with them. We are also indebted to the funding agencies of the TQM workshop series: Universidad de La Laguna, Ministerio de Educacioń y Cultura, Ministerio de Ciencia y Tecnolog´ıa,EuropeanUnion(throughthe Canadian– European Research Initiative on Nanostructures, CERION), and the Basque Government. Bilbao - La Laguna, J. Gonzalo Muga August 2001 Rafael Sala Mayato In˜igo L. Egusquiza Contents 1 Introduction J. Gonzalo Muga, Rafael Sala Mayato, and In˜igo L. Egusquiza......... 1 1.1 Role of Time in the Early Days of Quantum Theory ............. 1 1.2 The Thirties and Forties ..................................... 5 1.3 The Fifties ................................................. 7 1.4 The Sixties ................................................. 12 1.5 The Seventies: the Zeno Effect, TOA Distributions, POVMs ...... 16 1.6 Some Recent Trends ......................................... 18 1.7 Discussion.................................................. 24 References ...................................................... 25 2 Characteristic Times in One-Dimensional Scattering J. Gonzalo Muga................................................. 31 2.1 Introduction ................................................ 31 2.2 Scattering Theory in 1D...................................... 33 2.3 A Measure of the Collision Duration: The Dwell Time ........... 42 2.4 Importance of the Phases: Time Delays ........................ 46 2.5 Time Dependence of SurvivalProbability:ExponentialDecay and Deviations ................................................. 55 2.6 Other Characteristic Times of Wave Propagation ............... 61 References ...................................................... 69 3 The Time–Energy Uncertainty Relation Paul Busch...................................................... 73 3.1 Introduction ................................................ 73 3.2 The Three-fold Role of Time in Quantum Theory ............... 73 3.3 Relation Between External Time and Energy Spread............. 76 3.4 Relations Involving Intrinsic Time ............................. 82 3.5 Quantum Clock ............................................. 89 3.6 Relations Based on Time Observables.......................... 91 3.7 Conclusion .................................................103 References ......................................................104