Time in Quantum Mechanics

455 Pages·2007·9.12 MB·english

by  J. Gonzalo Muga

J.G. Muga
R. Sala Mayato
´
I.L. Egusquiza (Eds.)
Time in Quantum
Mechanics
Second Edition
Editors
J.G.Muga ´I.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.SalaMayatoand´I.L.Egusquiza(Eds.),TimeinQuantumMechanics,Lect.
NotesPhys.734(Springer,BerlinHeidelberg2008),DOI10.1007/978-3-540-73473-4
ThisfirsteditionofthisbookwasoriginallypublishedasVol.m72intheseriesLecture
NotesinPhysics
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ISBN978-3-540-43294-41stEditionSpringerBerlinHeidelbergNewYork
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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 Educaci´on y Ciencia (Grants BFM2003-01003,
FIS2006-10268-C03-01andFIS2004/05687),GobiernoAuto´nomodeCanarias
(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 quan-
tum 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 sub-
jects in the TQM workshop series. Even so, this book goes clearly beyond
these two seed subjects. Thorny issues such as the relations between quan-
tum 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,nomen-
clature ...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 “tun-
neling,” “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.P´erez,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 work-
shop series: Universidad de La Laguna, Ministerio de Educacio´n 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
XII Contents
4 Jump Time and Passage Time: The Duration
of a Quantum Transition
Lawrence S. Schulman............................................107
4.1 Introduction ................................................107
4.2 Jump Time.................................................109
4.3 Corroborationsof the Definition...............................111
4.4 Passage Time ...............................................116
4.5 Experimental Discrimination among Quantum Measurement
Theories and “Special States”.................................120
4.6 Discussion..................................................124
References ......................................................126
5 Bohm Trajectory Approach to Timing Electrons
C. Richard Leavens...............................................129
5.1 Introduction ................................................129
5.2 Motivation for Using a Trajectory Approach ....................130
5.3 Bohm’s Ontological Interpretation of Quantum Theory...........131
5.4 Conventional Approaches to Timing Quantum Particles from the
Perspective of Bohmian Mechanics ............................140
5.5 Spin-Dependent Arrival-Time Distributions for Nonrelativistic
Electrons...................................................147
5.6 Protective Measurements and Bohm Trajectories ...............150
5.7 Concluding Comments .......................................156
References ......................................................160
6 Decoherent Histories for Space–Time Domains
Jonathan J. Halliwell.............................................163
6.1 Introduction ................................................163
6.2 Decoherent Histories Approach to Quantum Theory .............167
6.3 Space–Time Coarse Grainings.................................170
6.4 Decoherence of Space–Time Coarse-Grained Histories in the
Quantum Brownian Motion Model.............................172
6.5 A Detector Model ..........................................178
6.6 A Comparison of the Decoherent Histories Result with the
Detector Result .............................................182
6.7 Timeless Questions in Quantum Theory........................184
6.8 Discussion..................................................187
6.9 An Update for the Second Edition.............................189
References ......................................................190
7 Quantum Traversal Time,PathIntegrals and‘Superluminal’
Tunnelling
Dmitri Sokolovski ................................................195
7.1 Introduction ................................................195
7.2 Path Decomposition. The Clocked Schro¨dinger Equation. Coarse
Graining ...................................................196
Contents XIII
7.3 Meters and Measurements. Uncertainty Relation ................200
7.4 Averages. Complex Times. Weak Measurements.................202
7.5 Examples: Free motion and Tunnelling ........................204
7.6 Semiclassical Limit. How Long Does it Take for a Particle to
Tunnel? ....................................................207
7.7 Larmor Clock as a Realistic Meter.............................208
7.8 TraversalTime Analysis......................................214
7.9 TraversalTime and the “Superluminal” Tunnelling ..............216
7.10 Relativistic TraversalTime ...................................220
7.11 “Superluminal” Paradox and the Speed of Information Transfer ...225
7.12 Concluding Remarks.........................................230
References ......................................................231
8 Quantum Clocks and Stopwatches
Rafael Sala Mayato, Daniel Alonso, and In˜igo L. Egusquiza ...........235
8.1 Introduction ................................................235
8.2 What is a Clock?............................................237
8.3 The Salecker–WignerClock...................................238
8.4 The Larmor Clock...........................................248
8.5 Other Clocks ...............................................256
8.6 Simple “Time-dependent” Clocks: the Kick Clock ...............263
8.7 Decoherence in Time ........................................270
References ......................................................275
9 The Local Larmor Clock, Partial Densities of States,
and Mesoscopic Physics
Markus Bu¨ttiker .................................................279
9.1 Introduction ................................................279
9.2 The Scattering Matrix and the Local Larmor Clock..............282
9.3 Absorption and Emission of Particles: Injectivities and Emissivities 286
9.4 Potential Perturbations ......................................289
9.5 Generalized Bardeen Formulae ................................290
9.6 Voltage Probe and Inelastic Scattering .........................291
9.7 AC Conductance of Mesoscopic Conductors.....................293
9.8 Transition from Capacitive to Inductive Response ...............294
9.9 Partial Density of States Matrix...............................296
9.10 Local Friedel Sum Rule ......................................300
9.11 Discussion..................................................301
References ......................................................301
10 “Standard” Quantum–Mechanical Approach
to Times of Arrival
In˜igo L. Egusquiza, J. Gonzalo Muga, and Andr´es D. Baute...........305
10.1 Introduction ................................................305
10.2 Kijowski’s Time-of-Arrival Distribution ........................306
10.3 POVMs ....................................................308
XIV Contents
10.4 The POVM of the Aharonov–BohmTime Operator..............315
10.5 Other Time Operators .......................................320
10.6 Arrival States...............................................323
10.7 Times of Arrival for Identical Particles .........................327
10.8 How can Kijowski’s distribution be measured? ..................329
10.9 Conclusions and Outlook.....................................329
References ......................................................330
11 Experimental Issues in Quantum–Mechanical
Time Measurement
Aephraim M. Steinberg ...........................................333
11.1 Time Operators Versus Real Measurements.....................333
11.2 Arrival-Time Measurements ..................................335
11.3 Dwell or Interaction Time Measurements .......................342
11.4 Weak Measurements .........................................347
11.5 Conclusion .................................................350
References ......................................................351
12 Microwave Experiments on Tunneling Time
Daniela Mugnai and Anedio Ranfagni ..............................355
12.1 An Overview of Theoretical Models of Tunneling in the
Electromagnetic Framework ..................................355
12.2 Sub cutoff Microwave Propagationin Waveguide ................374
12.3 Delay-Time Measurements in a Diffraction Experiment...........383
12.4 Tunneling Time in Frustrated Total Reflection ..................389
12.5 Concluding Remarks.........................................395
References ......................................................395
13 The Two-State Vector Formalism: An Updated Review
Yakir Aharonov and Lev Vaidman..................................399
13.1 Introduction ................................................399
13.2 Descriptions of Quantum Systems .............................399
13.3 Ideal Quantum Measurements................................405
13.4 Weak Measurements .......................................409
13.5 The Quantum Time-Translation Machine.......................424
13.6 Time Symmetry.............................................433
13.7 Protective Measurements.....................................441
13.8 TheTSVFandtheMany-WorldsInterpretationofQuantumTheory443
References ......................................................444
Index..........................................................449