Quantum mechanics: classical results, modern systems, and visualized examples

722 Pages·2006·6.617 MB·English

by  Richard Robinett

Quantum Mechanics
Classical Results,Modern Systems,and
Visualized Examples
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Quantum
Mechanics
Classical Results, Modern
Systems, and Visualized
Examples
Second Edition
Richard W. Robinett
Pennsylvania State University
1
3
GreatClarendonStreet,OxfordOX26DP
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Robinett,RichardW.(RichardWallace)
Quantummechanics:classicalresults,modernsystems,and
visualizedexamples/RichardW.Robinett.—2nded.
p.cm.
ISBN-13:978–0–19–853097–8(alk.paper)
ISBN-10:0–19–853097–8(alk.paper)
1.Quantumtheory. I. Title.
QC174.12.R62006
530.12—dc22 2006000424
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
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ISBN 0–19–853097–8 978–0–19–853097–8
10 9 8 7 6 5 4 3 2 1
Preface to the Second Edition
Oneofthehallmarksofscienceisthecontinualquesttorefineandexpandone’s
understandingandvisionof theuniverse,seekingnotonlynewanswerstoold
questions,but also proactively searching out new avenues of inquiry based on
pastexperience.Inmuchthesameway,teachersof science(includingtextbook
authors)canandshouldexplorethepedagogyoftheirdisciplinesinascientific
way,maintainingandstreamliningwhathasbeendocumentedtowork,butalso
improving,updating,andexpandingtheireducationalmaterialsinresponseto
newknowledgeintheirfields,inbasic,applied,andeducationalresearch.Forthat
reason,IamverypleasedtohavebeengiventheopportunitytoproduceaSecond
Editionofthistextbookonquantummechanicsattheadvancedundergraduate
level.
The First Edition of Quantum Mechanics had a number of novel features,
so it may be useful to first review some aspects of that work, in the context
of this Second Edition. The descriptive subtitle of the text, Classical Results,
ModernSystems,andVisualizedExamples,was,andstillis,intendedtosuggesta
numberoftheinter-relatedapproachestotheteachingandlearningofquantum
mechanicswhichhavebeenadoptedhere.
• Many of the expected familiar topics and examples (the Classical Results)
foundinstandardquantumtextsareindeedpresentinbotheditions,butwe
alsocontinuetofocusextensivelyontheclassical–quantumconnectionasone
ofthebestwaystohelpstudentslearnthesubject.Topicssuchasmomentum-
spaceprobabilitydistributions,time-dependentwavepacketsolutions,andthe
correspondenceprinciplelimitoflargequantumnumberscanallhelpstudents
usetheirexistingintuitiontomakecontactwithnewquantumideas;classical
wavephysicscontinuestobeemphasizedaswell,withitsownseparatechapter,
forthesamereason.Additionalexamplesof quantumwavepacketsolutions
havebeenincludedinthisnewEdition,aswellasaself-containeddiscussion
of theWignerquasi-probability(phase-space)distribution,designedtohelp
make contact with related ideas in statistical mechanics,classical mechanics,
andevenquantumoptics.
• An even larger number of examples of the application of quantum mech-
anics to Modern Systems is provided, including discussions of experimental
realizationsofquantumphenomenawhichhaveonlyappearedsincetheFirst
Edition.Advancesinsuchareasasmaterialsscienceandlasertrapping/cooling
vi PREFACETOTHESECONDEDITION
havemeantalargenumberofquantumsystemswhichhavehistoricallybeen
onlyconsideredas“textbook”exampleshavebecomephysicallyrealizable.For
example,the“quantum bouncer”,once discussed only in pedagogical journ-
als, has been explored experimentally in the Quantum states of neutrons in
É
theEarth’sgravitationalfield. Theproductionof atomicwavepacketswhich
Ê
exhibit the classical periodicity of Keplerian orbits is another example of a
ClassicalResult whichhasbecomeaModernSystem.
The ability to manipulate nature at the extremes of small distance (nano-
and even atomic-level) and low temperatures (as with Bose–Einstein con-
densates) implies that a knowledge of quantum mechanics is increasingly
important in modern physical science, and a number of new discussions of
applicationshavebeenaddedtoboththetextandtotheProblems,including
onesonsuchtopicsasexpanding/interferingBose–Einsteincondensates,the
quantumHalleffect,andquantumwavepacketrevivals,allinthecontextof
familiartextbooklevelexamples.
• We continue to emphasize the use of Visualized Examples (with 200 figures
included) to reinforce students’conceptual understanding of the basic ideas
andtoenhancetheirmathematicalfacilityinsolvingproblems.Thisincludes
notonlypictorialrepresentationsofstationarystatewavefunctionsandtime-
dependent wave packets, but also real data. The graphical representation of
suchinformationoftenprovidesthemapofthemeetinggroundofthesome-
timesarcaneformalismof atheorist,theobservationsof anexperimentalist,
andtherestof thescientificcommunity;theabilityto“followsuchmaps”is
animportantpartofaphysicseducation.
MotivatedinthisEdition(evenmorethanbefore)byresultsappearingfrom
Physics Education Research (PER), we still stress concepts which PER stud-
ies have indicated can pose difficulties for many students, such as notions of
probability, reading potential energy diagrams, and the time-development of
eigenstatesandwavepackets.
Aswithanytextbookrevision,theopportunitytostreamlinethepresentation
andpedagogy,basedonfeedbackfromactualclassroomuse,isoneof themost
importantaspectsofanewEdition,andIhavetakenthisopportunitytoremove
sometopics(movingthem,however,toanaccompanyingWebsite)andadding
newones.NewsectionsonTheWignerQuasi-ProbabilityDistribution(andmany
related problems), an Infinite Array of δ-functions: Periodic Potentials and the
DiracComb,Time-DependentPerturbationTheory,andTimescalesinBoundState
É
ThetitleofapaperbyV.V.Nesvizhevskyetal.(2002).Nature415,297.
Ê
SeeYeazelletal.(1989).
PREFACETOTHESECONDEDITION vii
Systems: Classical Period and Quantum Revival Times reflect suggestions from
various sources on (hopefully) useful new additions.A number of new in-text
Examples andend-of-chapterProblems havebeenaddedforsimilarreasons,as
wellasanexpandedsetofAppendices,ondimensionsandmathematicalmethods.
An exciting new feature of the Second Edition is the development of a Web
Ë
site insupportofthetextbook,forusebybothstudentsandinstructors,linked
Ì
from the Oxford University Press web page for this text. Students will find
manyadditional(extended)homeworkproblemsintheformofWorksheets on
bothformalandappliedtopics,suchas“slowlight”,femtosecondchemistry,and
quantumwavepacketrevivals.AdditionalmaterialintheformofSupplementary
Chaptersonsuchtopicsasneutrinooscillations,quantumMonteCarloapprox-
imationmethods,supersymmetryinquantummechanics,periodicorbittheory
ofquantumbilliards,andquantumchaosareavailable.
For instructors, copies of a complete Solutions Manual for the textbook, as
wellasWorksheetSolutions,willbeprovidedonamoresecureportionofthesite,
inadditiontocopiesoftheTransparencies forthefiguresinthetext.An85-page
GuidetothePedagogicalLiteratureonQuantumMechanicsisalsoavailablethere,
surveyingarticlesfromTheAmericanJournalofPhysics,TheEuropeanJournal
of Physics, and The Journal of Chemical Education from their earliest issues,
through the publication date of this text (with periodic updates planned.) In
addition,aquantummechanicsassessmenttest(theso-called QuantumMech-
anicsVisualizationInstrument orQMVI)isavailableattheInstructorssite,along
with detailed information on its development and sample results from earlier
educational studies. Given my long-term interest in the science, as well as the
pedagogy,of quantum mechanics,I trust that this site will continually grow in
bothsizeandcoverageasnewandupdatedmaterialsareadded.Informationon
accessingtheInstructorsareaof theWebsiteisavailablethroughthepublisher
attheOxfordUniversityPressWebsitedescribingthistext.
IamverygratefultoallthosefromwhomIhavehadhelpinlearningquantum
mechanics over the years, including faculty and fellow students in my under-
graduate, graduate, and postdoctoral days, current faculty colleagues (here at
PennStateandelsewhere),myownundergraduatestudentsovertheyears,and
numerousauthorsoftextbooksandbothresearchandpedagogicalarticles,many
of whomIhavenevermet,buttowhomIowemuch.Iwouldliketothankall
thosewhohelpedverydirectlyintheproductionof theSecondEditionof this
text,specifically including those who provided useful suggestions for improve-
ment or who found corrections, namely, J. Banavar, A. Bernacchi, B. Chasan,
Ë
Seerobinett.phys.psu.edu/qm
Ì
Seewww.oup.co.uk
viii PREFACETOTHESECONDEDITION
J. Edmonds,M. Cole,C. Patton,and J.Yeazell. I have truly enjoyed recent col-
laborations with both M. Belloni and M. A. Doncheski on pedagogical issues
relatedtoquantumtheory,andsomeofourrecentworkhasfounditswayinto
theSecondEdition(includingthecover)andIthankthemfortheirinsights,and
patience.
Noworkdoneinaprofessionalcontextcanbeseparatedfromone’spersonal
life (nor should it be) and so I want to thank my family for all of their help
and understanding over my entire career, including during the production of
thisnewEdition.TheFirstEditionofthistextwasthoroughlyproof-readbymy
mother-in-law(NancyMalone)whograciouslytriedtoteachmetheproperuse
oftheEnglishlanguage;herrecentpassinghassaddenedusall.Myownmother
(BettyRobinett)hasbeen,andcontinuestobe,thesinglemostimportantrole
model in my life—both personal and professional—and I am deeply indebted
toherfarmorethanIcaneverconvey.Finally,tomywife(Sarah)andchildren
(JamesandKatherine),Igivethankseverydayfortherichnessandjoytheybring
tomylife.
RichardRobinett
December,2005
StateCollege,PA
Contents
Part I The Quantum Paradigm 1
1 A First Look at Quantum Physics 3
1.1 HowthisBookApproachesQuantumMechanics 3
1.2 EssentialRelativity 8
1.3 QuantumPhysics:(cid:1)asaFundamentalConstant 10
1.4 SemiclassicalModeloftheHydrogenAtom 17
1.5 DimensionalAnalysis 21
1.6 QuestionsandProblems 23
2 Classical Waves 34
2.1 TheClassicalWaveEquation 34
2.2 WavePacketsandPeriodicSolutions 36
2.2.1 GeneralWavePacketSolutions 36
2.2.2 FourierSeries 38
2.3 FourierTransforms 43
2.4 InvertingtheFouriertransform:theDiracδ-function 46
2.5 DispersionandTunneling 51
2.5.1 VelocitiesforWavePackets 51
2.5.2 Dispersion 53
2.5.3 Tunneling 56
2.6 QuestionsandProblems 57
3 The Schrödinger Wave Equation 65
3.1 TheSchrödingerEquation 65
3.2 PlaneWavesandWavePacketSolutions 67
3.2.1 PlaneWavesandWavePackets 67
3.2.2 TheGaussianWavePacket 70
3.3 “Bouncing”WavePackets 75
3.4 NumericalCalculationofWavePackets 77
3.5 QuestionsandProblems 79