Introductory Quantum Mechanics: A Traditional Approach Emphasizing Connections with Classical Physics

641 Pages·2017·8.002 MB·English

by  Paul R. Berman

UNITEXT for Physics
Paul R. Berman
Introductory
Quantum
Mechanics
A Traditional Approach Emphasizing
Connections with Classical Physics
UNITEXT for Physics
Serieseditors
PaoloBiscari,Milano,Italy
MicheleCini,Roma,Italy
AttilioFerrari,Torino,Italy
StefanoForte,Milano,Italy
MortenHjorth-Jensen,Oslo,Norway
NicolaManini,Milano,Italy
GuidoMontagna,Pavia,Italy
OresteNicrosini,Pavia,Italy
LucaPeliti,Napoli,Italy
AlbertoRotondi,Pavia,Italy
UNITEXTforPhysicsseries,formerlyUNITEXTCollanadiFisicaeAstronomia,
publishestextbooksandmonographsinPhysicsandAstronomy,mainlyinEnglish
language, characterized of a didactic style and comprehensiveness. The books
publishedinUNITEXTforPhysicsseriesareaddressedtograduateandadvanced
graduate students, but also to scientists and researchers as important resources for
theireducation,knowledgeandteaching.
Moreinformationaboutthisseriesathttp://www.springer.com/series/13351
Paul R. Berman
Introductory Quantum
Mechanics
A Traditional Approach Emphasizing
Connections with Classical Physics
123
PaulR.Berman
UniversityofMichigan
AnnArbor,MI,USA
ISSN2198-7882 ISSN2198-7890 (electronic)
UNITEXTforPhysics
ISBN978-3-319-68596-0 ISBN978-3-319-68598-4 (eBook)
https://doi.org/10.1007/978-3-319-68598-4
LibraryofCongressControlNumber:2017954936
©SpringerInternationalPublishingAG2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Iwouldliketodedicatethisbookto
DebraBerman,mywifeof30+years,
forhersupportandpositiveoutlook onlife.
Preface
This book is based on junior and senior level undergraduate courses that I have
givenatbothNewYorkUniversityandtheUniversityofMichigan.Youmightask,
in heavens name, why anyone would want to write yet another introductory text
on quantum mechanics. And you would not be far off base with this assessment.
Therearemanyexcellentintroductoryquantummechanicstexts.Moreover,withthe
materialavailableontheinternet,youcanaccessalmostanytopicofyourchoosing.
Therefore,Imustagreethatthereareprobablynocompellingreasonstopublishthis
text.Ihaveundertakenthistaskmainlyattheurgingofmystudents,whofeltthatit
wouldbehelpfultostudentsstudyingquantummechanics.
For the most part, the approach taken is a traditional one. I have tried to
emphasizetherelationshipofthequantumresultswiththoseofclassicalmechanics
and classical electromagnetism. In this manner, I hope that students will be able
togainphysicalinsightintothenatureofthequantumresults.Forexample,inthe
studyofangularmomentum,youwillseethattheabsolutesquaresofthespherical
harmonics can be given a relatively simple physical interpretation. Moreover, by
using the effective potential in solving problems with spherical symmetry, I am
able to provide a physical interpretation of the probability distributions associated
with the eigenfunctions of such problems and to interpret the structures seen in
scatteringcrosssections.Ialsotrytostressthetime-dependentaspectsofproblems
in quantum mechanics, rather than focus simply on the calculation of eigenvalues
andeigenfunctions.
Thebookisintendedtobeusedinayear-longintroductorycourse.Chapters1–
13 or 1–14 can serve as the basis for a one-semester course. I do not introduce
Dirac notation until Chap.11. I do this so students can try to master the wave
function approach and its implications before engaging in the more abstract
Dirac formalism. Dirac notation is developed in the context of a more general
approach in which different representations, such as the position and momentum
representations, appear on an equal footing. Most topics are treated at a level
appropriatetoanundergraduatecourse.Sometopics,however,suchasthehyperfine
interactions described in the appendix of Chap.21, are at a more advanced level.
Theseareincludedforreferencepurposes,sincetheyarenottypicallyincludedin
vii
viii Preface
undergraduate (or graduate) texts. There is a web site for this book, http://www-
personal.umich.edu/~pberman/qmbook.html, that contains an Errata, Mathematica
subroutines,andsomeadditionalmaterial.
Theproblemsformanintegralpartofthebook.Manyarestandardproblems,but
thereareafewthatmightbeunique tothistext.Quantum mechanics isadifficult
subjectforbeginningstudents.Ioftentellthemthatfallingbehindinacoursesuch
as this is a disease from which it is difficult to recover. In writing this book, my
foremost task has been to keep the students in mind. On the other hand, I know
that no textbook is a substitute for a dedicated instructor who guides, excites, and
motivatesstudentstounderstandthematerial.
IwouldliketothankBillFord,AaronLeanhardt,PeterMilonni,MichaelRevzen,
AlbertoRojo,andRobinShakeshaftfortheirinsightfulcomments.Iwouldalsolike
toacknowledgethemanydiscussionsIhadwithDuncanSteelontopicscontained
in this book. Finally, I am indebted to my students for their encouragement and
positive (as well as negative) feedback over the years. I am especially grateful to
theFulbrightfoundationforhavingprovidedthesupportthatallowedmetooffera
courseinquantummechanicstostudentsattheCollegeofScienceandTechnology
attheUniversityofRwanda.Myinteractionswiththesestudentswillalwaysremain
anindeliblechapterofmylife.
AnnArbor,MI,USA PaulR.Berman
Contents
1 Introduction................................................................. 1
1.1 ElectromagneticWaves .............................................. 2
1.1.1 RadiationPulses........................................... 4
1.1.2 WaveDiffraction .......................................... 6
1.2 BlackBodySpectrum:OriginoftheQuantumTheory ............ 7
1.3 PhotoelectricEffect .................................................. 9
1.4 BohrTheory.......................................................... 10
1.5 DeBroglieWaves .................................................... 13
1.6 TheSchrödingerEquationandProbabilityWaves.................. 15
1.7 MeasurementandSuperpositionStates ............................. 17
1.7.1 What Is Truly Strange About Quantum
Mechanics:SuperpositionStates......................... 17
1.7.2 TheEPRParadoxandBell’sTheorem .................. 18
1.8 Summary.............................................................. 21
1.9 Appendix:BlackbodySpectrum..................................... 21
1.9.1 BoxNormalizationwithFieldNodesontheWalls...... 22
1.9.2 PeriodicBoundaryConditions............................ 23
1.9.3 Rayleigh-JeansLaw....................................... 24
1.9.4 Planck’sSolution.......................................... 24
1.9.5 ApproachtoEquilibrium ................................. 27
1.10 Problems.............................................................. 28
2 MathematicalPreliminaries............................................... 33
2.1 ComplexFunctionofaRealVariable............................... 33
2.2 FunctionsandTaylorSeries ......................................... 34
2.2.1 FunctionsofOneVariable................................ 34
2.2.2 ScalarFunctionsofThreeVariables...................... 36
2.2.3 VectorFunctionsofThreeVariables ..................... 37
2.3 VectorCalculus....................................................... 38
2.4 ProbabilityDistributions............................................. 39
2.5 FourierTransforms................................................... 43
ix
x Contents
2.6 DiracDeltaFunction................................................. 46
2.7 Problems.............................................................. 49
3 Free-ParticleSchrödingerEquation:WavePackets ................... 53
3.1 ElectromagneticWaveEquation:Pulses............................ 54
3.2 Schrödinger’sEquation .............................................. 55
3.2.1 WavePackets.............................................. 56
3.2.2 Free-ParticlePropagator .................................. 62
3.3 Summary.............................................................. 65
3.4 Problems.............................................................. 65
4 Schrödinger’sEquationwithPotentialEnergy:Introduction
toOperators................................................................. 69
4.1 HamiltonianOperator................................................ 69
4.2 Time-IndependentSchrödingerEquation........................... 72
4.3 Summary.............................................................. 73
4.4 Appendix:SchrödingerEquationinThreeDimensions............ 73
4.5 Problems.............................................................. 75
5 Postulates and Basic Elements of Quantum Mechanics:
PropertiesofOperators.................................................... 77
5.1 HermitianOperators:EigenvaluesandEigenfunctions............. 78
5.1.1 EigenvaluesReal.......................................... 80
5.1.2 Orthogonality.............................................. 81
5.1.3 Completeness.............................................. 85
5.1.4 ContinuousEigenvalues .................................. 86
5.1.5 RelationshipBetweenOperators ......................... 88
5.1.6 CommutatorofOperators................................. 90
5.1.7 UncertaintyPrinciple...................................... 94
5.1.8 ExamplesofOperators.................................... 96
5.2 BacktotheSchrödingerEquation................................... 99
5.2.1 HowtoSolvetheTime-DependentSchrödinger
Equation ................................................... 99
5.2.2 Quantum-MechanicalProbabilityCurrentDensity...... 102
5.2.3 OperatorDynamics........................................ 104
5.2.4 SumofTwoIndependentQuantumSystems ............ 105
5.3 Measurements in Quantum Mechanics: “Collapse”
oftheWaveFunction ................................................ 106
5.4 Summary.............................................................. 108
5.5 Appendix:FromDiscretetoContinuousEigenvalues.............. 109
5.6 Problems.............................................................. 110
6 ProblemsinOne-Dimension:GeneralConsiderations,Infinite
Well Potential, Piecewise Constant Potentials, and Delta
FunctionPotentials......................................................... 115
6.1 GeneralConsiderations .............................................. 115
6.1.1 PotentialsinwhichV.x/>0andV.˙1/(cid:2)0......... 116