Quantum Mechanics: An Enhanced Primer

407 Pages·2022·5.373 MB·English

by  Bruce Cameron Reed

Bruce Cameron Reed
Quantum
Mechanics
An Enhanced Primer
Second Edition
Quantum Mechanics
Bruce Cameron Reed
Quantum Mechanics
An Enhanced Primer
Second Edition
BruceCameronReed
DepartmentofPhysics(Emeritus)
AlmaCollege
Alma,MI,USA
ISBN 978-3-031-14019-8 ISBN 978-3-031-14020-4 (eBook)
https://doi.org/10.1007/978-3-031-14020-4
1stedition:©Jones&BartlettLearning2007
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Preface
This book is a revised edition of my Quantum Mechanics, which was originally
publishedbyJones&Bartlettin2008.ThisnewversiongestatedforlongerthanI
hadintended.FollowingtheJones&Bartlettedition,myprofessionalintereststurned
tootherdirections,andtheusualresponsibilitiesandcommitmentsofamid/senior-
levelacademicpositionkeptaccumulating.ButIalwayswantedtogetbacktothis
topic,whichhasfascinatedmefordecades.NotonlydidIwanttorefreshmyself,but
Ialsowantedtotaketheopportunitytofixerrorsandconfusingstatementsthathad
beenpointedouttomebymyownstudentsandotherdiligentreaders,addsomenew
problemsandsections,improvesomeofthefigures,andincorporatenewreferences.
Ihopethatreaders—particularlythestudentsforwhomitisintended—enjoyreading
andlearningfromthisrevisededitionasmuchasIenjoyedpreparingit.
Therationaleforthisbookremainsthesameasitwasin2008:Atextdirectedat
theneedsofphysicsandchemistrystudentsatsmallercollegesanduniversitieswho
areencounteringtheirfirstseriouscourseinquantummechanics(QM),typicallyat
aboutthejuniorlevel.Thecurriculaofsuchinstitutionsoftenleave roomforonly
onesemesterofQMasopposedtothetwoormorethatarecommonatlargerschools,
sosuchstudentsfacethetaskoftryingtoabsorbinashorttimeboththeunderlying
physicsandmathematicalformalismsofquantumtheory.Mygoalistogivestudents
asenseof“whereitallleads”beforetheyencounterthatfullmathematicalformalism
in,say,asenior-levelorfirstgraduatecoursebyprovidingatreatmentofthebasic
theoriesandresultsofwavemechanicsthatfallsintermediatebetweensophomore-
level“modernphysics”textsandthosetypicalofmoreadvancedtreatments.Students
shouldthinkofthisbookasbeingaprimertohelppreparethemformoreadvanced
work.
Backgroundpreparationassumedhereincludesadvancedcalculus(partialdiffer-
entiationandmultipleintegration),someexposuretobasicconceptsofdifferential
equations (particularly the series solution method), vector calculus in Cartesian,
spherical, and cylindrical coordinates, and the standard menu of physics courses:
mechanics, electricity and magnetism, and modern physics. Some previous expo-
suretothehistoricalcontext fromwhichquantum mechanics developed ishelpful
butnotstrictlynecessary;someofthisisreviewedinChap.1.
v
vi Preface
I am firmly of the belief that it is only by working through the derivations of
a number of classical problems and by doing numerous homework exercises can
studentscometodevelopafeelingfortheconceptualcontentofquantummechanics
andthenecessaryanalyticskillsinvolvedinapplyingit.Present-dayphysicsrepre-
sentsthecumulativeknowledgeofachainofreasoningandexperimentthatstretches
backovercenturies,andIbelievethatitisimportantforstudentstohaveasenseof
howwecametobewhereweare.Consequently,thisbookemphasizesthedevelop-
mentofexactorapproximateanalyticsolutionsofSchrödinger’sequation,thatis,
solutionsthatcanbeexpressedinalgebraicform.Tothisend,thelayoutofthisbook
followsafairlyconventionalordering.Chapter1reviewssomeofthedevelopments
of early twentieth-century physics that indicated the need for a radical re-thinking
offundamentalphysicallawsontheatomicscale:Planck’squantumhypothesis,the
Rutherford–Bohr atomic model, and de Broglie’s matter-wave concept. Chapter 2
develops Schrödinger’s equation, the fundamental “law” of quantum mechanics.
Chapter3exploresapplicationsofSchrödinger’sequationtoone-dimensionalprob-
lems,workinguptoatreatmentofalpha-decayasabarrier-penetrationphenomenon.
Chapter 4 is a sort of intermission wherein some of the mathematical formalisms
of quantum mechanics such as operators, expectation values, the uncertainty prin-
ciple, Ehrenfest’s theorem, the orthogonality theorem, the superposition principle,
thevirialtheorem,andDiracnotationareintroduced.
Chapter 5 returns to specific solutions of Schrödinger’s equation with an anal-
ysis of the harmonic oscillator potential. Extension of Schrödinger’s equation to
three dimensions appears in Chap. 6, along with a treatment of separation of vari-
ablesandangularmomentumoperatorsforcentralpotentials.Thismaterialsetsthe
stageforarigorousanalysisoftheCoulombpotentialinChap.7.Chapter8deals
with some more advanced aspects of angular momentum and can be considered
optional.Chapter9explorestechniquesforundertakingapproximateanalyticsolu-
tionsofSchrödinger’sequationinsituationswherefullanalyticsolutionsaredifficult
orimpossibletoachieve:theWKBmethod,perturbationtheory,andthevariational
method.Inviewofthecentralroleofcomputersinalmostallcontemporaryresearch,
Chap. 10 explores a simple algorithm for numerically integrating Schrödinger’s
equationwithaMicrosoftExcelspreadsheet.Thiscomeswithanimportantcaveat,
however:nophysicist,eitherbeginningstudentorseasonedresearcher,shouldever
let playing with the computer become a substitute for first exploring a problem
conceptuallyandanalytically.
Theemphasisinthisbookisonthetime-independentSchrödingerequation,but
someaspectsoftimedependencearetakenupbrieflyinChap.11.Afewparticularly
lengthymathematicalderivationsoflimitedphysicalcontentaregatheredtogether
in Appendix A. Answers to a number of the end-of-chapter problems appear in
Appendix B. Appendices C and D list a number of useful integrals and physical
constants.
The new material in this edition is spread among several chapters. Chapter 1
now includes a section on the classical electromagnetic-radiation atomic collapse
problem.AnewsubsectioninSect.3.3describeshowthefiniterectangularwellcan
be analyzed with matrix algebra, and a short new subsection in Sect. 3.10 gives a
Preface vii
moregeneralanalysisofresonantscatteringfrompotentialwells.Section4.8now
includesadiscussionofthetimeevolutionofthespreadingofaGaussianwavepacket,
andSect.4.10explorestheconceptofmomentum-spacewavefunctions.InSect.5.3,
thediscussionoftheterminationconditionforHermitepolynomialsintheharmonic
oscillator problem has been clarified, and a new Appendix, A.2, gives a detailed
derivationofthenormalizationofHermitepolynomials.Severalnewend-of-chapter
problemshavebeenadded.
Ineachchapter,Itrytogetdirectlytotheessentialphysicsandillustrateitwith
examplesthatcontainactualnumbersandwhichcanserveasvehiclestointroduce,
wherepracticable,powerfulancillarytechniquessuchasdimensionalanalysisand
numerical integration. Each chapter contains a number of problems; it would not
be unreasonable for students to attempt most of them throughout the course of a
semester.
For students, I have three pieces of advice. First,quantum mechanics is unique
amongthesubdisciplinesofphysicsinthatsomanyofitsessentialconceptsseem
contrarytoexperienceandintuition.Thereisnooneformulaordescriptionviawhich
you can comprehend it quickly. It will take time and thought: mull things over in
yourmind,discussthemwithyourclassmatesandprofessors,and,whenyoucome
tounderstandsomething,writeitdowninyourownwords.Inthisspirit,Ihavetried
tokeepthetoneofthisworkinformalwhilepreservingasensiblelevelofrigor.
Second,anumberofproblemsappearattheendofeachchapter.Onlybydoing
themforyourselfcanyoubecomefamiliarwiththetoolsofthetrade.Problemsare
classified as elementary (E), intermediate (I), or advanced (A), although these are
somewhatarbitrarydesignations.Forhandyreference,briefsummariesofimportant
conceptsandformulaeappearbeforeeachproblemset.
Third, there are likely many quantum mechanics texts in your school’s library.
Consult them. What may seem opaque as expressed by one author may be clearer
in the words of another. One source I have found particularly valuable over the
years is Introduction to Quantum Mechanics with Applications to Chemistry by
LinusPaulingandE.B.Wilson(NewYork:DoverPublications,1985).Originally
publishedin1935whenquantumphysicswasstillnew,thisvenerableworkcontains
detailed analyses of many classic problems. Another work I find appealing for its
lucid explanations is An Introduction to Quantum Physics by A. P. French and E.
F.Taylor(NewYork:W.W.Norton,1978).Anyseriousstudentofphysicsshould
alsoalwayshaveathandagoodreferencetothemanyspecialfunctionsthatcropup
inmathematicalphysics;HansWeberandGeorgeArfken’sEssentialMathematical
MethodsforPhysicists(Amsterdam:Elsevier,2004)canbestronglyrecommended
andisreferencedfrequentlyinthepresentwork.Also,agoodreferencefortechniques
of numerical analysis is indispensable; a standard work in this area is Numerical
Recipes: The Art of Scientific Computing by William H. Press, Brian P. Flannery,
SaulA.Tuekolsky,andWilliamT.Vetterling(Cambridge,UK:CambridgeUniversity
Press,1986).
Quantum mechanics and its applications are a vibrant, central part of much
present-day research. To give students a flavor of the diversity of current work,
references to semi-popular articles appearing in Physics Today (abbreviated Phys.
viii Preface
Today), a monthly publication of the American Institute of Physics, appear occa-
sionallythroughoutthetext.Thesearticlesaresometimesbrief“update”piecesand
sometimesfeature-lengthworks,butallshouldbelargelyaccessibletoundergradu-
atesandcontainreferencestotheoriginalresearchliterature.Studentsarestrongly
encouragedtoexplorethem.
Some sections in Chaps. 3, 6, and 7 have subsections. Citations are designated
withsquarebrackets,beginninganewwith[1]ineachchapter.
My interest in quantum mechanics was stimulated in my own student days by
a number of excellent teachers at both the University of Waterloo and Queen’s
University, and has only grown over the intervening years. A number of instruc-
tors, students, special friends, colleagues, and family members from those years
and on up to the present have supported and encouraged me. It gives me pleasure
to especially acknowledge John Altholz, Karen Ball, Dick Bowker, Peter Burns,
David Cassidy, John Coster-Mullen, Peter Dawson, Michael DeRobertis, Eugene
Deci, Carleen Dewit, Patrick Furlong, John Gibson, Dick Groves, Bob Hayward,
Miriam Hiebert, Lorraine Hill, Art Hobson, Lisa Jylänne, Patricia Kinnee, Tim
Koeth,VernKoslowsky,GillesLabrie,HarryLustig,LorneNelson,JohnPalimaka,
KlausRohe,JohnSchreiner,RaySmith,FrankSettle,UteStargardt,RogerStuewer,
George Wagner, and Pete Zimmerman. Unfortunately, some of these wonderful
people are no longer among us, but are fondly remembered: Requiesce in pace.
Also, I continue to be grateful to individuals who gave generously of their time
andexpertiseinreviewingfirst-editionchaptersofthisworkpriortopublication:Xi
Chen(CentralCollege),JamesClemens(MiamiUniversity),JospehGanem(Loyola
College), Noah Graham (Middlebury College), Rick McDaniel (Henderson State
University), Soma Mukherjee (University of Texas, Brownsville), David Olsgaard
(Simpson College), Vasilis Pagonis (McDaniel College), Harvey Picker (Trinity
College), Darrell Schroeter (Occidental College), Blair Tuttle (Pennsylvania State
University,Erie),andAnnWright(HendrixCollege).Iclaimexclusiveownership
ofanyerrorsthatremain.
ThankswouldnotbecompletewithoutatipofmyhattomyeditoratSpringer,
AngelaLahee,whomadethisbookapublishedreality.
Finally,thisworkisdedicatedtomywifeLaurie,whohasonceagainbornewith
mydistractionas“onelasttome”(Ipromise!)occupiedmytimeandthoughts.Our
variouscatsovertheyears,Fred,Leo,Stella,Cassie,Nyx,andNewton,haveamply
provenAldousHuxley’sadagethat“Ifyouwanttowrite,keepcats.”
Bedford,NovaScotia,Canada BruceCameronReed
2022
Contents
1 Foundations .................................................. 1
1.1 Faraday,Thomson,andElectrons ........................... 2
1.2 Spectra,Radiation,andPlanck ............................. 4
1.3 TheRutherford-BohrAtom ................................ 12
1.4 deBroglieMatterWaves .................................. 20
1.5 TheRadiativeCollapseProblem(Optional) .................. 24
References .................................................... 32
2 Schrödinger’sEquation ........................................ 35
2.1 TheClassicalWaveEquation .............................. 36
2.2 TheTime-IndependentSchrödingerEquation ................. 41
2.3 TheTime-DependentSchrödingerEquation .................. 44
2.4 Interpretationofψ:ProbabilitiesandBoundaryConditions ..... 49
References .................................................... 56
3 SolutionsofSchrödinger’sEquationinOneDimension ........... 57
3.1 ConceptofaPotentialWell ................................ 58
3.2 TheInfinitePotentialWell ................................. 60
3.3 TheFinitePotentialWell .................................. 67
3.3.1 AMatrixApproachtotheFinitePotentialWell ........ 75
3.4 FinitePotentialWell-EvenSolutions ........................ 78
3.5 NumberofBoundStatesinaFinitePotentialWell ............ 80
3.6 SketchingWavefunctions .................................. 83
3.7 PotentialBarriersandScattering ............................ 87
3.8 PenetrationofArbitrarily-ShapedBarriers ................... 92
3.9 Alpha-DecayasaBarrierPenetrationEffect .................. 94
3.10 ScatteringbyOne-DimensionalPotentialWells ............... 100
References .................................................... 109
4 Operators, Expectation Values, and Various Quantum
Theories ...................................................... 111
4.1 PropertiesofOperators .................................... 112
ix
x Contents
4.2 ExpectationValues ....................................... 115
4.3 TheUncertaintyPrinciple ................................. 123
4.4 CommutatorsandUncertaintyRelations ..................... 127
4.5 Ehrenfest’sTheorem ...................................... 130
4.6 TheOrthogonalityTheorem ............................... 132
4.7 TheSuperpositionTheorem ................................ 134
4.8 ConstructingaTime-DependentWavePacket ................ 136
4.9 TheVirialTheorem ....................................... 141
4.10 Momentum-SpaceWavefunctions .......................... 147
References .................................................... 156
5 TheHarmonicOscillator ...................................... 157
5.1 ALessoninDimensionalAnalysis .......................... 158
5.2 TheAsymptoticSolution .................................. 161
5.3 TheSeriesSolution ....................................... 163
5.4 Hermite Polynomials and Harmonic Oscillator
Wavefunctions ........................................... 170
5.5 Comparing the Classical and Quantum Harmonic
Oscillators .............................................. 173
5.6 RaisingandLoweringOperators ............................ 176
Reference ..................................................... 186
6 Schrödinger’s Equation in Three Dimensions
andtheQuantumTheoryofAngularMomentum ................ 187
6.1 SeparationofVariables:CartesianCoordinates ............... 188
6.2 SphericalCoordinates ..................................... 195
6.3 AngularMomentumOperators ............................. 197
6.4 SeparationofVariablesinSphericalCoordinates:Central
Potentials ............................................... 203
6.5 AngularWavefunctionsandSphericalHarmonics ............. 205
6.5.1 Solutionofthe(cid:3)Equation .......................... 206
6.5.2 Solutionofthe(cid:4)Equation .......................... 208
6.5.3 SphericalHarmonics ............................... 213
References .................................................... 223
7 CentralPotentials ............................................. 225
7.1 Introduction ............................................. 225
7.2 TheInfiniteSphericalWell ................................ 228
7.3 TheFiniteSphericalWell .................................. 230
7.4 TheCoulombPotential .................................... 233
7.5 HydrogenAtomProbabilityDistributions .................... 245
7.5.1 The(1,0,0)StateofHydrogen ...................... 246
7.5.2 The(2,0,0)andOtherStatesofHydrogen ............ 249