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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 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer NatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland 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 principle, Ehrenfest’s theorem, the orthogonality theorem, the superposition principle, thevirialtheorem,andDiracnotationareintroduced. Chapter 5 returns to specific solutions of Schrödinger’s equation with an analysis 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