Topics in Atomic Physics PDF
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Topics in Atomic Physics Charles E. Burkhardt DepartmentofPhysics,St.LouisCommunityCollege, St.Louis,Missouri Jacob J. Leventhal DepartmentofPhysics,UniversityofMissouri—St.Louis, St.Louis,Missouri Topics in Atomic Physics With75Illustrations CharlesE.Burkhardt JacobJ.Leventhal DepartmentofPhysics DepartmentofPhysics St.LouisCommunityCollege UniversityofMissouri—St.Louis St.Louis,MO63121 St.Louis,MO63121 USA USA LibraryofCongressControlNumber:2005926818 ISBN10:0-387-25748-9 ISBN13:978-0387-25748-8 Printedonacid-freepaper. (cid:1)C 2006SpringerScience+BusinessMedia,Inc. 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(TB/EB) 9 8 7 6 5 4 3 2 1 springeronline.com Helen,Charlie,Sarah,andMichelle Bette,Steve,Andy,Dan,andTina Preface The importance of the field of atomic physics to modern technology cannot be overemphasized.Atomicphysicsservedasamajorimpetustothedevelopmentof the quantum theory of matter in the early part of the twentieth century and, due totheavailabilityofthelaserasalaboratorytool,ithastakenusintothetwenty- first century with an abundance of new and exciting phenomena to understand. Ourintentioninwritingthisbookistoprovideafoundationforstudentstobegin researchinmodernatomicphysics.Asthetitleimplies,itisnot,norwasitintended tobe,anall-inclusivetomecoveringeveryaspectofatomicphysics. Any specialized textbook necessarily reflects the predilection of the authors towardcertainaspectsofthesubject.Thisoneisnoexception.Itreflectsourbe- liefthatathoroughunderstandingoftheuniquepropertiesofthehydrogenatom is essential to an understanding of atomic physics. It also reflects our fascina- tionwiththedistinguishedpositionthatMotherNaturehasbestowedonthepure CoulombandNewtonianpotentials,andthushydrogenatomsandKeplerianor- bits.Therefore,wehavedevotedalargeportionofthisbooktothehydrogenatom toemphasizethisdistinctiveness.Weattempttostresstheuniquenessoftheattrac- tive1/rpotentialwithoutdelvingintogrouptheory.Itisourbeliefthat,oncean understanding of the hydrogen atom is achieved, the properties of multielectron atomscanbeunderstoodasdeparturesfromhydrogenicproperties. From the beginning, it was our intention to include information in this book thatisnoteasilylocatedelsewhere.Thus,whilethebookcanbeusedasatext,it ishopedthatitwillalsobeausefulreference.Tothisend,wehaveincorporated derivationsthataredifficulttofindinotherbooksand,indeed,evenintheliterature. Forexample,thequantummechanicalLenzvectoroperatorisnotoftendiscussed in books on quantum mechanics and atomic physics. When it is discussed, it is usually stated that it commutes with the hydrogen atom Hamiltonian, but it is difficulttoprove.However,thisbookgivesthisproofinsomedetail.Infact,one ofthegeneralfeaturesofourbookisthatweoftenincludemorealgebraicsteps than are traditionally given in textbooks. By doing this, we wish to relieve the readerofthetediumofreproducingalgebraand,thus,permitconcentrationonthe physics. vii viii Preface Thismaterialisintendedtobesuitableforaone-semestergraduateoradvanced undergraduate-levelcourseinatomicphysics.Itisassumedthatthestudenthashad atleastintroductoryquantummechanics,althoughpertinenttopicssuchaspertur- bationtheoryandvariationaltechniquesarebrieflyreviewed.Chapter1presents somebackgroundmaterialwhich,inlargepart,isreview.Inthischaptertheutility oftheBohratomisdiscussedandthederivationperformedasBohrdidit.Thisis incontrasttomostmodernpresentations.Chapters2and3arestandardreviewsof angularmomentumalgebrawithemphasisonaspectspertinenttoatomicphysics. Chapter4isadiscussionofthequantummechanicalhydrogenatomandthesep- aration of the Schrödinger equation in both spherical and parabolic coordinates. Emphasis is placed on the conditions that force quantization. This chapter also includes an attempt to clarify the difference between two commonly used defi- nitions of Laguerre and associated Laguerre polynomials. In most treatments of thequantummechanicalhydrogenatom,nomentionismadeofalternatedefini- tionsofthesespecialfunctions.Chapter5isadiscussionoftheclassicalhydrogen atom.NaturallyitdrawsheavilyonKeplerianorbitsandtheconsequencesofthe additionalconstantoftheclassicalmotion,theLenzvector.Chapter6discusses theaccidentaldegeneracyofthehydrogenatomanditscausesinthecontextofthe quantum mechanical analog of the Lenz vector. To our knowledge, the material in Chapter 6 is not covered in any modern text. Chapter 7 discusses the break- ingoftheaccidentaldegeneracyviafinestructure,theLambshift,andhyperfine structure.Thetreatmentisfairlystandard.Chapter8treatsthehydrogenatomin externalfields.ThedescriptionoftheZeemaneffectisstandard,buttheweakfield Stark effect is described in quantum mechanically and classically. The classical treatmentleansheavilyontheresultsofChapter5whilethequantummechanical treatmentexploitstheoperatorformalismdevelopedinChapter6. Chapters9and10arediscussionsofmultielectronatoms,beginningwithhelium inChapter9.Thepresentationisquitestandard.Chapter11presentsthequantum defect in a way that is seldom seen in texts. In keeping with the theme of this book,thequantumdefectisrelatedtoclassicalconceptsandthecorrespondence principle.Chapter12dealswithmultielectronatomsinexternalfields.Hereagain the Zeeman effect is treated in the standard manner, but the Stark effect is pre- sentedinawaythatleansheavilyonthematerialpresentedinChapter11.Finally, in Chapter 13, radiation is discussed at length. We emphasize how the concept of a stationary state is not at odds with the classical concept of radiation from acceleratingcharges.Otherwise,thepresentationinthischapterisstandard,but, wehope,thorough. SIunitsareusedexceptinthoseinstancesforwhichwebelievethatatomicunits areconsiderablymoreconvenient.Forinstance,theZeemaneffectistreatedusing SIunitsbecausetheBohrmagnetontimesthemagneticinductionfieldprovides aconvenientmeasureoftheZeemanenergy.However,wefindatomicunitstobe moreconvenientinthetreatmentoftheStarkeffect. There are more than eighty problems listed at the ends of chapters, with se- lected answers given at the end of the book. Some are merely exercises, but others are more challenging. A few are derivations of results that are used later Preface ix in the book. A detailed solutions manual, in many cases showing more than one way to solve a given problem, is available to qualified instructors: visit www.springeronline.com/0-387-25748-9. AlistofcorrectionstothebookisavailableontheInternetat: http://www.umsl.edu/jjl/homepage/ and http://www.stlcc.edu/fv/physics/CBurkhardt/ Comments or previously unreported errors can be transmitted to the authors throughtheseWebsites. We wish to thank our graduate students Joseph F. Baugh, Marco Ciocca, and Lucy Wenzhong He. Thanks are also due to the numerous undergraduates who have worked in the UMSL Atomic Physics Laboratory over the past 40 years. We also wish to acknowledge the contributions of our many collaborators and ourUMSLcolleagues,Ta-PeiCheng,WayneP.Garver,andPhilipB.Jameswith whomwehadmanyvaluablediscussionsovertheyears. ThanksalsotoWai-YimChingofUMKCformanyvaluablecommentsonthe manuscriptduringitspreparation. CharlesE.Burkhardt JacobJ.Leventhal Contents Preface................................................................................. vii Chapter1 Background............................................. 1 1.1 Introduction..................................................................... 1 1.2 TheBohrModeloftheAtom................................................ 1 1.3 NumericalValuesandtheFineStructureConstant...................... 7 1.4 AtomicDimensions—Isa aReasonableAtomicDiameter?......... 8 0 1.5 LocalizingtheElectron:IsaPointParticleReasonable?............... 10 1.6 TheClassicalRadiusoftheElectron....................................... 11 1.7 AtomicUnits.................................................................... 11 Chapter2 AngularMomentum.................................. 14 2.1 Introduction..................................................................... 14 2.2 Commutators.................................................................... 18 2.3 AngularMomentumRaisingandLoweringOperators................. 20 2.4 AngularMomentumCommutationRelationswith VectorOperators............................................................... 25 2.5 MatrixElementsofVectorOperators...................................... 26 2.6 EigenfunctionsofOrbitalAngularMomentumOperators............. 29 2.7 Spin............................................................................... 33 2.8 TheStern–GerlachExperiment.............................................. 41 Chapter3 AngularMomentum—TwoSources.............. 46 3.1 Introduction..................................................................... 46 3.2 TwoSetsofQuantumNumbers—UncoupledandCoupled............ 46 3.3 VectorModelofAngularMomentum...................................... 51 3.4 ExamplesofCalculationoftheClebsch–GordanCoefficients........ 55 3.5 HyperfineSplittingintheHydrogenAtom................................ 61 xi xii Contents Chapter4 TheQuantumMechanicalHydrogenAtom.... 73 4.1 TheRadialEquationforaCentralPotential.............................. 73 4.2 SolutionoftheRadialEquationinSpherical Coordinates—TheEnergyEigenvalues.................................... 75 4.3 TheAccidentalDegeneracyoftheHydrogenAtom..................... 77 4.4 SolutionoftheHydrogenAtomRadialEquationin SphericalCoordinates—TheEnergyEigenfunctions.................... 79 4.5 TheNatureoftheSphericalEigenfunctions.............................. 82 4.6 SeparationoftheSchro¨dingerEquationin ParabolicCoordinates......................................................... 82 4.7 SolutionoftheSeparatedEquationsinParabolic Coordinates—TheEnergyEigenvalues.................................... 85 4.8 SolutionoftheSeparatedEquationsinParabolic Coordinates—TheEnergyEigenfunctions................................ 87 Chapter5 TheClassicalHydrogenAtom..................... 92 5.1 Introduction..................................................................... 92 5.2 TheClassicalDegeneracy.................................................... 95 5.3 AnotherConstantoftheMotion—TheLenzVector..................... 97 Chapter6 TheLenzVectorandthe AccidentalDegeneracy............................................... 105 6.1 TheLenzVectorinQuantumMechanics.................................. 105 6.2 LenzVectorLadderOperators;ConversionofaSpherical EigenfunctionintoAnotherSphericalEigenfunction................... 109 6.3 ApplicationofLenzVectorLadderOperatorstoa GeneralSphericalEigenfunction............................................ 114 6.4 ANewSetofAngularMomentumOperators............................ 116 6.5 EnergyEigenvalues............................................................ 118 6.6 RelationsBetweentheParabolicQuantumNumbers................... 120 6.7 RelationshipBetweentheSphericaland ParabolicEigenfunctions..................................................... 122 6.8 AdditionalSymmetryConsiderations...................................... 123 Chapter7 BreakingtheAccidentalDegeneracy............ 126 7.1 Introduction..................................................................... 126 7.2 RelativisticCorrectionfortheElectronicKineticEnergy.............. 127 7.3 Spin-OrbitCorrection......................................................... 128 7.4 TheDarwinTerm.............................................................. 130 7.5 EvaluationoftheTermsThatContributetotheFine-Structure ofHydrogen..................................................................... 130 Contents xiii 7.6 TheTotalFine-StructureCorrection........................................ 135 7.7 TheLambShift................................................................. 137 7.8 HyperfineStructure............................................................ 139 7.9 TheSolutionoftheDiracEquation......................................... 142 Chapter8 TheHydrogenAtominExternalFields......... 145 8.1 Introduction..................................................................... 145 8.2 TheZeemanEffect—TheHydrogenAtominaConstant MagneticField.................................................................. 146 8.3 WeakElectricField—TheQuantumMechanicalStarkEffect........ 159 8.4 WeakElectricField—TheClassicalStarkEffect........................ 171 Chapter9 TheHeliumAtom..................................... 178 9.1 IndistinguishableParticles.................................................... 178 9.2 TheTotalEnergyoftheHeliumAtom..................................... 180 9.3 EvaluationoftheGroundStateEnergyoftheHeliumAtom UsingPerturbationTheory................................................... 183 9.4 TheVariationalMethod....................................................... 186 9.5 ApplicationoftheVariationalPrincipletotheGround StateofHelium................................................................. 187 9.6 ExcitedStatesofHelium..................................................... 189 9.7 DoublyExcitedStatesofHelium:Autoionization....................... 192 Chapter10 MultielectronAtoms................................ 196 10.1 Introduction..................................................................... 196 10.2 ElectronConfiguration........................................................ 196 10.3 TheDesignationofStates—LSCoupling................................. 198 10.4 TheDesignationofStates—jjCoupling................................... 207 Chapter11 TheQuantumDefect............................... 214 11.1 Introduction..................................................................... 214 11.2 EvaluationoftheQuantumDefect.......................................... 216 11.3 ClassicalFormulationoftheQuantumDefectandthe CorrespondencePrinciple.................................................... 220 11.4 TheConnectionBetweentheQuantumDefectandthe RadialWaveFunction......................................................... 225 Chapter12 MultielectronAtomsinExternalFields....... 230 12.1 TheStarkEffect................................................................ 230 12.2 TheZeemanEffect............................................................. 238
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