Theoretical Atomic Physics PDF
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Graduate Texts in Physics Harald Friedrich Theoretical Atomic Physics Fourth Edition Graduate Texts in Physics Serieseditors KurtH.Becker,PolytechnicSchoolofEngineering,Brooklyn,USA Jean-MarcDiMeglio,UniversitéParisDiderot,Paris,France SadriHassani,IllinoisStateUniversity,Normal,USA BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan RichardNeeds,UniversityofCambridge,Cambridge,UK WilliamT.Rhodes,FloridaAtlanticUniversity,BocaRaton,USA SusanScott,AustralianNationalUniversity,Acton,Australia H.EugeneStanley,BostonUniversity,Boston,USA MartinStutzmann,TUMünchen,Garching,Germany AndreasWipf,Friedrich-Schiller-UnivJena,Jena,Germany GraduateTexts inPhysics Graduate Texts in Physics publishes core learning/teachingmaterial for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions,derivations,experimentsandapplications(asrelevant)fortheirmastery and teaching, respectively. International in scope and relevance, the textbooks correspondtocoursesyllabisufficientlytoserveasrequiredreading.Theirdidactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scientists entering, or requiring timely knowledgeof,aresearchfield. Moreinformationaboutthisseriesathttp://www.springer.com/series/8431 Harald Friedrich Theoretical Atomic Physics Fourth Edition 123 HaraldFriedrich FachbereichPhysikT30 TUMuRnchen Garching,Germany ISSN1868-4513 ISSN1868-4521(electronic) GraduateTextsinPhysics ISBN978-3-319-47767-1 ISBN978-3-319-47769-5 (eBook) DOI10.1007/978-3-319-47769-5 LibraryofCongressControlNumber:2017936054 1st,2ndedition:©Springer-VerlagBerlinHeidelbergNewYork1991,1998 3rdedition:©Springer-VerlagBerlinHeidelberg2006 4thedition:©SpringerInternationalPublishingAG2017 ©SpringerInternationalPublishingAG2017 Originallypublishedintheseries:AdvancedTextsinPhysics Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Fourth Edition The first edition of Theoretical Atomic Physics was written more than a quarter of a century ago, with the aim of providing graduate students and researchers in atomicphysicswiththe“kindofadvancedquantummechanicsneededforpractical applications in modern atomic physics”. Since then, the unbroken advancement of improvedexperimentaltechniquesand computationalpowerhasbroadenedthe range of fascinating effects that can be studied in the laboratory and modelled in theoretical analyses. It includes the study of individual atoms in electromagnetic traps, where fundamental postulates of quantum mechanics can be tested, of degeneratequantumgasesofultracoldatoms(ormolecules)andofcomplexsystems with chaotic classical dynamics, where semiclassical theories have experienced a revival and found many applications of practical relevance in the atomic domain. Theaimformulatedforthefirsteditionremainsvalidinthiscontext.Theemphasis on theory should enable the reader to appreciate the fundamental assumptions underlyingstandardtheoreticalconstructsand to embark on independentresearch projects. The production of and experimentation with Bose–Einstein condensates of atomic gases is now routine in many laboratories, and this has helped to make cold and ultracold atoms (and molecules) a field of rapidly growing interest. The interaction of atoms close to the threshold between weakly bound diatomic molecular states and low-energyscattering states is importantin this context, and so,manyconceptsofnear-thresholdscatteringtheoryareusedbyresearchersinthe field.Theobservationthatmanycolleagueswerenotawareoftheoriginandcould not appreciate the precise meaning of such concepts as, e.g. “scattering length”, motivatedmetowriteamonographonscatteringtheory,withaspecialfocusonthe relevanceforcold-atomphysics[H.Friedrich,ScatteringTheory,LectureNotesin Physics872,Springer,Berlin, Heidelberg,2013,2nd.Edition2016].In the fourth edition of Theoretical Atomic Physics, I have updated and expanded the sections and subsections involving scattering theory and/or near-threshold phenomena by incorporating the corresponding contributions from the monograph. Hence, the treatment of scattering and near-threshold phenomena has become more sophis- ticated. Special attention is given to the quantization of weakly bound states just v vi PrefacetotheFourthEdition belowthecontinuumthresholdandtolow-energyscatteringandquantumreflection just above. Particular emphasis is laid on the fundamental differences between long-ranged Coulombic potentials, where the continuum threshold represents the (semi-)classical limit of the Schrödinger equation, and shorter-ranged potentials falling off faster than 1=r2 at large distances r, where the threshold corresponds totheanticlassical,extremequantumlimit.Modifiedeffectiverangeexpansionsare given, even for potentials with attractive inverse-cube tails, a result derived only recentlybyMüller[Phys.Rev.Lett.110(2013)260401],see(4.113)inSect.4.1.8. A newsectiononscatteringintwospatialdimensionsisincluded;itisrelevant not only for genuinely two-dimensional systems but also for 3D systems with translationalinvarianceinonedegreeoffreedom,suchasanatominteractingwitha cylindricalnanotube.Thereisalsoanewsectionontunablenear-thresholdFeshbach resonances, a subject that was treated poorly in the third edition. The appendix on special mathematical functions has been expanded in order to accommodate formulas occurring in the extended treatment of scattering and near-threshold phenomena. Itisapleasuretothankmanycolleagueswhoinspiredmewithnumerousdiscus- sionsinvolvingatomicphysics,quantummechanicsandsemiclassicalconnections, in particular Robin Côté at the University of Connecticut, Wolfgang Domcke and Manfred Kleber at the Technical University of Munich, Gerhard Rempe and StephanDürrattheMaxPlanckInstituteforQuantumOpticsinGarchingandJan- MichaelRostattheMax PlanckInstituteforComplexSystemsinDresden.Some postdocsandseveralformerstudentsproducedresultsthatIhaveusedinthebook, in particular Florian Arnecke, JohannesEiglsperger,ChristopherEltschka, Martin Fink, Georg Jacoby, Alexander Jurisch, Alexander Kaiser, Petra Meerwald, Carlo Meister,EskenderMesfin,JavierMadroñero,MichaelJ.Moritz,Tim-OliverMüller, Thomas Purr, Patrick Raab, Sebastian Schröter, Frauke Schwarz and Johannes Trost. I am grateful for the technical assistance provided by Stefan Recksiegel, our IT expertat the Physik-Departmentin Garching.I also thank Ute Heuser and Birgit Münch and Dr. Thorsten Schneider at Springer for their efficient help and cooperation. Schließlich möchte ich mich bei meiner Familie bedanken, die mir immer den ZugangzuralltäglichenWeltjenseits derPhysikoffengehaltenhat.Vor allembei meinerFrau Elfi, dieübermehrals vierJahrzehntemeineArbeitmitErmutigung, Geduld und Flexibilität unterstützt hat. Dazu hat uns in den letzten zweieinhalb JahrendasGlückdreiEnkelkinderbeschert,Lorenz,AlexanderundJohann,diemit ihrerauthentischenLebensfreudealleHerzenhöherschlagenlassen. Garching,Germany HaraldFriedrich September2016 Preface to the Third Edition TheoneandahalfdecadessincethepublicationofthefirsteditionofTheoretical AtomicPhysicshaveseenacontinuationofremarkableanddramaticexperimental breakthroughs. With the help of ultrashort laser pulses, special states of atoms and molecules can now be prepared and their time evolution studied on time scales shorter than femtoseconds. Trapped atoms and molecules can be cooled to temperatures on the order of a few nano-Kelvin and light fields can be used to guide and manipulate atoms, for example, in optical lattices formed as standing waves by counterpropagating laser beams. After the first production of Bose– Einsteincondensatesofultracoldatomicgasesin1995,degeneratequantumgases ofultracoldatomsandmoleculesare nowpreparedandstudiedroutinelyin many laboratories around the world. Such progress in atomic physics has been well received and appreciated in the general academic community and was rewarded withtworecentNobelPrizesforphysics.The1997prizewasgiventoStevenChu, ClaudeCohen-TannoudjiandWilliamPhillipsfortheirworkoncoolingatoms,and only 4 years later Eric Cornell, Wolfgang Ketterle and Carl Wieman received the 2001prizefortherealizationoftheBose–Einsteincondensatesmentionedabove. Theprominenceofmodernexperimentalatomicphysicsestablishesfurtherneed for a deeper understanding of the underlying theory. The continuing growth in quality and quantity of available computer power has substantially increased the effectivity of large-scale numerical studies in all fields, including atomic physics. This makes it possible to obtain some standard results such as the properties of low-lying states in many-electron atoms with good accuracy using generally applicable program packages. However, largely due to the dominant influence of long-ranged Coulomb forces, atomic systems are rather special. They can reveal a wide range of interesting phenomena in very different regimes—from near- classicalstatesofhighlyexcitedatoms,whereeffectsofnonlinearityandchaosare important,totheextremequantumregimeofultracoldatoms,wherecounterintuitive nonclassicaleffectscanbeobserved.Thetheoreticalsolutionoftypicalproblemsin modernatomicphysicsrequiresproficiencyinthepracticalapplicationofquantum mechanicsatanadvancedlevel,andagoodunderstandingofthelinkstoclassical mechanicsisalmostalwayshelpful.TheaimofTheoreticalAtomicPhysicsremains vii viii PrefacetotheThirdEdition to provide the reader with a solid foundation of this sort of advanced quantum mechanics. In preparing the third edition, I have again tried to do justice to the rapid development of the field. I have included references to important new work wheneverthisseemedappropriateandeasytodo.Chapter1nowincludesasection on processes involving (wave packets of) continuum states and also an expanded treatment of the semiclassical approximation. Chapter 3 begins with a section illuminatingthe characteristicdifferencesin the near-thresholdpropertiesoflong- rangedandshorter-rangedpotentials,andthefirstsectionofChap.4containsamore elaboratediscussion of scattering lengths.As a further“SpecialTopic” in Chap.5 thereisa sectiondescribingsomeaspectsofatomoptics,includingdiscussionsof theinteractionsofatomswithmaterialsurfacesandwithlightfields.Theappendix on special mathematical functions has been slightly expanded to accommodate a fewresultsthatIrepeatedlyfoundtobeuseful. I am grateful to many colleagues who continue to inspire me with numer- ous discussions involving atomic physics, quantum mechanics and semiclassical connections, in particular Robin Côté at the University of Connecticut, Manfred KleberattheTechnicalUniversityMunichandJan-MichaelRostattheMaxPlanck Institute for Complex Systems in Dresden. Several current and former graduate studentsproducednewresultsthatIhaveusedinthebook,inparticularChristopher Eltschka, Georg Jacoby, Alexander Jurisch, Michael J. Moritz, Thomas Purr and Johannes Trost. I thank them all for the effort and enthusiasm with which they contributed to the various projects. I also thank Thomas Mehnert for helpful comments on the previous editions. A sabbatical term at the Australian National UniversityinCanberraduringthesouthernsummer2002/2003establishedafruitful connection to Ken Baldwin and Stephen Gibson in the Atomic and Molecular Physics Laboratories, and I am grateful to Brian Robson and Erich Weigold who madethisvisitpossible.Finally,IwishtothankmywifeElfiwho(again)endured a hard-workingand preoccupiedhusbandduringthe final stages of preparationof thisthirdedition. Garching,Germany HaraldFriedrich June2005 Preface to the First Edition In the first few decades of this century, atomic physics and quantum mechanics developed dramatically from early beginnings to maturity and a degree of com- pleteness. After about 1950 fundamental research in theoretical physics focussed increasingly on nuclear physics and high energy physics, where new conceptual insightswereexpectedtobemoreprobable.Afurtherfieldofgrowingimportance wastheoreticalsolidstatephysics,whichledtooraccompaniedmanyrevolutionary technological developments. In this environment the role of atomic physics as an independent discipline of theoretical physics became somewhat subdued. In the last two decades, however, high precision experimental techniques such as high resolution laser spectroscopy have opened up new and interesting fields in atomic physics. Experiments can now be performed on individual atoms and ions in electromagnetic traps, and the dependence of their properties on their environmentcan bestudied.Effectsandphenomenawhichusedto be regardedas smallperturbationsorexperimentallyirrelevantexceptionalcaseshavemovedinto the centre of attention. At the same time it has become clear that interesting and intricateeffectscanoccureveninseeminglysimplesystemswithonlyfewdegrees offreedom. The successful description and interpretation of such effects usually requires the solution of a non-trivial Schrödinger equation, and perturbative methods are often inadequate. Most lectures and textbooks which go beyond an introductory “Quantum Mechanics I” are devoted to many-body theories and field theories at a high level of abstraction. Not enough attention is given to a more practical kind of advanced quantum mechanics as required by modern atomic physics. In order to meet this demand I have taught several courses on Theoretical Atomic Physics at the Munich Universities since 1984. The present book grew out of these lectures. It is an updated version of the textbook Theoretische Atomphysik, whichappearedinGermanin September1990,andcontainsthekindofadvanced quantum mechanics needed for practical applications in modern atomic physics. The level of abstraction is deliberately kept low—almost all considerations start with the Schrödinger equation in coordinate representation. The book is intended asa textbookforstudentswho havehada firstintroductorycontactwith quantum ix
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