TheoreticalAtomicPhysics Harald Friedrich Theoretical Atomic Physics ABC ProfessorDr.HaraldFriedrich PhysicsDepartmentT30 TechnischeUniversitätMünchen James-Franck-Straße1 85747Garching Germany E-mail:[email protected] LibraryofCongressControlNumber:2005927229 ISBN-10 3-540-25644-XSpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-25644-1SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springeronline.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandTechBooksusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:10960592 55/TechBooks 543210 Preface to the Third Edition The one and a half decades since the publication of the first edition of Theo- retical Atomic Physics have seen a continuation of remarkable and dramatic experimental 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 mole- cules can be cooled to temperatures on the order of a few nano–Kelvin and lightfieldscanbeusedtoguideandmanipulateatoms,forexampleinoptical lattices formed as standing waves by counterpropagating laser beams. After the first production of Bose–Einstein condensates of ultracold atomic gases in 1995, degenerate quantum gases of ultracold atoms and molecules are now prepared and studied routinely in many laboratories around the world. Such progress in atomic physics has been well received and appreciated in the gen- eral academic community and was rewarded with two recent Nobel Prizes for physics. The 1997 prize was given to Steven Chu, Claude Cohen-Tannoudji andWilliamPhillipsfortheirworkoncoolingatoms,andonlyfouryearslater Eric Cornell, Wolfgang Ketterle and Carl Wieman received the 2001 prize for the realization of the Bose–Einstein condensates mentioned above. The prominence of modern experimental atomic physics establishes fur- therneedforadeeperunderstandingoftheunderlyingtheory.Thecontinuing growth in quality and quantity of available computer power has substantially increasedtheeffectivityoflarge-scalenumericalstudiesinallfields,including atomicphysics.Thismakesitpossibletoobtainsomestandardresultssuchas the properties of low-lying states in many-electron atoms with good accuracy using generally applicable programme packages. However, largely due to the dominant influenceof long-ranged Coulomb forces,atomic systemsarerather special.Theycanrevealawiderangeofinterestingphenomenainverydiffer- entregimes—fromnear-classicalstatesofhighlyexcitedatoms,whereeffects of nonlinearity and chaos are important, to the extreme quantum regime of ultracold atoms, where counterintuitive nonclassical effects can be observed. Thetheoreticalsolutionoftypicalproblemsinmodernatomicphysicsrequires proficiencyinthepracticalapplication ofquantummechanicsatanadvanced VI Preface to the Third Edition level, and a good understanding of the links to classical mechanics is almost alwayshelpful.TheaimofTheoretical Atomic Physicsremainstoprovidethe 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 whenever this seemed appropriate and easy to do. Chapter 1 now includes a section on processes involving (wave packets of) continuum states and also an expanded treatment of the semiclassical approximation. Chapter 3 begins withasectionilluminatingthecharacteristicdifferencesinthenear-threshold properties of long-ranged and shorter-ranged potentials, and the first section of Chap. 4 contains a more elaborate discussion of scattering lengths. As a further “special topic” in Chap. 5 there is a section describing some aspects ofatomoptics,includingdiscusionsoftheinteractionsofatomswithmaterial surfacesandwithlightfields.Theappendixonspecialmathematicalfunctions has been slightly expanded to accommodate a few results that I repeatedly found to be useful. I am grateful to many colleagues who continue to inspire me with numer- ous discussions involving atomic physics, quantum mechanics and semiclassi- cal connections, in particular Robin Coˆt´e at the University of Connecticut, Manfred Kleber at the Technical University Munich and Jan-Michael Rost at the Max–Planck–Institute for Complex Systems in Dresden. Several current and former graduate students produced new results that I have used in the book, in particular Christopher Eltschka, Georg Jacoby, Alexander Jurisch, MichaelJ.Moritz,ThomasPurrandJohannesTrost.Ithankthemallforthe effort and enthusiasm with which they contributed to the various projects. I alsothankThomasMehnertforhelpfulcommentsonthepreviouseditions.A sabbaticaltermattheAustralianNationalUniversityinCanberraduringthe southernsummer2002/2003establishedafruitfulconnectiontoKenBaldwin andStephenGibsonintheAtomicandMolecularPhysicsLaboratories,andI amgrateful toBrian Robson and Erich Weigold whomadethisvisitpossible. Finally, I wish to thank my wife Elfi who (again) endured a hard-working and preoccupied husband during the final stages of preparation of this third edition. Garching, June 2005 Harald Friedrich Preface to the First Edition In the first few decades of this century atomic physics and quantum mechan- ics developed dramatically from early beginnings to maturity and a degree of completeness. After about 1950 fundamental research in theoretical physics focussed increasingly on nuclear physics and high energy physics, where new conceptual insights were expected to be more probable. A further field of growing importance was theoretical solid state physics, which led to or ac- companied many revolutionary technological developments. In this environ- ment 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 environment can be studied. Effects and phenomena which used to be regarded as small perturbations or experimentally irrelevant exceptional cases have moved into the centre of at- tention At the same time it has become clear that interesting and intricate effects can occur even in seemingly simple systems with only few degrees of freedom. The successful description and interpretation of such effects usually re- quires the solution of a non-trivial Schro¨dinger equation, and perturbative methods are often inadequate. Most lectures and textbooks which go beyond an introductory “Quantum Mechanics I” are devoted to many-body theo- ries 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 sev- eralcourseson“TheoreticalAtomicPhysics”attheMunichUniversitiessince 1984. The present book grew out of these lectures. It is an updated version of the textbook Theoretische Atomphysik, which appeared in German in Sep- tember 1990, and contains the kind of advanced quantum mechanics needed for practical applications in modern atomic physics. The level of abstraction isdeliberatelykeptlow–almostallconsiderationsstartwiththeSchro¨dinger VIII Preface to the First Edition equationincoordinaterepresentation.Thebookisintendedasatextbookfor students who have had a first introductory contact with quantum mechanics. Ihave,however,aimedataself-containedpresentationwhichshould–atleast in principle – be understandable without previous knowledge. Thebookcontainsfivechapters,thefirsttwoofwhichpresentmostlycon- ventional material as can be found in more detail in available textbooks on quantum mechanics and atomic physics. The first chapter contains a concise review of quantum mechanics and the second chapter a deliberatel brief sum- mary of traditional atomic theory. I have taken pains to treat bound states andcontinuumstatesonthesamefooting.Thisenablestheinclusionofacom- paratively straightforward introduction to quantum defect theory (Chap. 3), which hasbecomeapowerful and widely usedtoolforanalyzing atomic spec- tra and which, up to now, has not been treated at such a basic level in a student textbook. The scope of the reaction theory presented in Chap. 4 is thatof“simplereactions”inducedbythecollisionofasingleelectronwithan atom or ion. This avoids many complications otherwise occurring in the defi- nitionsofcoordinates,channelsandpotentials.Ontheotherhand,important concepts such as cross sections, scattering matrix, transition operator, reac- tance matrix, polarization effects, Born approximation, break-up channels, etc. can already be discussed in this simple framework. The last chapter contains a selection of special topics which are currently subject to intense and sometimes controversial discussion. The interest in multiphoton processes has grown strongly with the availability of high-power lasersandunderlinestheimportanceofnon-perturbativemethodsinquantum mechanics. The possibility of using very short laser pulses to study spatially andtemporallylocalizedexcitationsofindividualatomshasrevivedinterestin the relation between classical mechanics and quantum mechanics. The final section discusses “chaos”, which is currently one of the most popular and rapidly growing subfields in almost all fields of physics. While most specific investigations of chaos are numerical experiments on model systems, there are a few prominent examples in atomic physics of simple but real systems, which can be and have been observed in the laboratory and which have all the properties currently causing excitement in connection with chaos. It is a pleasure to thank the many colleagues and friends who unselfishly helped me in the course of writing this book. Special thanks are due to Karl Blum,WolfgangDomcke,Berthold-GeorgEnglert,ChristianJungen,Manfred Kleber, Achim Weiguny and Dieter Wintgen, who read through individual chapters and/or sections and suggested several improvements of the original manuscript. Valuable suggestions and hints were also provided by John S. Briggs, Hubert Klar and Peter Zoller. Gerd Handke and Markus Draeger conscientiously checked more than a thousand formulae and helped to avoid disaster. The original drawings were produced with the competent help of Mrs. I. Kuchenbecker and a plot program specially tailored for the purpose by Markus Draeger. Special thanks are also dur to Dr. H.-U. Daniel from Springer-Verlag. His experience and competence contributed significantly to Preface to the First Edition IX the success of the project. Finally I would like to thank my wife Elfi, who not only read through the German and the English manuscript word by word, butalsosupportedmyworkwithpatienceandencouragementduringthelast three years. Garching Harald Friedrich June 1991 Contents 1 Review of Quantum Mechanics ............................ 1 1.1 Wave Functions and Equations of Motion................... 1 1.1.1 States and Wave Functions ......................... 1 1.1.2 Linear Operators and Observables ................... 3 1.1.3 The Hamiltonian and Equations of Motion ........... 7 1.2 Symmetries............................................. 9 1.2.1 Constants of Motion and Symmetries ................ 9 1.2.2 The Radial Schro¨dinger Equation ................... 12 1.2.3 Example: The Radially Symmetric Harmonic Oscillator 14 1.3 Bound States and Unbound States......................... 16 1.3.1 Bound States ..................................... 16 1.3.2 Unbound States................................... 19 1.3.3 Examples ........................................ 23 1.3.4 Normalization of Unbound States ................... 28 1.4 Processes Involving Unbound States ....................... 30 1.4.1 Wave Packets ..................................... 30 1.4.2 Transmission and Reflection ........................ 33 1.4.3 Time Delays and Space Shifts....................... 35 1.5 Resonances and Channels ................................ 40 1.5.1 Channels ......................................... 41 1.5.2 Feshbach Resonances .............................. 43 1.5.3 Potential Resonances .............................. 48 1.6 Methods of Approximation ............................... 50 1.6.1 Time-independent Perturbation Theory .............. 50 1.6.2 Ritz’s Variational Method .......................... 54 1.6.3 Semiclassical Approximation........................ 57 1.6.4 Inverse Power-Law Potentials ....................... 67 1.7 Angular Momentum and Spin............................. 72 1.7.1 Addition of Angular Momenta ...................... 74 1.7.2 Spin ............................................. 75 1.7.3 Spin-Orbit Coupling ............................... 77 XII Contents Problems ................................................... 79 References .................................................. 83 2 Atoms and Ions............................................ 85 2.1 One-Electron Systems.................................... 85 2.1.1 The Hydrogen Atom............................... 85 2.1.2 Hydrogenic Ions................................... 87 2.1.3 The Dirac Equation ............................... 88 2.1.4 Relativistic Corrections to the Schro¨dinger Equation... 93 2.2 Many-Electron Systems .................................. 95 2.2.1 The Hamiltonian .................................. 95 2.2.2 Pauli Principle and Slater Determinants.............. 96 2.2.3 The Shell Structure of Atoms .......................100 2.2.4 Classification of Atomic Levels ......................103 2.3 The N-Electron Problem .................................107 2.3.1 The Hartree-Fock Method ..........................107 2.3.2 Correlations and Configuration Interaction ...........112 2.3.3 The Thomas−Fermi Model .........................115 2.3.4 Density Functional Methods ........................118 2.4 Electromagnetic Transitions ..............................120 2.4.1 Transitions in General, “Golden Rule” ...............121 2.4.2 The Electromagnetic Field..........................124 2.4.3 Interaction Between Atom and Field.................129 2.4.4 Emission and Absorption of Photons.................130 2.4.5 Selection Rules....................................135 2.4.6 Oscillator Strengths, Sum Rules.....................138 Problems ...................................................140 References ..................................................142 3 Atomic Spectra............................................145 3.1 Long-Ranged and Shorter-Ranged Potentials................146 3.1.1 Very-Long-Ranged Potentials .......................146 3.1.2 Shorter-Ranged Potentials..........................147 3.1.3 The Transition From a Finite Number to Infinitely Many Bound States, Inverse-Square Tails.............152 3.1.4 Example: Truncated Dipole Series in the H− Ion ......158 3.2 One Electron in a Modified Coulomb Potential ..............164 3.2.1 Rydberg Series, Quantum Defects ...................164 3.2.2 Seaton’s Theorem, One-Channel Quantum Defect Theory...........................................171 3.2.3 Photoabsorption und Photoionization ................172 3.3 Coupled Channels .......................................177 3.3.1 Close-Coupling Equations ..........................177 3.3.2 Autoionizing Resonances ...........................181 3.3.3 Configuration Interaction, Interference of Resonances ..186