Theoretical Atomic Physics

512 Pages·2006·5.853 MB·English

by  Professor Dr. Harald Friedrich (auth.)

TheoreticalAtomicPhysics
Harald Friedrich
Theoretical Atomic
Physics
ABC
ProfessorDr.HaraldFriedrich
PhysicsDepartmentT30
TechnischeUniversitätMünchen
James-Franck-Straße1
85747Garching
Germany
E-mail:[email protected]
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ISBN-10 3-540-25644-XSpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-25644-1SpringerBerlinHeidelbergNewYork
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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