Theoretical Atomic Physics (Graduate Texts in Physics)

This expanded and updated well-established textbook contains an advanced presentationof quantum mechanics adapted to the requirements of modern atomic physics. Itincludes topics of current interest such as semiclassical theory, chaos, atom optics andBose-Einstein condensation in atomic gases. In order to facilitate the consolidationof the material covered, various problems are included, together with completesolutions. The emphasis on theory enables the reader to appreciate the fundamentalassumptions underlying standard theoretical constructs and to embark on independentresearch projects.The fourth edition of Theoretical Atomic Physics contains anupdated treatment of the sections involving scattering theory and near-thresholdphenomena manifest in the behaviour of cold atoms (and molecules). Special attentionis given to the quantization of weakly bound states just below the continuum thresholdand to low-energy scattering and quantum reflection just above. Particular emphasisis laid on the fundamental differences between long-ranged Coulombic potentialsand shorter-ranged potentials falling off faster than 1/__r__2 at large distances __r__. The newsections on tunable near-threshold Feshbach resonances and on scattering in two spatialdimensions also address problems relevant for current and future research in the fieldof cold (and ultra-cold) atoms. Graduate students and researchers will find this book avaluable resource and comprehensive reference alike. Preface to the Fourth Edition 6 Preface to the Third Edition 8 Preface to the First Edition 10 Contents 13 1 Review of Quantum Mechanics 18 1.1 Wave Functions and Equations of Motion 18 1.1.1 States and Wave Functions 18 1.1.2 Linear Operators and Observables 21 1.1.3 The Hamiltonian and Equations of Motion 25 1.2 Symmetries 28 1.2.1 Constants of Motion and Symmetries 28 1.2.2 The Radial Schrödinger Equation 31 1.2.3 Example: The Radially Symmetric Harmonic Oscillator 33 1.3 Bound States and Unbound States 35 1.3.1 Bound States 35 1.3.2 Unbound States 38 1.3.3 Examples 43 1.3.3.1 Sharp-Step Potential 43 1.3.4 Attractive Coulomb Potential 45 1.3.5 Normalization of Unbound States 49 1.4 Processes Involving Unbound States 50 1.4.1 Wave Packets 50 1.4.2 Transmission and Reflection 55 1.4.3 Time Delays and Space Shifts 56 1.5 Resonances and Channels 62 1.5.1 Channels 63 1.5.2 Feshbach Resonances 66 1.5.3 Potential Resonances 71 1.6 Methods of Approximation 73 1.6.1 Time-Independent Perturbation Theory 73 1.6.2 Ritz's Variational Method 78 1.6.3 Semiclassical Approximation 81 1.6.4 Inverse Power-Law Potentials 91 1.7 Angular Momentum and Spin 98 1.7.1 Addition of Angular Momenta 98 1.7.2 Spin 100 1.7.3 Spin-Orbit Coupling 102 Problems 104 References 108 2 Atoms and Ions 110 2.1 One-Electron Systems 110 2.1.1 The Hydrogen Atom 110 2.1.2 Hydrogenic Ions 112 2.1.3 The Dirac Equation 113 2.1.4 Relativistic Corrections to the Schrödinger Equation 119 2.2 Many-Electron Systems 121 2.2.1 The Hamiltonian 121 2.2.2 Pauli Principle and Slater Determinants 123 2.2.3 The Shell Structure of Atoms 127 2.2.4 Classification of Atomic Levels 130 2.3 The N-Electron Problem 135 2.3.1 The Hartree-Fock Method 135 2.3.2 Correlations and Configuration Interaction 140 2.3.3 The Thomas-Fermi Model 142 2.3.4 Density Functional Methods 147 2.4 Electromagnetic Transitions 149 2.4.1 Transitions in General, ``Golden Rule'' 149 2.4.2 The Electromagnetic Field 153 2.4.3 Interaction Between Atom and Field 159 2.4.4 Emission and Absorption of Photons 160 2.4.4.1 Spontaneous Emission 162 2.4.4.2 Induced Emission 163 2.4.4.3 Absorption 164 2.4.4.4 Photoionization 165 2.4.5 Selection Rules 166 2.4.6 Oscillator Strengths, Sum Rules 168 Problems 171 References 173 3 Atomic Spectra 175 3.1 Long-Ranged and Shorter-Ranged Potentials 176 3.1.1 Very-Long-Ranged Potentials 176 3.1.2 Shorter-Ranged Potentials 178 3.1.3 Example 1. The Lennard-Jones Potential 196 3.1.4 Example 2. The H2+ Molecular Ion 199 3.1.5 The Transition From a Finite Number to Infinitely Many Bound States, Inverse-Square Tails 203 3.1.6 Example: Truncated Dipole Series in the H - Ion 210 3.2 One Electron in a Modified Coulomb Potential 216 3.2.1 Rydberg Series, Quantum Defects 216 3.2.2 Seaton's Theorem, One-Channel Quantum Defect Theory 222 3.2.3 Photoabsorption and Photoionization 225 3.3 Coupled Channels 230 3.3.1 Close-Coupling Equations 230 3.3.2 Autoionizing Resonances 235 3.3.3 Configuration Interaction, Interference of Resonances 239 3.3.4 Perturbed Rydberg Series 247 3.4 Multichannel Quantum Defect Theory (MQDT) 250 3.4.1 Two Coupled Coulomb Channels 250 3.4.2 The Lu-Fano Plot 258 3.4.3 More Than Two Channels 260 3.5 Atoms in External Fields 270 3.5.1 Atoms in a Static, Homogeneous Electric Field 271 3.5.2 Atoms in a Static, Homogeneous Magnetic Field 281 3.5.3 Atoms in an Oscillating Electric Field 293 Problems 296 References 300 4 Simple Reactions 303 4.1 Elastic Scattering 303 4.1.1 Elastic Scattering by a Shorter-Ranged Potential 303 4.1.2 Partial-Waves Expansion 307 4.1.3 Scattering Phase Shifts 308 4.1.4 Radial Lippmann-Schwinger Equation 311 4.1.5 S-Matrix 313 4.1.6 Determination of the Scattering Phase Shifts 314 4.1.7 Near-Threshold Behaviour of the Scattering Phase Shifts 316 4.1.8 Modified Effective-Range Expansions 327 4.1.9 Levinson's Theorem 330 4.1.10 An Example 334 4.1.11 Semiclassical Description of Elastic Scattering 336 4.1.12 Elastic Scattering by a Pure Coulomb Potential 343 4.1.13 Elastic Scattering by a Modified Coulomb Potential, DWBA 345 4.1.14 Feshbach Projection. Optical Potential 348 4.2 Scattering in Two Spatial Dimensions 351 4.2.1 Scattering Amplitude and Scattering Cross Section 352 4.2.2 Lippmann-Schwinger Equation and BornApproximation 353 4.2.3 Partial-Waves Expansion and Scattering Phase Shifts 354 4.2.4 Near-Threshold Behaviour of the Scattering Phase Shifts 358 4.2.5 The Case m=0, s-Waves in Two Dimensions 359 4.2.6 Rutherford Scattering in Two Dimensions 361 4.3 Spin and Polarization 364 4.3.1 Consequences of Spin-Orbit Coupling 364 4.3.2 Application to General Pure Spin States 367 4.3.3 Application to Mixed Spin States 370 4.4 Inelastic Scattering 373 4.4.1 General Formulation 373 4.4.2 Coupled Radial Equations 379 4.4.3 Threshold Effects 384 4.4.4 An Example 388 4.5 Exit Channels with Two Unbound Electrons 390 4.5.1 General Formulation 391 4.5.2 Application to Electrons 398 4.5.3 Example 402 4.5.4 Threshold Behaviour of Ionization Cross Sections 407 Problems 410 References 415 5 Special Topics 418 5.1 Multiphoton Absorption 419 5.1.1 Experimental Observations on Multiphoton Ionization 420 5.1.2 Calculating Ionization Probabilities via Volkov States 423 5.1.3 Calculating Ionization Probabilities via Floquet States 428 5.2 Classical Trajectories and Wave Packets 430 5.2.1 Phase Space Densities 432 5.2.2 Coherent States 437 5.2.3 Coherent Wave Packets in Real Systems 443 5.3 Regular and Chaotic Dynamics in Atoms 447 5.3.1 Chaos in Classical Mechanics 447 5.3.2 Traces of Chaos in Quantum Mechanics 454 5.3.3 Semiclassical Periodic Orbit Theory 460 5.3.4 Scaling Properties for Atoms in External Fields 466 (a) Classical Mechanics 466 (b) Quantum Mechanics 472 (c) Scaled-Fields Spectroscopy 474 (d) Time-dependent Potentials 476 5.3.5 Examples 477 (a) Ionization of the Hydrogen Atom in a Microwave Field 477 (b) Hydrogen Atom in a Uniform Magnetic Field 480 (c) The Helium Atom 488 5.4 Bose-Einstein Condensation in Atomic Gases 492 5.4.1 Quantum Statistics of Fermions and Bosons 492 5.4.2 The Effect of Interactions in Bose-Einstein Condensates 496 5.4.3 Realization of Bose-Einstein Condensation in Atomic Gases 501 5.5 Near-Threshold Quantization and Scattering for Deep Shorter-Ranged Potentials 504 5.5.1 Connecting Near-Threshold Quantization and Near-Threshold Scattering 504 5.5.2 Nonvanishing Angular Momentum 513 5.5.3 Summary 515 5.5.4 Relation to Other Approaches 517 5.6 Near-Threshold Feshbach Resonances 519 5.6.1 Motivation 519 5.6.2 Threshold-Insensitive Parametrization of a Feshbach Resonance 521 5.6.3 Influence on the Scattering Length 525 5.6.4 Influence on the Bound-State Spectrum 528 5.6.5 Relation to the Empirical Formula (5.244) 531 5.7 Some Aspects of Atom Optics 532 5.7.1 Atom-Wall Interactions 533 5.7.2 Evanescent-Wave Mirrors 541 5.7.3 Quantum Reflection 546 5.7.3.1 Rephrasing (5.217) in Terms of the Amplitudes for Transmission and Quantum Reflection 558 5.7.3.2 Observation of Quantum Reflection 559 5.7.3.3 Nonplanar Surfaces 563 5.7.4 Quantum Reflection and Near-Threshold Quantization in Two Spatial Dimensions 564 Problems 566 References 569 Solutions to the Problems 576 References 628 A Special Mathematical Functions 629 A.1 Legendre Polynomials, Spherical Harmonics 629 A.2 Laguerre Polynomials 631 A.3 Gamma Function 632 A.4 Bessel Functions 634 A.5 Confluent Hypergeometric Functions, Coulomb Functions, Whittaker's Function 639 References 642 Index 643 This expanded and updated well-established textbook contains an advanced presentation of quantum mechanics adapted to the requirements of modern atomic physics. It includes topics of current interest such as semiclassical theory, chaos, atom optics and Bose-Einstein condensation in atomic gases. In order to facilitate the consolidation of the material covered, various problems are included, together with complete solutions. The emphasis on theory enables the reader to appreciate the fundamental assumptions underlying standard theoretical constructs and to embark on independent research projects. The fourth edition of Theoretical Atomic Physics contains anupdated treatment of the sections involving scattering theory and near-threshold phenomena manifest in the behaviour of cold atoms (and molecules). Special attention is given to the quantization of weakly bound states just below the continuum threshold and to low-energy scattering and quantum reflection just above. Particular emphasis is laid on the fundamental differences between long-ranged Coulombic potentials and shorter-ranged potentials falling off faster than 1/r2 at large distances r. The new sections on tunable near-threshold Feshbach resonances and on scattering in two spatial dimensions also address problems relevant for current and future research in the field of cold (and ultra-cold) atoms. Graduate students and researchers will find this book a valuable resource and comprehensive reference alike. This expanded and updated well-established textbook contains an advanced presentation of quantum mechanics adapted to the requirements of modern atomic physics. It includes topics of current interest such as semiclassical theory, chaos, atom optics and Bose-Einstein condensation in atomic gases. In order to facilitate the consolidation of the material covered, various problems are included, together with complete solutions. The emphasis on theory enables the reader to appreciate the fundamental assumptions underlying standard theoretical constructs and to embark on independent research projects. The fourth edition of Theoretical Atomic Physics contains an updated treatment of the sections involving scattering theory and near-threshold phenomena manifest in the behaviour of cold atoms (and molecules). Special attention is given to the quantization of weakly bound states just below the continuum threshold and to low-energy scattering and quantum reflection just above. Particular emphasis is laid on the fundamental differences between long-ranged Coulombic potentials and shorter-ranged potentials falling off faster than 1/r2 at large distances r. The new sections on tunable near-threshold Feshbach resonances and on scattering in two spatial dimensions also address problems relevant for current and future research in the field of cold (and ultra-cold) atoms. Graduate students and researchers will find this book a valuable resource and comprehensive reference alike This expanded and updated well-established textbook contains an advanced presentationof quantum mechanics adapted to the requirements of modern atomic physics. Itincludes topics of current interest such as semiclassical theory, chaos, atom optics andBose-Einstein condensation in atomic gases. In order to facilitate the consolidationof the material covered, various problems are included, together with completesolutions. The emphasis on theory enables the reader to appreciate the fundamentalassumptions underlying standard theoretical constructs and to embark on independentresearch projects. The fourth edition of Theoretical Atomic Physics contains anupdated treatment of the sections involving scattering theory and near-thresholdphenomena manifest in the behaviour of cold atoms (and molecules). Special attentionis given to the quantization of weakly bound states just below the continuum thresholdand to low-energy scattering and quantum reflection just above. Particular emphasisis laid on the fundamental differences between long-ranged Coulombic potentialsand shorter-ranged potentials falling off faster than 1/ r 2 at large distances r . The newsections on tunable near-threshold Feshbach resonances and on scattering in two spatialdimensions also address problems relevant for current and future research in the fieldof cold (and ultra-cold) atoms. Graduate students and researchers will find this book avaluable resource and comprehensive reference alike. After a brief review of quantum mechanics and a summary of conventional atomic theory, H. Friedrich discusses the structure of atomic spectra on the basis of quantum defect theory, which is treated for the first time at such a basic level in a textbook. Special attention is given to highly excited states and to the influence of external fields, which can cause intricate and interesting effects in seemingly simple systems. After a chapter on reaction theory the final chapter treats special topics such as multiphoton absorption and chaos. The book contains the kind of advanced quantum mechanics needed for practical applications in modern atomic physics. The presentation is kept deliberately simple and avoids abstract formalism as far as possible. Front Matter....Pages i-xvii Review of Quantum Mechanics....Pages 1-92 Atoms and Ions....Pages 93-157 Atomic Spectra....Pages 159-286 Simple Reactions....Pages 287-401 Special Topics....Pages 403-560 Back Matter....Pages 561-642