Lectures on Atomic Physics Walter R Johnson Department of Physics, University of Notre Dame Notre Dame, Indiana 46556, U S A January 4, 2006 Contents Preface xi 1 Angular Momentum 1 1 1 Orbital Angula[.]
Lectures on Atomic Physics Walter R Johnson Department of Physics, University of Notre Dame Notre Dame, Indiana 46556, U.S.A January 4, 2006 Contents Preface xi Angular Momentum 1.1 Orbital Angular Momentum - Spherical Harmonics 1.1.1 Quantum Mechanics of Angular Momentum 1.1.2 Spherical Coordinates - Spherical Harmonics 1.2 Spin Angular Momentum 1.2.1 Spin 1/2 and Spinors 1.2.2 Infinitesimal Rotations of Vector Fields 1.2.3 Spin and Vectors 1.3 Clebsch-Gordan Coefficients 1.3.1 Three-j symbols 1.3.2 Irreducible Tensor Operators 1.4 Graphical Representation - Basic rules 1.5 Spinor and Vector Spherical Harmonics 1.5.1 Spherical Spinors 1.5.2 Vector Spherical Harmonics Central-Field Schră odinger Equation 2.1 Radial Schrăodinger Equation 2.2 Coulomb Wave Functions 2.3 Numerical Solution to the Radial Equation 2.3.1 Adams Method (adams) 2.3.2 Starting the Outward Integration (outsch) 2.3.3 Starting the Inward Integration (insch) 2.3.4 Eigenvalue Problem (master) 2.4 Quadrature Rules (rint) 2.5 Potential Models 2.5.1 Parametric Potentials 2.5.2 Thomas-Fermi Potential 2.6 Separation of Variables for Dirac Equation 2.7 Radial Dirac Equation for a Coulomb Field 2.8 Numerical Solution to Dirac Equation 2.8.1 Outward and Inward Integrations (adams, outdir, indir) i 1 7 10 11 15 17 19 21 21 23 25 25 27 31 33 36 38 39 41 44 45 46 51 52 57 57 ii CONTENTS 2.8.2 2.8.3 Eigenvalue Problem for Dirac Equation (master) Examples using Parametric Potentials Self-Consistent Fields 3.1 Two-Electron Systems 3.2 HF Equations for Closed-Shell Atoms 3.3 Numerical Solution to the HF Equations 3.3.1 Starting Approximation (hart) 3.3.2 Refining the Solution (nrhf) 3.4 Atoms with One Valence Electron 3.5 Dirac-Fock Equations 60 61 63 63 69 79 79 81 84 88 Atomic Multiplets 4.1 Second-Quantization 4.2 6-j Symbols 4.3 Two-Electron Atoms 4.4 Atoms with One or Two Valence Electrons 4.5 Particle-Hole Excited States 4.6 9-j Symbols 4.7 Relativity and Fine Structure 4.7.1 He-like ions 4.7.2 Atoms with Two Valence Electrons 4.7.3 Particle-Hole States 97 97 101 104 108 112 115 117 117 121 122 Hyperfine Interaction & Isotope Shift 5.1 Hyperfine Structure 5.2 Atoms with One Valence Electron 5.2.1 Pauli Approximation 5.3 Isotope Shift 5.3.1 Normal and Specific Mass Shifts 5.4 Calculations of the SMS 5.4.1 Angular Decomposition 5.4.2 Application to One-Electron Atom 5.5 Field Shift 125 125 130 132 134 135 136 137 139 140 Radiative Transitions 6.1 Review of Classical Electromagnetism 6.1.1 Electromagnetic Potentials 6.1.2 Electromagnetic Plane Waves 6.2 Quantized Electromagnetic Field 6.2.1 Eigenstates of Ni 6.2.2 Interaction Hamiltonian 6.2.3 Time-Dependent Perturbation Theory 6.2.4 Transition Matrix Elements 6.2.5 Gauge Invariance 6.2.6 Electric Dipole Transitions 143 143 144 145 146 147 148 149 150 154 155 CONTENTS 6.3 iii 6.2.7 Magnetic Dipole and Electric Quadrupole Transitions 161 6.2.8 Nonrelativistic Many-Body Amplitudes 167 Theory of Multipole Transitions 171 Introduction to MBPT 7.1 Closed-Shell Atoms 7.1.1 Angular Momentum Reduction 7.1.2 Example: 2nd-order Energy in Helium 7.2 B-Spline Basis Sets 7.2.1 Hartree-Fock Equation and B-splines 7.2.2 B-spline Basis for the Dirac Equation 7.2.3 Application: Helium Correlation Energy 7.3 Atoms with One Valence Electron 7.3.1 Second-Order Energy 7.3.2 Angular Momentum Decomposition 7.3.3 Quasi-Particle Equation and Brueckner-Orbitals 7.3.4 Monovalent Negative Ions 7.4 Relativistic Calculations 7.4.1 Breit Interaction 7.4.2 Angular Reduction of the Breit Interaction 7.4.3 Coulomb-Breit Many-Electron Hamiltonian 7.4.4 Closed-Shell Energies 7.4.5 One Valence Electron 7.5 CI Calculations 7.5.1 Relativistic CI Calculations 179 181 183 186 187 190 191 192 193 194 195 196 198 199 201 202 205 205 206 209 210 MBPT for Matrix Elements 8.1 Second-Order Corrections 8.1.1 Angular Reduction 8.2 Random-Phase Approximation 8.2.1 Gauge Independence of RPA 8.2.2 RPA for hyperfine constants 8.3 Third-Order Matrix Elements 8.4 Matrix Elements of Two-particle Operators 8.4.1 Two-Particle Operators: Closed-Shell Atoms 8.4.2 Two-Particle Operators: One Valence Electron Atoms 8.5 CI Calculations for Two-Electron Atoms 8.5.1 E1 Transitions in He 213 213 215 216 218 219 220 223 223 224 227 228 A Exercises A.1 Chapter A.2 Chapter A.3 Chapter A.4 Chapter A.5 Chapter A.6 Chapter 231 231 233 237 237 239 240 248 BIBLIOGRAPHY C Eckart The application of group theory to the quantum dynamics of monatomic systems Rev Mod Phys., 2:305–380, 1930 A R Edmonds Angular Momentum in Quantum Mechanics Princeton University Press, Princeton, New Jersey, 1974 J P Elliott Proc Roy Soc A, 218:345, 1953 J A Gaunt Proc Roy Soc (London), A122:513, 1929 A E S Green, D L Sellin, and A S Zachor Phys Rev., 184:1, 1969 D R Hartree The Calculation of Atomic Structures J Wiley, New York, 1957 G Huber, F Touchard, S Bă uttgenbach, C Thibault, R Klapisch, H T Duong, S Liberman, J Pinard, J L Vialle, P Juncar, and P Jacquinot Spins, magnetic 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Mittleman Configuration-space hamiltonian for heavy atoms and correction to the breit interaction Phys Rev A, 5:2395–2401, 1972 M H Mittleman Theory of relativistic effects on atoms: Configuration-space... Reduction of the Breit Interaction 7.4.3 Coulomb-Breit Many-Electron Hamiltonian 7.4.4 Closed-Shell Energies 7.4.5 One Valence Electron 7.5 CI Calculations ... calculations of transition amplitudes in the helium isoelectronic sequence Advances in Atomic, Molecular and Optical Physics, 35, 1995 W R Johnson and G Soff Lamb shift in high z atoms Atomic Data