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286 pages, January 4, 2006

Abstract:

The author states at the outset that [the]: "aim in this course was to provide opportunities for hands-on practice in the calculation of atomic wave functions and energies. The lectures start with a review of angular momentum theory including the formal rules for manipulating angular momentum operators, a discussion of orbital and spin angular momentum, Clebsch-Gordan coefficients and three-j symbols... More advanced material on angular momentum needed in atomic structure calculations followed, including an introduction to graphical methods, irreducible tensor operators, spherical spinors and vector spherical harmonics. The lectures on angular momentum were followed by an extended discussion of the central--field Schrodinger equation. The Schrodinger equation was reduced to a radial differential equation and analytical solutions for Coulomb wave functions were obtained. Again, this reduction should have been familiar from first--year Quantum Mechanics. This preliminary material was followed by an introduction to the finite difference methods used to solve the radial Schrodinger equation. A subroutine to find eigenfunctions and eigenvalues of the Schrodinger equation was developed. This routine was used together with parametric potentials to obtain wave functions and energies for alkali atoms. The Thomas--Fermi theory was introduced and used to obtain approximate electron screening potentials. Next, the Dirac equation was considered. The bound--state Dirac equation was reduced to radial form and Dirac--Coulomb wave functions were determined analytically. Numerical solutions to the radial Dirac equation were considered and a subroutine to obtain the eigenvalues and eigenfunctions of the radial Dirac equation was developed. In the third part of the course, many electron wave functions were considered and the ground-state of a two--electron atom was determined variationally. This was followed by a discussion of Slater--determinant wave functions and a derivation of the Hartree--Fock equations for closed--shell atoms. Numerical methods for solving the HF equations were described. The HF equations for atoms with one--electron beyond closed shells were derived and a code was developed to solve the HF equations for the closed--shell case and for the case of a single valence electron. Finally, the Dirac--Fock equations were derived and discussed.The final section of the material began with a discussion of second- quantization. This approach was used to study a number of structure problems in first--order perturbation theory, including excited states of two--electron atoms, excited states of atoms with one or two electrons beyond closed shells and particle--hole states. Relativistic fine--structure effects were considered usingthe "no--pair" Hamiltonian. A rather complete discussion of the magnetic--dipole and electric quadrupole hyperfine structure from the relativistic point of view was given, and nonrelativistic limiting forms were worked out in the Pauli approximation. Fortran subroutines to solve the radial Schrodinger equation and the Hartree--Fock equations were handed out to be used in connection with weekly homework assignments. Some of these assigned exercises required the student to write or use Fortran codes to determine atomic energy levels or wave functions. Other exercises required the student to write maple routines to generate formulas for wave functions or matrix elements. Additionally, more standard "pencil and paper" exercises on Atomic Physics were assigned." Further details follow from the textbook's contents: "1 Angular Momentum 1; 1.1 Orbital Angular; Central-Field Schroedinger Equation; Spherical Spinors p. 21; 1.5.2 Vector Spherical Harmonics; Central-Field Schroedinger Equation;2.6 Separation of Variables for Dirac Equation 51 2.7 Radial Dirac Equation for a Coulomb Field 52 2.8 Numerical Solution to Dirac Equation 57 2.8.1 Outward and Inward Integrations (adams, outdir, indir) Self-Consistent Fields;Numerical Solution to the HF Equations 79 3.3.1 Starting Approximation (hart) 79 3.3.2 Refining the Solution (nrhf) 81 3.4 Atoms with One Valence Electron 84 3.5 Dirac-Fock Equation;4 Atomic Multiplets 97 4.1 Second-Quantization 97 4.2 6-j Symbols;Relativity and Fine Structure 117 4.7.1 He-like ions 117 4.7.2 Atoms with Two Valence Electrons 121 4.7.3 Particle-Hole States 5 Hyperfine Interaction and Isotope Shift 125 5.1 Hyperfine Structure; 6 Radiative Transitions 143 6.1 Review of Classical Electromagnetism 6.2 Quantized Electromagnetic Field 146 6.2.1 Eigenstates of Ni 147 6.2.2 Interaction Hamiltonian 148 6.2.3 Time-Dependent Perturbation Theory 149 6.2.4 Transition Matrix Elements 150 6.2.5 Gauge Invariance 154 6.2.6 Electric Dipole Transitions 155 6.2.7 Magnetic Dipole and Electric Quadrupole Transitions 161 6.2.8 Nonrelativistic Many-Body Amplitudes 167 6.3 Theory of Multipole Transitions 171 7 Introduction to MBPT 179 7.1 Closed--Shell Atoms 7.3.3 Quasi--Particle Equation and Brueckner-Orbitals 196 7.3.4 Monovalent Negative Ions 7.5.1 Relativistic CI Calculations 210 8 MBPT for Matrix Elements 213 8.2.2 RPA for hyperfine constants p.219 8.3 Third-Order Matrix Elements 220 8.4 Matrix Elements of Two-particle Operators 223 8.4.1 Two-Particle Operators: Closed-Shell Atoms 223 8.4.2 Two-Particle Operators: One Valence Electron Atoms 224 8.5 CI Calculations for Two-Electron Atoms 227 8.5.1 E1 Transitions in He 228 A Exercises 231; A.1 Chapters 1 to 8. " http://www.nd.edu/~johnson/Publications/book.pdf

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Open access: http://www.nd.edu/~johnson/Publications/book.pdf

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