TOPICS IN ATOMIC PHYSICS
C. E. Burkhardt Department of Physics St. Louis Community College St. Louis, MO 63135 & J. J. Leventhal Department of Physics University of Missouri - St. Louis St. Louis, MO 63121
CHAPTER 1 - BACKGROUND ...... 1 1.1 Introduction ...... 1 1.2 The Bohr model of the atom...... 2 1.3 Numerical values and the fine structure constant ...... 9 1.4 Atomic dimensions – is a0 a reasonable atomic diameter? ...... 11 1.5 Localizing the electron: Is a point particle reasonable? ...... 13 1.6 The classical radius of the electron...... 14 1.7 Atomic units...... 15
CHAPTER 2 - ANGULAR MOMENTUM...... 18 2.1 Introduction ...... 18 2.2 Commutators ...... 23 2.3 Angular momentum raising and lowering operators...... 26 2.4 Angular momentum commutation relations with vector operators...... 33 2.5 Matrix elements of Vector operators ...... 35 2.6 Eigenfunctions of orbital angular momentum operators...... 40 2.7 Spin...... 46
CHAPTER 3 - ANGULAR MOMENTUM - TWO SOURCES...... 62 3.1 Introduction ...... 62 3.2 Two sets of quantum numbers - uncoupled and coupled...... 63 3.3 Vector model of angular momentum ...... 69 3.4 Examples of calculation of the Clebsch-Gordan coefficients ...... 73 3.5 Hyperfine splitting in the hydrogen atom ...... 81
CHAPTER 4 - THE QUANTUM MECHANICAL HYDROGEN...... 95 4.1 The radial equation for a central potential ...... 95 4.2 Solution of the radial equation in spherical coordinates - the energy eigenvalues ...... 98 4.3 The accidental degeneracy of the hydrogen atom...... 101 4.4 Solution of the hydrogen atom radial equation in spherical coordinates - the energy eigenfunctions ...... 103 4.5 The nature of the spherical eigenfunctions ...... 107 4.6 Separation of the Schrödinger equation in parabolic coordinates...... 109 4.7 Solution of the separated equations in parabolic coordinates - the energy eigenvalues..... 112 4.8 Solution of the separated equations in parabolic coordinates - the energy eigenfunctions 115
CHAPTER 5 - THE CLASSICAL HYDROGEN ATOM ...... 120 5.1 Introduction ...... 120 5.2 The classical degeneracy ...... 125 5.3 Another constant of the motion - the Lenz vector ...... 127
CHAPTER 6 - THE LENZ VECTOR AND THE ACCIDENTAL DEGENERACY 136 6.1 The Lenz vector in quantum mechanics ...... 136 6.2 Lenz vector ladder operators; conversion of a spherical eigenfunction into another spherical eigenfunction ...... 142 6.3 Application of Lenz vector ladder operators to a general spherical eigenfunction...... 148 6.4 A new set of angular momentum operators ...... 152 6.5 Energy eigenvalues...... 154 6.6 Relations between the parabolic quantum numbers...... 157 6.7 Relationship between the spherical and parabolic eigenfunctions...... 160
CHAPTER 7 - BREAKING THE ACCIDENTAL DEGENERACY...... 164 7.1 Introduction ...... 164 7.2 Relativistic correction for the electronic kinetic energy ...... 166 7.3 Spin-Orbit Correction ...... 168 7.4 The Darwin Term ...... 170 7.5 Evaluation of the terms that contribute to the fine-structure of hydrogen ...... 171 7.6 The total fine structure correction...... 178 7.7 The Lamb shift...... 180 7.8 Hyperfine structure ...... 183 7.9 The solution of the Dirac equation...... 187
CHAPTER 8 - THE HYDROGEN ATOM IN EXTERNAL FIELDS ...... 190 8.1 Introduction ...... 190 8.2 The Zeeman effect – the hydrogen atom in a constant magnetic field...... 191 8.3 Weak electric field - the quantum mechanical Stark effect ...... 208 8.4 Weak electric field - the classical Stark effect...... 223
CHAPTER 9 - THE HELIUM ATOM...... 230 9.1 Indistinguishable particles ...... 230 9.2 The total energy of the helium atom...... 233 9.3 Evaluation of the ground state energy of the helium atom using perturbation theory ...... 237 9.4 The variational method ...... 240 9.5 Application of the variational principle to the ground state of helium ...... 242 9.6 Excited states of helium...... 245 9.7 Doubly excited states of helium: autoionization...... 251
CHAPTER 10 - MULTIELECTRON ATOMS ...... 254 10.1 Introduction ...... 254 10.2 Electron Configuration ...... 255 10.3 The designation of states - LS coupling ...... 257 10.4 The designation of states – jj coupling ...... 268
CHAPTER 11 - THE QUANTUM DEFECT ...... 276 11.1 Introduction ...... 276 11.2 Evaluation of the quantum defect ...... 280 11.3 Classical formulation of the quantum defect and the correspondence principle...... 286 11.4 The connection between the quantum defect and the radial wave function...... 293
CHAPTER 12 - MULTIELECTRON ATOMS IN EXTERNAL FIELDS...... 297 12.1 The Stark effect...... 297 12.2 The Zeeman effect ...... 307
CHAPTER 13 - INTERACTION OF ATOMS WITH RADIATION ...... 316 13.1 Introduction ...... 316 13.2 Time dependence of the wave function ...... 318 13.3 Interaction of an atom with a sinusoidal electromagnetic field ...... 320 13.4 A two state system – the rotating wave approximation ...... 323 13.5 Stimulated absorption and stimulated emission...... 327 13.6 Spontaneous emission...... 335 13.7 Angular momentum selection rules ...... 342 13.8 Selection rules for hydrogen atoms...... 344 13.9 Transitions in multi-electron atoms ...... 349