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