Topiczs in atomic physics/ Charlie Burkhardt, Jacob J. Leventhal.

Chapter 1 Background......................................... 1
1.1 Introduction .............................. . ...... ............ ........... .. 1
1.2 The Bohr Model of the Atom ............................................... 1
1.3 Numerical Values and the Fine Structure Constant..................... 7
1.4 Atomic Dimensions-Is ao a Reasonable Atomic Diameter?......... 8
1.5 Localizing the Electron: Is a Point Particle Reasonable? ............ 10
1.6 The Classical Radius of the Electron...................................... 11
1.7 Atomic Units ................... ...... ................... .......... ..... . 11
Chapter 2 Angular Momentum................................ 14
2.1 Introduction ................................................................. .. 14
2.2 Commutators................................................................... 18
2.3 Angular Momentum Raising and Lowering Operators ............... 20
2.4 Angular Momentum Commutation Relations with
Vector Operators .............................................................. 25
2.5 Matrix Elements of Vector Operators ..................................... 26
2.6 Eigenfunctions of Orbital Angular Momentum Operators ............. 29
2.7 Spin ............................................ ................... ............. . 33
2.8 The Stern-Gerlach Experiment........................................ 41
Chapter 3 Angular Momentum-Two Sources.............. 46
3.1 Introduction ................................................ .................... 46
3.2 Two Sets of Quantum Numbers-Uncoupled and Coupled.......... 46
3.3 Vector Model of Angular Momentum.................................. 51
3.4 Examples of Calculation of the Clebsch-Gordan Coefficients....... 55
3.5 Hyperfine Splitting in the Hydrogen Atom.............................. 61
Chapter 4 The Quantum Mechanical Hydrogen Atom.... 73
4.1 The Radial Equation for a Central Potential ............................. 73
4.2 Solution of the Radial Equation in Spherical
Coordinates-The Energy Eigenvalues................................. 75
4.3 The Accidental Degeneracy of the Hydrogen Atom.................... 77
4.4 Solution of the Hydrogen Atom Radial Equation in
Spherical Coordinates-The Energy Eigenfunctions................... 79
4.5 The Nature of the Spherical Eigenfunctions........................... 82
4.6 Separation of the Schr6dinger Equation in
Parabolic Coordinates....................................................... 82
4.7 Solution of the Separated Equations in Parabolic
Coordinates-The Energy Eigenvalues................................. 85
4.8 Solution of the Separated Equations in Parabolic
Coordinates-The Energy Eigenfunctions............................. 87
Chapter 5 The Classical Hydrogen Atom................... 92
5.1 Introduction .......................... ................... ....... ............... 92
5.2 The Classical Degeneracy ................................................... 95
5.3 Another Constant of the Motion-The Lenz Vector.................... 97
Chapter 6 The Lenz Vector and the
Accidental Degeneracy......... .................................. 105
6.1 The Lenz Vector in Quantum Mechanics................................ 105
6.2 Lenz Vector Ladder Operators; Conversion of a Spherical
Eigenfunction into Another Spherical Eigenfunction.................. 109
6.3 Application of Lenz Vector Ladder Operators to a
General Spherical Eigenfunction.................. ....................... 114
6.4 A New Set of Angular Momentum Operators............................ 116
6.5 Energy Eigenvalues....................................................... ... 118
6.6 Relations Between the Parabolic Quantum Numbers ................ 120
6.7 Relationship Between the Spherical and
Parabolic Eigenfunctions............... .................................. . 122
6.8 Additional Symmetry Considerations................................... 123
Chapter 7 Breaking the Accidental Degeneracy ......... 126
7.1 Introduction ................. ..................... ................. ......... 126
7.2 Relativistic Correction for the Electronic Kinetic Energy .............. 127
7.3 Spin-Orbit Correction...................................................... 128
7.4 The Darwin Term ............................................................ 130
7.5 Evaluation of the Terms That Contribute to the Fine-Structure
of Hydrogen ..... .............................. . ... ..................... 130
7.6 The Total Fine-Structure Correction....................................... 135
7.7 The Lamb Shift............................................................... 137
7.8 Hyperfine Structure........................................................... 139
7.9 The Solution of the Dirac Equation........................................ 142
Chapter 8 The Hydrogen Atom in External Fields......... 145
8.1 Introduction ................. ...... ................. .......... ............... 145
8.2 The Zeeman Effect-The Hydrogen Atom in a Constant
Magnetic Field................................................................. 146
8.3 Weak Electric Field-The Quantum Mechanical Stark Effect ........ 159
8.4 Weak Electric Field-The Classical Stark Effect....................... 171
Chapter 9 The Helium Atom ........... ................ ......... 178
9.1 Indistinguishable Particles................................................... 178
9.2 The Total Energy of the Helium Atom.................................... 180
9.3 Evaluation of the Ground State Energy of the Helium Atom
Using Perturbation Theory .............................. ..... .......... 183
9.4 The Variational Method.................................................... 186
9.5 Application of the Variational Principle to the Ground
State of Helium ................................................................ 187
9.6 Excited States of Helium................................................. 189
9.7 Doubly Excited States of Helium: Autoionization...................... 192
Chapter 10 Multielectron Atoms................................ 196
10.1 Introduction .......................... ....... ......... ..... ... ............. ... 196
10.2 Electron Configuration....................................................... 196
10.3 The Designation of States-LS Coupling ................................ 198
10.4 The Designation of States-jj Coupling................................ 207
Chapter 11 The Quantum Defect ............................... 214
11.1 Introduction ............................... ....... .. ....... ... ................... 214
11.2 Evaluation of the Quantum Defect........................................ 216
11.3 Classical Formulation of the Quantum Defect and the
Correspondence Principle.................................................. 220
11.4 The Connection Between the Quantum Defect and the
Radial Wave Function....................................................... 225
Chapter 12 Multielectron Atoms in External Fields....... 230
12.1 The Stark Effect.............................................................. 230
12.2 The Zeeman Effect.......................................................... 238
Chapter 13 Interaction of Atoms with Radiation......... 246
13.1 Introduction ............................. .. ........ ..... ....... ............... 246
13.2 Time Dependence of the Wave Function............................... 248
13.3 Interaction of an Atom with a Sinusoidal
Electrom agnetic Field........................................................ 249
13.4 A Two-State System-The Rotating Wave Approximation............ 251
13.5 Stimulated Absorption and Stimulated Emission ...................... 254
13.6 Spontaneous Emission ....................................................... 260
13.7 Angular Momentum Selection Rules..................................... 266
13.8 Selection Rules for Hydrogen Atoms ................................... 267
13.9 Transitions in Multielectron Atoms........................................ 272
Answers to Selected Problems............................................ 279