TOPICS IN ATOMIC PHYSICS
C. E. Burkhardt
Department of Physics
St. LouisCommunity College
St. Louis, MO63135
J. J. Leventhal
Department of Physics
University of Missouri - St. Louis
St. Louis, MO63121
TABLE OF CONTENTS
(The discussion in this chapter reviews some elementary concepts and theories.)
CHAPTER 1 - BACKGROUND...... 2
1.1 Introduction...... 2
1.2 The Bohr model of the atom...... 4
1.3 Numerical values and the fine structure constant...... 15
1.4 Atomic dimensions – is a reasonable atomic diameter?...... 18
1.5 Localizing the electron: Is a point particle reasonable?...... 21
1.6 The classical radius of the electron...... 23
1.7 Atomic units...... 25
(This is discussion of angular momentum of a single quantum mechanical particle.)
CHAPTER 2 - ANGULAR MOMENTUM...... 2
2.1 Introduction...... 2
2.2 Commutators...... 11
2.3 Angular momentum raising and lowering operators...... 15
2.4 Angular momentum commutation relations with vector operators...... 27
2.5 Matrix elements of Vector operators...... 29
2.6 Eigenfunctions of orbital angular momentum operators...... 37
2.7 Spin...... 46
(This is a discussion of angular momentum algebra for the addition of two angular momenta, It is important throughout the text.)
CHAPTER 3 - ANGULAR MOMENTUM - TWO SOURCES...... 2
3.1 Introduction...... 2
3.2 Two sets of quantum numbers - uncoupled and coupled...... 4
3.3 Vector model of angular momentum...... 14
3.4 Examples of calculation of the Clebsch-Gordan coefficients...... 20
3.5 Hyperfine splitting in the hydrogen atom...... 32
(This contains the standard treatment of the hydrogen atom in spherical coordinates with emphasis on the conditions that force quantization. Also contains the separation in parabolic coordinates.)
CHAPTER 4 - THE QUANTUM MECHANICAL HYDROGEN ATOM....2
4.1 The radial equation for a central potential...... 2
4.2 Solution of the radial equation in spherical coordinates - the energy eigenvalues 7
4.3 The accidental degeneracy of the hydrogen atom...... 12
4.4 Solution of the hydrogen atom radial equation in spherical coordinates - the energy eigenfunctions 15
4.5 The nature of the spherical eigenfunctions...... 20
4.6 Separation of the Schrödinger equation in parabolic coordinates...... 22
4.7 Solution of the separated equations in parabolic coordinates - the energy eigenvalues 26
4.8 Solution of the separated equations in parabolic coordinates - the energy eigenfunctions 31
(This is not covered in any text of which we are aware. It is based on papers written by us and published in AJP. We believe this is a unique feature of this book.)
CHAPTER 5 - THE CLASSICAL HYDROGEN ATOM...... 2
5.1 Introduction...... 2
5.2 The classical degeneracy...... 9
5.3 Another constant of the motion - the Lenz vector...... 12
(This is not covered in any text of which we are aware. It is based on papers written by us and published in AJP. We believe this is a unique feature of this book. Emphasis is also on the consequences of the separability in parabolic coordinates and the relationship between the spherical eigenfunctions and the parabolic eigenfunctions. This is not covered in most Atomic Physics texts nor is it covered in most Quantum Mechanics texts )
CHAPTER 6 - THE LENZ VECTOR AND THE ACCIDENTAL DEGENERACY 2
6.1 The Lenz vector in quantum mechanics...... 2
6.2 Lenz vector ladder operators; conversion of a spherical eigenfunction into another spherical eigenfunction 10
6.3 Application of Lenz vector ladder operators to a general spherical eigenfunction 20
6.4 A new set of angular momentum operators...... 26
6.5 Energy eigenvalues...... 29
6.6 Relations between the parabolic quantum numbers...... 34
6.7 Relationship between the spherical and parabolic eigenfunctions...... 40
6.8 Additional symmetry considerations...... 42
(This is a standard treatment of hydrogenic fine structure. While there is more discussion of the Darwin term than is found in most texts there is really nothing unique here. It must, however, be included for completeness.)
CHAPTER 7 - BREAKING THE ACCIDENTAL DEGENERACY...... 2
7.1 Introduction...... 2
7.2 Relativistic correction for the electronic kinetic energy...... 5
7.3 Spin-Orbit Correction...... 8
7.4 The Darwin Term...... 11
7.5 Evaluation of the terms that contribute to the fine-structure of hydrogen.12
7.6 The total fine structure correction...... 22
7.7 The Lamb shift...... 26
7.8 Hyperfine structure...... 30
7.9 The solution of the Dirac equation...... 36
(The treatment of the Zeeman effect is standard. The quantum mechanical treatment of the Stark effect in hydrogen is extended from the usual treatment to include solution using parabolic coordinates. Additonal uniqueness if this chapter is in the comparison between the quantum mechanical treatment of the Stark effect with the classical treatment.)
CHAPTER 8 - THE HYDROGEN ATOM IN EXTERNAL FIELDS
8.1 Introduction
8.2 The Zeeman effect – the hydrogen atom in a constant magnetic field
8.3 Weak electric field - the quantum mechanical Stark effect
8.4 Weak electric field - the classical Stark effect
(This chapter contains standard treatments of the helium atom energy levels and their relationship to the hydrogen atom. Methods of computing the helium energies are compared and the variational principle utilized and discussed extensively. The variational principle is usually omitted from most quantum mechanics courses, but the treatment here is standard.)
CHAPTER 9 - THE HELIUM ATOM...... 2
9.1 Indistinguishable particles...... 2
9.2 The total energy of the helium atom...... 7
9.3 Evaluation of the ground state energy of the helium atom using perturbation theory 14
9.4 The variational method...... 19
9.5 Application of the variational principle to the ground state of helium....22
9.6 Excited states of helium...... 27
9.7 Doubly excited states of helium: autoionization...... 35
(This chapter are fairly standard, although the discussion of jj-coupling is lengthier than is found in most texts.)
CHAPTER 10 - MULTIELECTRON ATOMS...... 2
10.1 Introduction...... 2
10.2 Electron Configuration...... 3
10.3 The designation of states - LS coupling...... 6
10.4 The designation of states – jj coupling...... 24
(This chapter presents the quantum defect in a way that is seldom seen in texts. In keeping with the theme of this work the quantum defect is related to classical concepts and the correspondence principle. )
CHAPTER 11 - THE QUANTUM DEFECT
11.1 Introduction
11.2 Evaluation of the quantum defect
11.3 Classical formulation of the quantum defect and the correspondence principle
11.4 The connection between the quantum defect and the radial wave function
(The treatment of the Zeeman effect is standard. The treatment of the Stark effect rests on unique material presented in the Chapter 11. )
CHAPTER 12 - MULTIELECTRON ATOMS IN EXTERNAL FIELDS
12.1 The Stark effect
12.2 The Zeeman effect
(This chapter is fairly standard, although an effort is made to reconcile the classical concept that accelerating charges radiate with the quantum concept of stationary states. Again, appeal is made to classical physics.)
CHAPTER 13 - INTERACTION OF ATOMS WITH RADIATION
13.1 Introduction
13.2 Time dependence of the wave function
13.3 Interaction of an atom with a sinusoidal electromagnetic field
13.4 A two state system – the rotating wave approximation
13.5 Stimulated absorption and stimulated emission
13.6 Spontaneous emission
13.7 Angular momentum selection rules
13.8 Selection rules for hydrogen atoms
13.9 Transitions in multi-electron atoms
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