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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
TABLE OF CONTENTS
(The discussion in this chapter reviews some elementary concepts and theories.)
CHAPTER 1 - BACKGROUND ....................... Error! Bookmark not defined.
1.1 Introduction .................................................. Error! Bookmark not defined.
1.2 The Bohr model of the atom ........................ Error! Bookmark not defined.
1.3 Numerical values and the fine structure constantError! Bookmark not defined.
1.4 Atomic dimensions – is a0 a reasonable atomic diameter?Error! Bookmark not
defined.
1.5 Localizing the electron: Is a point particle reasonable?Error! Bookmark not
defined.
1.6 The classical radius of the electron. ............. Error! Bookmark not defined.
1.7 Atomic units. ................................................ Error! Bookmark not defined.
(This is discussion of angular momentum of a single quantum mechanical particle.)
CHAPTER 2 - ANGULAR MOMENTUM ...... Error! Bookmark not defined.
2.1 Introduction .................................................. Error! Bookmark not defined.
2.2 Commutators ............................................... Error! Bookmark not defined.
2.3 Angular momentum raising and lowering operatorsError! Bookmark not defined.
2.4 Angular momentum commutation relations with vector operators. .... Error!
Bookmark not defined.
2.5 Matrix elements of Vector operators ........... Error! Bookmark not defined.
2.6 Eigenfunctions of orbital angular momentum operatorsError! Bookmark not
defined.
2.7 Spin .............................................................. Error! Bookmark not defined.
(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 SOURCESError! Bookmark not
defined.
3.1 Introduction .................................................. Error! Bookmark not defined.
3.2 Two sets of quantum numbers - uncoupled and coupledError! Bookmark not
defined.
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3.3 Vector model of angular momentum ........... Error! Bookmark not defined.
3.4 Examples of calculation of the Clebsch-Gordan coefficientsError! Bookmark not
defined.
3.5 Hyperfine splitting in the hydrogen atom .... Error! Bookmark not defined.
(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 ATOMError!
Bookmark not defined.
4.1 The radial equation for a central potential ... Error! Bookmark not defined.
4.2 Solution of the radial equation in spherical coordinates - the energy eigenvalues
........................................................................... Error! Bookmark not defined.
4.3 The accidental degeneracy of the hydrogen atomError! Bookmark not defined.
4.4 Solution of the hydrogen atom radial equation in spherical coordinates - the energy
eigenfunctions .................................................... Error! Bookmark not defined.
4.5 The nature of the spherical eigenfunctions . Error! Bookmark not defined.
4.6 Separation of the Schrödinger equation in parabolic coordinatesError! Bookmark
not defined.
4.7 Solution of the separated equations in parabolic coordinates - the energy
eigenvalues......................................................... Error! Bookmark not defined.
4.8 Solution of the separated equations in parabolic coordinates - the energy
eigenfunctions .................................................... Error! Bookmark not defined.
(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 ATOMError! Bookmark not
defined.
5.1 Introduction .................................................. Error! Bookmark not defined.
5.2 The classical degeneracy ............................. Error! Bookmark not defined.
5.3 Another constant of the motion - the Lenz vectorError! Bookmark not defined.
(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
........................................................................... Error! Bookmark not defined.
6.1 The Lenz vector in quantum mechanics ...... Error! Bookmark not defined.
6.2 Lenz vector ladder operators; conversion of a spherical eigenfunction into another
spherical eigenfunction ...................................... Error! Bookmark not defined.
6.3 Application of Lenz vector ladder operators to a general spherical eigenfunction
........................................................................... Error! Bookmark not defined.
6.4 A new set of angular momentum operators . Error! Bookmark not defined.
6.5 Energy eigenvalues ...................................... Error! Bookmark not defined.
6.6 Relations between the parabolic quantum numbersError! Bookmark not defined.
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6.7 Relationship between the spherical and parabolic eigenfunctionsError! Bookmark
not defined.
6.8 Additional symmetry considerations ........... Error! Bookmark not defined.
(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 DEGENERACYError! Bookmark
not defined.
7.1 Introduction .................................................. Error! Bookmark not defined.
7.2 Relativistic correction for the electronic kinetic energyError! Bookmark not
defined.
7.3 Spin-Orbit Correction .................................. Error! Bookmark not defined.
7.4 The Darwin Term......................................... Error! Bookmark not defined.
7.5 Evaluation of the terms that contribute to the fine-structure of hydrogenError!
Bookmark not defined.
7.6 The total fine structure correction ................ Error! Bookmark not defined.
7.7 The Lamb shift ............................................. Error! Bookmark not defined.
7.8 Hyperfine structure ...................................... Error! Bookmark not defined.
7.9 The solution of the Dirac equation .............. Error! Bookmark not defined.
(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 ..... Error!
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8.1 Introduction .................................................. Error! Bookmark not defined.
8.2 The Zeeman effect – the hydrogen atom in a constant magnetic field Error!
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8.3 Weak electric field - the quantum mechanical Stark effectError! Bookmark not
defined.
8.4 Weak electric field - the classical Stark effectError! Bookmark not defined.
(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 .............. Error! Bookmark not defined.
9.1 Indistinguishable particles ........................... Error! Bookmark not defined.
9.2 The total energy of the helium atom ........... Error! Bookmark not defined.
9.3 Evaluation of the ground state energy of the helium atom using perturbation theory
........................................................................... Error! Bookmark not defined.
9.4 The variational method ................................ Error! Bookmark not defined.
9.5 Application of the variational principle to the ground state of helium Error!
Bookmark not defined.
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9.6 Excited states of helium ............................... Error! Bookmark not defined.
9.7 Doubly excited states of helium: autoionizationError! Bookmark not defined.
(This chapter are fairly standard, although the discussion of jj-coupling is lengthier than
is found in most texts.)
CHAPTER 10 - MULTIELECTRON ATOMS .... Error! Bookmark not defined.
10.1 Introduction ................................................ Error! Bookmark not defined.
10.2 Electron Configuration .............................. Error! Bookmark not defined.
10.3 The designation of states - LS coupling ..... Error! Bookmark not defined.
10.4 The designation of states – jj coupling ...... Error! Bookmark not defined.
(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........ Error! Bookmark not defined.
11.1 Introduction ................................................ Error! Bookmark not defined.
11.2 Evaluation of the quantum defect .............. Error! Bookmark not defined.
11.3 Classical formulation of the quantum defect and the correspondence principle
........................................................................... Error! Bookmark not defined.
11.4 The connection between the quantum defect and the radial wave functionError!
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(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 .. Error!
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12.1 The Stark effect .......................................... Error! Bookmark not defined.
12.2 The Zeeman effect ..................................... Error! Bookmark not defined.
(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 RADIATIONError! Bookmark
not defined.
13.1 Introduction ................................................ Error! Bookmark not defined.
13.2 Time dependence of the wave function ..... Error! Bookmark not defined.
13.3 Interaction of an atom with a sinusoidal electromagnetic fieldError! Bookmark
not defined.
13.4 A two state system – the rotating wave approximationError! Bookmark not
defined.
13.5 Stimulated absorption and stimulated emissionError! Bookmark not defined.
13.6 Spontaneous emission ................................ Error! Bookmark not defined.
13.7 Angular momentum selection rules ........... Error! Bookmark not defined.
13.8 Selection rules for hydrogen atoms ........... Error! Bookmark not defined.
13.9 Transitions in multi-electron atoms ........... Error! Bookmark not defined.
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