Quantum Coulomb Systems
Lectures
on Monday and Wednesday 14:00-16:00
at the TUM in Room 03.10.011
This page will be continuously edited, so please consult it on a weekly
basis.
The goal of the course is to give
an introduction to mathematical problems related to
the constitution of matter, i.e., quantum mechanical charged particles
that interact with each other through Coulomb forces. The
fundamental equation governing matter is the Schroedinger equation
supplemented by the Pauli exclusion principle. Needless to say that the
range of phenomena described by this equation is huge, e.g., it
includes
all of chemistry. Hence we have to limit ourselves to a few questions
that we can treat in a mathematically reasonable fashion.
Among the physical topics to be investigated are:
The stability and instability of matter under various
circumstances such as normal (non-relativistic) matter, relativistic
matter as it occurs in stars, matter that interacts with fields; the
radiation field is a particularly important case.
Binding of atoms under various physical circumstances,
non-relativistic, relativistic and in the presence of fields.
Existence and basic properties of thermodynamic functions.
The Schroedinger equation is often much too difficult to handle and can
be approximated by considerably simpler models, such as Thomas-Fermi
theory, Hartree and Hartree-Fock theory. These models are very
important in that they provide excellent intuitions for the
understanding of the Schroedinger equation. Generally, the construction
of `effective equations' is an important approach in modeling.
The main mathematical tool in the investigation of these topics is
functional analysis. The theory will be developed throughout the course
in a practical fashion. Among the tools used will be measure theory,
L^p
spaces and the various topologies for analyzing optimization problems.
Likewise an elementary theory of partial differential equations
together with the relevant
inequalities will be developed as we go along. While this course is not
a systematic exposition
of analysis, you will develop a feeling for this material that will
guide you, hopefully with success, through subsequent studies.
The key prerequisites for this course is a good knowledge of analysis
in finite dimensional spaces and linear algebra. Kowledge of physics,
in
particular, quantum mechanics is not expected, but a serious interest
in
physical phenomena is required.
In addition to the lectures I will run a
seminar in which students will make presentations
on some topic. I will furnish a list of suitable topics and will help
to prepare the talks. The point about this activity is to encourage you
to do some work on your own and develop some expertise. Last but not
least we get to know each other.
References: 1) Analysis, Elliott
H. Lieb and Michael Loss, Graduate
Studies in Mathematics Vol. 14, Second
Edition,
American Mathematical
Society, Provicence, 2000.
2) The stability of
matter: from atoms to stars, Selecta of Elliott H. Lieb, Fourth
Edition,
Springer 2005.
Below are titles of lectures. These will eventually
be
converted into links pointing to lecture notes in PDF format.
Lectures
Note:
These notes are not yet
polished and hence there are a number
of inaccuracies and the references are not complete.
Copyright by Michael Loss. The use of these notes for non- commercial
purposes is
permitted.
1. Some
historical comments
2. The
hydrogen atom, quantum mechanics in a nutshell
3. Sobolev's inequality (last
edit Tuesday, April 26, 2005)
4. Summary on Sobolev spaces
(last edit Wednesday, May 4 , 2005)
5.The single particle Schroedinger
equation for general potentials (this is a copy of
the relevant chapter of `Analysis' by Lieb and Loss)
6. Higher eigenvalues and the Pauli
Exclusion prinicples (last edit Saturday, May 7, 2005)
7. Some heuristical ideas about
Semi-classical estimate (last edit Monday, May 9, 2005)
8. Lieb-Thirring inequality
(last edit Monday May 9, 2005)
9. Stability of Matter
(last edit Wednesday May 18, 2005)
10. Thomas-Fermi Theory (last edit Wednesday May
23, 2005)
11. The no-binding theorem
and stability (last edit Monday May 30, 2005)
12. Improved constants
for Lieb-Thirring inequalities (last edit Wednesday June 1, 2005)
13. Relativistic systems
(last edit, Wednesday June 8)
14. Estimate on the indirect
part of the Coulomb energy (these notes are joint work with E. Lieb)
15. Localization of the
kinetic energy (last edit, Monday June 20)
16. An electrostatic
inequality (last edit, Monday June 20)
17. Putting the whole thing
together; relativistic stability (last edit, Monday June 20)
18. Magnetic fields (last edit,
Wednesday June 22)
19. The Pauli operator (last edit,
Monday June 27)
20. Stability
of matter with magnetic fields
21. A short introduction to
statistical mechanics (last edit, Wednesday July 6)
22. The thermodynamics limit
of a classical system (last edit, Wednesday July 6)
23. Quantum statistical
mechanics (last edit, Wednesday July 6)
24. The thermodynamic limit
for matter (last edit, Wednesday July 13)
25. The Cheese Theorem (last edit
Wednesday July 13)
Here is a concatenated version
containing all the lectures except Section 5. Thanks go
to Max Lein for arranging this