The Quantum Three-Body Problem
In a nice analogy with planetary motion, the quantum two-body problem
is exactly solvable: the Schrödinger equation for hydrogen
is a textbook problem in every undergraduate quantum course. Attempts
to solve for the eigenstates of the helium wave function exactly
were not nearly as
persistent
as they were for the classical three body problem. The positively
charged hydrogen molecule, the negatively charged hydrogen atom, and
heavier nuclei with two electrons are other quantum three-body problems
which are classic problems on which one can test approximate quantum
methods. (I got involved in one of the more
bizarre approaches, the
Fock expansion.)
There is another nice analogy between the quantum and classical problems.
We understand the qualitative behavior of planetary motion using the
``non-interacting planet'' approximation. That is, the orbits of the
planets are ellipses, plus small perturbations from other planets.
Quantum chemistry, and much of solid-state physics, is qualitatively
understood in the ``non-interacting electron'' approximation. We think
of the electrons in an atom as filling orbitals: two in the 1S shell,
two in the 2S shell and six in the 2P orbitals, ... This makes perfect
sense if the electron-electron repulsion is ignored. Why does it make
any sense at all, once we include the electron-electron repulsion?
Experimentally, of course, the shell model for atoms is a powerful
tool: it's our best explanation for the periodic table!
Questions
- Find out what the current explanation for the success of the
non-interacting electron approximation is in the context of quantum
chemistry. (In solid-state physics, Fermi Liquid Theory and Density
Functional Theory provide competing explanations.)
- Arnol'd (appendix 8, Math Methods of Classical Mechanics) describes
an approximation for the Earth's motion where the other planets are
spread over their elliptical orbits proportionally
to the time spent in travelling each piece of the orbit. Is there
a corresponding ``mean-field'' approximation in atomic physics,
where the charge-density of the other electrons produces an effective
single-electron potential? (Hartree-Fock theory?)
Jupiter:
How to Get Jupiter
Jupiter is available
for Windows 95, Windows NT, Macintosh, and several Unix platforms
(the IBM RS6000, Sun Sparc, Dec Alpha (courtesy Kamal Bhattacharya),
Linux, and the PowerPC running AIX4.1).
The files are available without charge by anonymous FTP
(ftp.lassp.cornell.edu) or
via
the World Wide Web.
Last modified: May 19, 1996
James P. Sethna,
sethna@lassp.cornell.edu.
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).