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

  1. 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.)
  2. 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).