Subject: Summary: 3 body problems

From: "Shobe, Dave" <[email-masked]>
Date: Wed, 12 Dec 2001 17:45:34 -0500

Thanks to the many people who answered my question about quantum 3-bodied
problems.

> Maybe someone here knows this.
> 
> It is well known that the Schrödinger equation has only been analytically
> solved for two-body problems: H, He+, Li++, ...
> 
> My question (which may sound like hair splitting to some, but is actually
a
> profound question) is: whether the 3-body problems He and (H2)+ are
unsolved
> because people have not been clever enough to find the solution, or if
they
> are unsolved because they CAN'T be solved using the usual functions
> (logarithms, sines, cosines, etc.)  Just as certain integrals can't be
> expressed, and one has to make up new functions like the Bessel functions
to
> "solve" the integration problem.

Apparently, even classical-physics analogues like Sun-Earth-Moon or
Sun-Earth-Jupiter are insoluble.  A web site that was provided discusses
this in about the level of detail I wanted :-)
http://www.physics.cornell.edu/sethna/teaching/sss/jupiter/Web/Rest3Bdy.htm.
And as one of you points out, the correspondence principle implies that if
the general classical case is insoluble, the general quantum case is
insoluble also.

As for (H2)+, i.e. a hydrogen-molecule radical-cation, whether it is
solvable depends on how one defines the problem.  It turns out that *for a
given internuclear distance*, the problem can be solved using confocal
elliptical coordinates, and this solution is discussed in some quantum
chemistry textbooks.

--David Shobe
Süd-Chemie Inc.
phone (502) 634-7409
fax     (502) 634-7724
email  [email-masked]

Don't bother flaming me: I'm behind a firewall.