[time 44] N-body problem

Stephen P. King (stephenk1@home.com)
Mon, 22 Mar 1999 08:20:05 -0500

Dear Hitoshi,

        I'm looking into the n-body problem:


"Many-body problems in quantum mechanics

     An important idea used in our approach to the many-body problem is
to exploit the non-individuality of identical particles to relate the
N-body problem to certain specially constructed 2-body problems. We
study systems of N identical particles bound by pair potentials. We let
H be the Hamiltonian for the system, with the centre-of-mass KE removed.
     Since the symmetrization postulate (for bosons or fermions) is in
the individual particle indices, the expression of the Bose or Fermi
symmetry in the translation-invariant many-body wave function may be
quite complicated. In suitable units, and for either species of
particle, we have <H> = (N-1)<K + (N/2)V>, where K and V are (reduced)
two-body operators. Thus there is a close relation between H/(N-1) and
the corresponding (reduced) two-body Hamiltonian H = K + (N/2)V. This
relation allows one to derive general lower-bound energy formulas for
the N-body problem: for strongly bound systems, these lower bounds are
often surprisingly good; for harmonic oscillators, they may be exact. In
the case of fermions, the attempt to express the anti-symmetry of the
many-body wave function in expansions over two-body states leads to the
use of non-orthogonal relative coordinates. These methods can also be
applied to the excited N-body states."

I sent Dr. Richard L. Hall
(http://www-cicma.concordia.ca/faculty/rhall/rhall.html#top) e-mail
asking for a copy of his paper: Spectral geometry and the N-body
problem, Richard L. Hall Phys. Rev. A 51, 3499-3505 (1995).

Looking further, I do remember that Prigogine's work is motivated by the
problem of computing n-body minimizations...



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