  We consider an inverse $N$-body scattering problem of determining two
potentials---an external potential acting on all particles and a pair
interaction potential---from the scattering particles. This paper finds that
the time-dependent Hartree-Fock approximation for a three-dimensional inverse
$N$-body scattering in quantum mechanics enables us to recover the two
potentials from the scattering states with high-velocity initial states. The
main ingredient of mathematical analysis in this paper is based on the
asymptotic analysis of the scattering operator defined in terms of a scattering
solution to the Hartree-Fock equation at high energies. We show that the
leading part of the asymptotic expansion of the scattering operator uniquely
reconstructs the Fourier transform of the pair interaction, and the second term
of the expansion uniquely reconstructs the $X$-ray transform of the external
potential.
