  We present an exact quantum mechanical analysis of collinear four-wave mixing
in a multicomponent Bose-Einstein condensate consisting of sodium atoms in the
F=1 ground state. Technically, this is achieved by taking advantage of the
conservation laws of the system to represent its Hamiltonian in terms of
angular momentum operators. We discuss explicitly the build-up of matter-wave
side-modes from noise, as well as the correlations between these modes. We show
the appearance of a strong quantum entanglement between hyperfine states. We
also demonstrate that for finite atomic numbers, the system exhibits periodic
collapses and revivals in the exchange of atoms between different spin states.
