  We show that a system of four particles in a one-dimensional box with a
two-particle harmonic interaction can by described by means of the symmetry
point group $O_h$. Group theory proves useful for the discussion of both the
small-box and large-box regimes. We apply perturbation theory and obtain the
corrections of first order for the lowest states. We carry out a simple
Rayleigh-Ritz variational calculation with basis sets adapted to the symmetries
of the system. We also obtain alternative variational results for the first
three lowest energy levels that are more suitable for larger box sizes.
