  We study a quantum-mechanical system of three particles in a one-dimensional
box with two-particle harmonic interactions. The symmetry of the system is
described by the point group $D_{3d}$. Group theory greatly facilitates the
application of perturbation theory and the Rayleigh-Ritz variational method. A
great advantage is that every irreducible representation can be treated
separately. Group theory enables us to predict the connection between the
states for the small box length and large box length regimes of the system. We
discuss the crossings and avoided crossings of the energy levels as well as
other interesting features of the spectrum of the system.
