On the symmetry of three identical interacting particles in a one-dimensional box

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$D_{3d}$. 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.