Energy Spectrum of a Particle Within a Confined Double Well Potential

The energy levels of a particle within a confined double well potential are studied in this work. The spectrum of the particle can be obtained by solving the corresponding Schrödinger equation but, for practical purposes, we have used a numerical approach based in the diagonalization of the matrix related to the Hamiltonian when the wavefunction is represented as an expansion in terms of "a particle-in-a-box" basis functions. The results show that, in the symmetric confining case, the energy levels are degenerate and a regular pairwise association between them is observed, similarly as it occurs in the free case. Moreover, when the confining is asymmetric, the degeneration is partially lifted but the pairwise association of the energy levels becomes irregular. The lifting of the degeneration in the latter case is addressed to the lack of symmetry or distortion of the system, namely, to a sort of Jahn-Teller effect which is common in the energy levels of diatomic molecules, to which a double well potential can be crudely associated. In the symmetric case, the states with nodes at the origin are recognized to be the same as those of the harmonic oscillator confined by two impenetrable walls, in such a way that the system presented in this work would be interpreted as half the solution of the problem of a particle within a confined four well potential. The latter suggests the existence of a sort of hidden symmetry which remains to be studied in a more detailed way.