Confined one-dimensional harmonic oscillator as a two-mode system

The one-dimensional harmonic oscillator in a box is possibly the simplest example of a two-mode system. This system has two exactly solvable limits, the harmonic oscillator and a particle in a (one-dimensional) box. Each of the limits has a characteristic spectral structure describing the two different excitation modes of the system. Near these limits perturbation theory can be used to find an accurate description of the eigenstates. Away from the limits it is necessary to do a matrix diagonalization because the basis-state mixing that occurs is typically large. An alternative to formulating the problem in terms of one or the other basis set is to use an "oblique" basis that uses both sets. We study this alternative for the example system and then discuss the applicability of this approach for more complex systems, such as the study of complex nuclei where oblique-basis calculations have been successful. (C) 2006 American Association of Physics Teachers.