  The one-dimensional harmonic oscillator in a box problem 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 two limits has a characteristic spectral structure describing the two
different excitation modes of the system. Near each of these limits, one can
use perturbation theory to achieve an accurate description of the eigenstates.
Away from the exact limits, however, one has to carry out a matrix
diagonalization because the basis-state mixing that occurs is typically too
large to be reproduced in any other way. An alternative to casting the problem
in terms of one or the other basis set consists of using an "oblique" basis
that uses both sets. Through a study of this alternative in this
one-dimensional problem, we are able to illustrate practical solutions and
infer the applicability of the concept for more complex systems, such as in the
study of complex nuclei where oblique-basis calculations have been successful.
Keywords: one-dimensional harmonic oscillator, particle in a box, exactly
solvable models, two-mode system, oblique basis states, perturbation theory,
coherent states, adiabatic mixing.
