Confined One Dimensional Harmonic Oscillator as a Two-Mode System

TL;DR

This paper investigates a one-dimensional harmonic oscillator in a box as a two-mode system, exploring spectral structures, perturbation theory, and oblique basis methods to understand eigenstates and their applicability to complex systems.

Contribution

It introduces an oblique basis approach combining harmonic oscillator and particle-in-a-box states for better eigenstate description, with implications for complex nuclear systems.

Findings

Oblique basis effectively describes eigenstates away from exact limits.

Perturbation theory works near the harmonic oscillator and box limits.

Matrix diagonalization is necessary for intermediate regimes.

Abstract

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.