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European Journal of Physics

One-dimensional oscillator in a box

Paolo Amore, Francisco Marcelo Fernández

Published Nov 4, 2009• 13 citations
Publisher

Abstract

We discuss a quantum-mechanical model of two particles that interact by means of a harmonic potential and are confined to a one-dimensional box with impenetrable walls. We apply perturbation theory to the cases of different and equal masses and analyse the symmetry of the states in the latter case. We compare the approximate perturbation results with accurate numerical ones.

DOI
10.1088/0143-0807/31/1/007

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We discuss a quantum-mechanical model of two particles that interact by means of a harmonic potential and are confined to a one-dimensional box with impenetrable walls. We apply perturbation theory to the cases of different and equal masses and analyse the symmetry of the states in the latter case. We compare the approximate perturbation results with accurate numerical ones.
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.
We study the size effect on the energy levels of the D-dimensional isotropic harmonic oscillator confined within a box of radius rcr_crc​ with impenetrable walls. Two different approaches are used to obtain the energy eigenvalues and eigenfunctions for D=1,2,...,5. In the first we solve the Schroedinger equation exactly. In the second we use a series expansion of the wave function. The numerical results obtained are extremely accurate; these values are reported with 50 decimal places.
We generalize a recently developed perturbation approach [F. M. Fern\'andez and E. A. Castro, Phys. Rev. A 46, 7288 (1992)], extending its applicability to a wider variety of separable quantum-mechanical problems. As an illustrative example, we consider a particle in a one-dimensional box with a finite potential wall perturbed by a linear potential and obtain the perturbation corrections to the wave function and energy of an arbitrary state in terms of the corresponding unperturbed quantities.