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arXiv:math-ph/0512019 · American Journal of Physics

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

V. G. Gueorguiev, A. R. P. Rau, Jerry Paul Draayer

Published Apr 14, 2006• 37 citations

Abstract

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.

DOI: 10.1119/1.2173270

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Abstract This article examines a particle in a one-dimensional potential well V a (x) ∝ x a , where a ≥ 2 is an even integer. The system is referred to as a harmonic oscillator when a = 2 and as a particle in a box when a → ∞. The finite difference method is applied to solve the Schrödinger equation, determining the ground-state energies and wave functions. Giving accurate description in the cases of a = 2 and a → ∞, the numerical method allows us to explore the system for intermediate values of a, revealing how the energies and wave functions evolve between these two limits. The study includes discussions on the virial theorem, the uncertainty principle, excited states, and the limitation of the numerical method.

An exactly solvable model of the one-dimensional quantum harmonic oscillator confined in a box with infinite walls is constructed. We have found explicit expressions of the non-equidistant energy spectrum as well as stationary states wavefunctions in both momentum and position configuration spaces. It is shown that they are expressed through continuous q-Hermite polynomials. We have also found an explicit expression for the kernel of the finite-continuous Fourier transform between these two representation spaces.

Courses on undergraduate quantum mechanics usually focus on solutions of the Schrödinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.