Confined One Dimensional Harmonic Oscillator as a Two-Mode System

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.

Figures (9)

• Figure 1: Two-mode toy system consisting of a particle in a one-dimensional box subject to a central harmonic oscillator restoring force.
• Figure 2: Exact energies of a two--mode system with $m=\hbar =2L/\pi =1$ and $\omega =4$ compared to the spectrum of the one-dimensional harmonic oscillator (left), spectrum of the free particle in a 1D box (right), and spectrum as calculated within a first order perturbation theory of a free particle in a 1D box perturbed by a 1D HO potential. The lowest three eigenenergies of the two--mode system nearly coincide with the 1D HO eigenenergies, while higher energy states are better described as perturbations of the other limit of a free particle in a 1D box.
• Figure 3: Spreading of the wave functions: harmonic oscillator wave functions spread outside the harmonic oscillator potential into the classically forbidden region; particle in a box wave functions are zero at and outside of the box boundary.
• Figure 4: Harmonic-oscillator trial wave functions (dark gray) adjusted with respect to the one-dimensional box problem: (a) adjusted according to the potential width $E_{n}^{1D}=\omega _{n}^{2}L^{2}/2\Rightarrow \omega _{n}=\frac{ \hbar }{L^{2}}\left( 1+2n\right)$, (b) nodally adjusted (first three are deliberately phase shifted), (c) boundary adjusted using $\Psi (q)\rightarrow \Psi (q)-\Psi \left( L\right) (1+q/L)/2-\Psi \left( -L\right) (1-q/L)/2$. The exact wave functions (light gray) for a particle in a box are zero at $\pm 1$, as clearly seen in the (a) graphs.
• Figure 5: Absolute deviations of variously calculated energies from the exact energy eigenvalues for $\omega=16$, $L=\pi /2$, $\hbar =m=1$ as a function of $n$. Circles represent deviation of the exact energy eigenvalue from the corresponding harmonic oscillator eigenvalue ($\Delta E=E^{exact}_n-E^{HO}_n$), the diamonds are the corresponding deviation from the energy spectrum of a particle in a box ($\Delta E=E^{exact}_n-E^{1D}_n$), and the squares are the first-order perturbation theory estimates.