^{1,a)}, A. R. P. Rau

^{2}and J. P. Draayer

^{2}

### 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.

This work was supported by the National Science Foundation under Grants Nos. PHY 0140300 and PHY 0243473, and the Southeastern Universities Research Association. One of the authors (A.R.P.R) thanks the Research School of Physical Sciences and Engineering at the Australian National University for its hospitality during the writing of this paper. This work was partially performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

I. INTRODUCTION

II. HARMONIC OSCILLATOR IN A ONE-DIMENSIONAL BOX

III. SPECTRAL STRUCTURE AT DIFFERENT ENERGY SCALES

IV. HILBERT SPACE OF THE BASIS WAVE FUNCTIONS

A. Harmonic oscillator in the one-dimensional box basis

B. Particle in a box in the harmonic oscillator basis

C. Oblique basis for the two-mode system

V. DISCUSSION OF THE TOY MODEL CALCULATIONS AND RESULTS

VI. CONCLUSION

VII. STUDENT PROBLEMS

### Key Topics

- Oscillators
- 77.0
- Wave functions
- 56.0
- Perturbation theory
- 14.0
- Free oscillations
- 13.0
- Eigenvalues
- 11.0

## Figures

Two-mode toy system consisting of a particle in a one-dimensional box subject to a central harmonic oscillator restoring force ().

Two-mode toy system consisting of a particle in a one-dimensional box subject to a central harmonic oscillator restoring force ().

Exact energies of a two-mode system with and compared to the spectrum of the one-dimensional harmonic oscillator (left), spectrum of the free particle in a one-dimensional box (right), and the spectrum as calculated within a first order perturbation theory of a free particle in a one-dimensional box perturbed by a one-dimensional harmonic oscillator potential. The lowest three eigenenergies of the two-mode system nearly coincide with the one-dimensional harmonic oscillator eigenenergies, while higher energy states are better described as perturbations of the other limit of a free particle in a one-dimensional box.

Exact energies of a two-mode system with and compared to the spectrum of the one-dimensional harmonic oscillator (left), spectrum of the free particle in a one-dimensional box (right), and the spectrum as calculated within a first order perturbation theory of a free particle in a one-dimensional box perturbed by a one-dimensional harmonic oscillator potential. The lowest three eigenenergies of the two-mode system nearly coincide with the one-dimensional harmonic oscillator eigenenergies, while higher energy states are better described as perturbations of the other limit of a free particle in a one-dimensional box.

The harmonic oscillator wave functions spread outside the harmonic oscillator potential into the classically forbidden region; the particle in a box wave functions are zero at and outside the box boundary.

The harmonic oscillator wave functions spread outside the harmonic oscillator potential into the classically forbidden region; the particle in a box wave functions are zero at and outside the box boundary.

Harmonic oscillator trial wave functions in blue (dark gray) adjusted (a) according to the potential width , (b) nodally adjusted (first three are phase shifted), and (c) boundary adjusted using . The exact wave functions in red (light gray) for a particle in a box are zero at , as clearly seen in (a).

Harmonic oscillator trial wave functions in blue (dark gray) adjusted (a) according to the potential width , (b) nodally adjusted (first three are phase shifted), and (c) boundary adjusted using . The exact wave functions in red (light gray) for a particle in a box are zero at , as clearly seen in (a).

Absolute deviations of variously calculated energies from the exact energy eigenvalues for and as a function of eigenstate number . The blue circles represent the deviation of the exact energy eigenvalue from the corresponding harmonic oscillator eigenvalue, , the red diamonds are the corresponding deviation from the energy spectrum of a particle in a box, , and the green squares are the first-order perturbation theory calculations.

Absolute deviations of variously calculated energies from the exact energy eigenvalues for and as a function of eigenstate number . The blue circles represent the deviation of the exact energy eigenvalue from the corresponding harmonic oscillator eigenvalue, , the red diamonds are the corresponding deviation from the energy spectrum of a particle in a box, , and the green squares are the first-order perturbation theory calculations.

As in Fig. 5 but for relative deviations from the exact energy eigenvalues of the three calculations.

As in Fig. 5 but for relative deviations from the exact energy eigenvalues of the three calculations.

Nonzero components of the 105th exact eigenvector in the basis of a free particle in a one-dimensional box. The parameters are and .

Nonzero components of the 105th exact eigenvector in the basis of a free particle in a one-dimensional box. The parameters are and .

Nonzero components of the third harmonic oscillator (even parity) eigenvector expanded in the basis of a free particle in a one-dimensional box. The parameters are and .

Nonzero components of the third harmonic oscillator (even parity) eigenvector expanded in the basis of a free particle in a one-dimensional box. The parameters are and .

The coherent structure with respect to the nonzero components of the 25th, 27th, and 29th exact eigenvector in the basis of a free particle in a one-dimensional box. The parameters are and .

The coherent structure with respect to the nonzero components of the 25th, 27th, and 29th exact eigenvector in the basis of a free particle in a one-dimensional box. The parameters are and .

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