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Confined one-dimensional harmonic oscillator as a two-mode system
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10.1119/1.2173270
/content/aapt/journal/ajp/74/5/10.1119/1.2173270
http://aip.metastore.ingenta.com/content/aapt/journal/ajp/74/5/10.1119/1.2173270
View: Figures

Figures

Image of Fig. 1.
Fig. 1.

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

Image of Fig. 2.
Fig. 2.

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.

Image of Fig. 3.
Fig. 3.

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.

Image of Fig. 4.
Fig. 4.

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).

Image of Fig. 5.
Fig. 5.

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.

Image of Fig. 6.
Fig. 6.

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

Image of Fig. 7.
Fig. 7.

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

Image of Fig. 8.
Fig. 8.

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 .

Image of Fig. 9.
Fig. 9.

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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/content/aapt/journal/ajp/74/5/10.1119/1.2173270
2006-05-01
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Confined one-dimensional harmonic oscillator as a two-mode system
http://aip.metastore.ingenta.com/content/aapt/journal/ajp/74/5/10.1119/1.2173270
10.1119/1.2173270
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