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.
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).
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.
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 .
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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