Demos.MorseOscillator.Gaussian1D
Starting from a Gaussian wavepacket, which is displaced from the equilibrium position, the wavefunction immediately changes its shape and undergoes rather complex dynamics. While the position/momentum densities appear completely unstructured, some structures are still visible in the phase space (Wigner) representation. At later times, (full or partial) revivals of the initial wavepacket occur.
Dephasing
| Method | Visualization |
|---|---|
| Quant.-mech. wavefunction |
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| Classical trajectories |
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Soon after the start of the wavepacket propagation, position and momentum wavefunctions become apparently unstructured. In contrast, the Wigner function remains a little more structured before it eventually becomes smeared all over the energetically accessible portion of phase space.
Revivals
| Method | Visualization |
|---|---|
| Quant.-mech. wavefunction |
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| Classical trajectories |
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At much longer times, however, regular, periodic motion is resumed. This rephasing can be explained in terms of quantum beats of the (initially populated) quantum levels. For an account of the theory of (full and fractional!) revivals, see work by Averbukh and Perelman or by Leichtle, Averbukh, and Schleich. In the present case, the full revival time 2π/(χω) is approximately 15,365 atomic units. An (almost) complete restoration of the initial shape of the wavepacket occurs also at half the revival time (7682.5). Further partial revivals are found at 1/3, 1/4, etc. of the revival time, see also the peaks in the corresponding wavepacket autocorrelation function.
Note that the Chebychev approximation scheme is the most suitable numerical propagator for the long propagation time required here. The expansion of the time evolution operator in terms of Chebychev polynomials is known to be highly accurate because of the exponential fall-off of the expansion coefficients beyond a certain threshold.
