Details

The guiding equation for the particle velocity is , which is calculated from the gradient of the phase of the wavefunction in the energy representation. In the eikonal form the total wavefunction becomes

.

In this case, the amplitude and the phase reduce to

and

,

from which the velocity is derived by differentiating the total phase function

with

and

,

where , and so on.

In the case that the phase function does not depend on the variable , the velocity becomes

with

and

,

which could be simplified to

.

Local singularities in the velocity do appear to exist for certain and values (), which makes the trajectories very unstable.

In the program, to make the results more accurate, increase PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps.

References

[1] P. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.

[2] D. Bohm, Quantum Theory, New York: Prentice-Hall, 1951.

[3] I. Marzoli, F. Saif, I. Bialynicki-Birula, O. M. Friesch, A. E. Kaplan, and W. P. Schleich, "Quantum Carpets Made Simple." arXiv:quant-ph/9806033.