Ayymmetric Shaped Wave Packet in a Circular Parabolic Potential

This animation owes its speed and accuracy to an algorithm that iterates the imaginary part first and then used the modified complex wave function to iterate the real part. This was shown to reduce many errors in the following paper: P.B. Visscher, A Fast Explicit Algorithm for the Time-Dependent Schr¨odinger Equation, Computers In Physics, 596–598 (Nov/Dec 1991). Without this method the Integral of psi*psi would tend to diverge at much shorter iteration time increments.
This animation shows the two dimensional (2D) propagation of an asymmetric gaussian wave packet as governed by the Schrodinger equation. The reason for the asymmetry is that I wanted to show that the packet shape inverts along both x and y axes when it goes through a turn around. This is a result of the fundamental first in/first out behavior of the wave function when it goes through a maximum in potential which is the cause of the turn around. Isopotentials at multiples of the initial kinetic energy are shown as circles. Many parameters are variable in this animation and they are pretty well defined by the titles of the sliders.

I have deliberately allowed the learner to adjust the variables beyond the ranges where valid propagation results are obtained. Important parameter for this are the pair (Gridding Half Height and kx) which, if made small enough, will easily run the wave packet beyond the gridded vertical half widths. Effectively, beyond the gridded width and height, the potential is infinite and results in strong reflections. Just be aware that when you see significant reflections from the bounds of the gridded region, that the plot data is no longer valid.

The total desktop computation time for completion of a period of motion is usually about 5 minutes. At any time the learner has access to starting a movie that is much faster than the computation time.