2D Wave Packet Propagation in a Circular Parabolic Potential

This animation shows the two dimensional (2D) movement of a gaussian envelope wave packet as governed by the Schrodinger equation. For comaprison it includes the 1D motion using the just the kx and x Offset parameters of the 2D version. The potential used is parabolic and circular in the (x,y) plane. The wave packet propagation in a parabolic potential has the very special quality that it is perfectly periodic with the same period as a classical particle. In fact the (x,y) centroid of the absolute value squared of the wave packet follows the trajectory of the classic perticle. These qualities makes the study of propagation in a parabolic potential a very educational experience. The time dependent Schrodinger equation has spatial second derivatives that needs to be evaluated in order to iterate its time depemdence. The problem of defining the wave function, psi(x,y), and its second derivative, d2psi/dx2+d2psi/dy2, using finite difference equations is addressed here. Many parameters are variable in this animation.

I have deliberately allowed the learner to adjust the variables beyond the ranges where valid propagation results are obtained. The most important parameters for this are kx, ky, xOffset, yOffset which if made large enough will easily run the wave packet beyond the gridded half widths. Effectively, beyond the gridded width, the potential is infinite and results in strong reflections

The 2D animation is much richer in diversity than the the 1D version since in 2D we can have an infinite variety of angular momentum components. Note that, when plotting real, imagniary, or phase, the wavefronts are always perpendicular to the trajectory as they must be. Also note that the diameter of the wave packet shrinks and grows as it proceeds along the trajectory. The total desktop computer time for completion of a 1D period is about 12 seconds while the time for a 2D period is about 30 seconds.