Quantum Lens Focusing by Iterating the Schrodinger Equation

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 s gaussian envelope wave packet as governed by the Schrodinger equation. In this case I have chosen the initial packet to be rectangular with gaussian boundaries. The lens is a shaped potential which is either double concave or double convex. The wave packet and the lens are color coded. The color of the initial wave packet is associated with the value of psi*psi in the packet. The color of the bulk of the lens is the ratio of the potential energy to the kinetic energy of the packet. These energies are adjustable using the kx and the Mirror Potential Sliders. Note that the lens potential must be less than the packet potential energy if transmission is desired. The color bars at the left show the learner the value of psi*psi and potential energy/kinetic energy. Thus the lens potential color as indicated on the potential energy color bar must be less than 1 or a shorter wavelength than red if the lens is to transmit the wave packet. Otherwise the lens becomes a mirror. 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 parameters for this are the pair (Iteration Time Step and kx) which, if made large enough, will easily run the diffracting wave packet beyond the gridded width and height. Effectively, beyond the gridded width and height, the potential is infinite and results in strong reflections. Just be aware that when you see reflections from the bounds of the gridded region that the plot data is no longer valid.

After experimenting with the animation is becomes obvious that a concave lens gives better focusing than a convex one. The total desktop computer time for completion of the focus is usually about 2 minutes.