Animating the 2-D Schroedinger equation.

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Animating the 2-D Schroedinger equation.*From*: TDR at vaxc.cc.monash.edu.au*Date*: 10 Mar 1993 16:55:31 +1000

Dear MathGroup, A week or so ago I said I'd post a Notebook to MathSource on how to generate some animations of the one-dimensional time-dependent Schroedinger equation. This I did last week, although I haven't received an acknowledgement yet, so it could be a few more days before it is obtainable from there. As a bonus I've also produced another Notebook that can numerically solve the TWO-dimensional time-dependent Schroedinger equation, and I have posted it to MathSource too. The new Notebook contains some wonderful density-plot animations. With the Notebook, various interactive experiments can done and the probability field gets displayed as a density plot. In one experiment I took a wave-packet (a fuzzy blob in one corner) that moves across the screen to get scattered off a potential column -- it's a very neat animation that illustrates what a symbolic/numeric interface such as InterCall can achieve. By using the InterCall package I was able to write the potential function in Mathematica and yet pass it as an argument to a parabolic partial differential equation solver within in C. This made interactive experiments very easy and also fast. Here is the abstract for the new Notebook that I have sent to MathSource -- I hope someone at wri will put the Notebook in the MathSource directory soon ;-) ++++++++++++++++++++ Animating Schroedinger's equation in two-dimensions. An efficient method for solving parabolic partial differential equations is implemented in Mathematica using InterCall and an external C routine. As an application, the two-dimensional time-dependent Schroedinger's equation is solved for various initial conditions and potential functions. Four different numerical experiments are given: scattering of a particle off a cylindrical potential barrier; a double slit experiment; interaction of wave-packets; and stirring a wave-packet with a potential 'stick'. The resulting animations make excellant demonstrations of the properties of Schroedinger's equation. Also the technique used can be applied to other similar parabolic partial differential equations. ++++++++++++++++++++ Terry.