DATA-log

Studying wave equation discretization has personally led me to understand and appreciate the physics which rely heavily on this mathematics, namely electrodynamics and field theories. I think this sort of ‘visual intuition’ is a important part of learning when dealing with dynamics which can be extremely complicated and complex, at least it has been exactly that  for me. Studying the computability of these equations has definetly paved some way in these theoretically dense subjects.

I recently made few basic computer models of the Schrödinger wave equation with MATLAB for computational physics course work.

Here is a two dimensional version of the discretisized time-dependent wave equation calculated using a clever leapfrog integration algorithm scheme by Visscher.

Things start to get visually more interesting when integrating the time-dependant equation in three dimensions.

Here is another run with a slightly different value for the momentum of the wavepacket.

Unfortunately MATLAB seriously lacks in the volumetric plotting department so we’ll have to do with phong shaded isosurfaces instead of a more appropriate voxel based plot. Don’t worry, this will not end up forming black holes and destroying Vulcan.

Having studied the fundamentals of quantum mechanics from the perspective of numerical computation it has become obvious that stability is a major drawback in most fast numerical approaches. The best scheme to tackle with the issue is by Visscher whose explicit algorithm is both fast and stable. Accuracy is not my biggest concern since I’m mostly interested in visualization of the quantum mechanical world but the algorithm holds very well. The algorithm is second order accurate, there is some dispersion of the wave packet however. It is also straightforward to extend it to n-dimensional space.

My processing example applet of the 1D time-dependent Schrödinger wave equation with the source code is at http://www.punainen.org/~biotek/visscher1d

Reference to the original paper: P.B. Visscher “A fast explicit algorithm for the time-dependent Schrödinger equation”, Computers in Physics Nov/Dec 1991, 596-8