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.