In our recent paper, my supervisor and I developed a method for generating the electronic band structure of one-dimensional periodic potentials using matrix mechanics. We used the Kronig-Penney model, which has known analytical solutions, as our benchmark and then extended the approach to potentials that had no analytical solutions.
Recent I made use of MATLAB’s ability to quickly generate animated GIFs by repeatedly writing new figures as frames. So now I can make animations like this:
This is the band structure of the Kronig-Penney model as you gradually turn on the potential (the black rectangles) and see the colored bands begin to split. The horizontal axis is in K-space, not real space, so the spacing of the barriers and wells isn’t meaningful in an absolute sense and is merely for decoration. The height of the barriers in terms of the vertical energy scale is correct, as well as the relative sizing of the barriers and wells.