Quoting Eugene Salamin <gene_salamin@yahoo.com>:
This method we have been discussing appears capable of analysing in great detail the electronic structure of one dimensional crystals, at least under the constraint of a given fixed potential. We have relied upon the theorem that the solution space is two dimensional, corresponding to the two arbitrary constants in the general solution of the ODE. How do we handle the three dimensional crystal? The Schroedinger equation becomes a PDE, and the general solution possesses arbitrary functions.
I am not aware of any "momentum gaps" in solid state physics, and I will try to see whether this phenomenon is possible in three dimensions.
The results might be a little easier to visualize in two dimensions, where the complexity inherent in three dimensions is already present. You get Brillouin zones and Fermi levels and such like. I am not sure whether there are separable potentials to get nice, even if unrealistic, models. Generally, I think, matrix Hamiltonians are used rather than the partial differential Schroedinger equation. I'm not competent to discuss this in any further detail, but I do know that bands and gaps and all that are calculable and work out quite nicely. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos