Creating bands using square well eigenstates.

Now that we have the single square well bound state problem pretty well under control, let's use those states to create bands. One thing you can do to start is a conceptual drawing. In this we are guided by the Bloch form of the crystal eigenstate and the approximation we used before, where only $I_{11}$ and maybe another overlap integral, one that is unitless, were kept. Keep it simple. Don't expand the approach beyond that.

1st suggestion:  Use specific states and actual numbers for A, B, D, k and kappa.  Anything you do with symbols will probably inconclusive and unhelpful. My preference is for the 250 eV deep, 1A well, but any of them are okay. Note that for the 1A wells E3 is the highest energy bound state.*

2nd suggestion: Pick a good separation. One with some overlap but not too much. Maybe about 1 A? or 0.5 A?

3rd suggestion: Sketch the integrands. Pay attention to signs. Remember that $I_{11}$ is an overlap integral with units of energy and that the q dependence is separate from that. The sign of $I_{11}$ effects the qualitative nature of the q dependence.


* I think my original idea of a 4 well state may have been an over-reach and that the numbers you would get from that will be "awkward", but I am really not sure. In 1D, excited states tend to get really far apart in energy. Or maybe it is because it is a square well? Anyway, i got concerned about the huge depth of that one and the values that would ensue for the $I_{11}$ integral.

PS. Nice work so far everybody. It is the time you wasted on it that makes it important.