Posted by Steve Simon on April 10, 2014, 4:48 pm, in reply to "Re: Confusion about Brillouin Zones"
I think TNR did a fairly good job of clarifying (and I'm not entirely sure what the question was) but let me add a few comments which might make a few things more crisp: (1) In fig 13.5 there are no fermi surfaces drawn  just the brillouin zones shaded in extended zone scheme. You could translate things by G in which case each BZ would occupy the same kstates as the first BZ. (2) If we were to have electrons in infinitely weak periodic potential, we would be tempted to just draw a parabolic dispersion E proportional to kx^2 + ky^2 at all k, oblivious to any zone boundaries, and you wouldn't think of anything periodic and there would be a single eigenstate at each value of (kx,ky). This does not show the fact that the spectrum remains unchanged when we shift momentum by a reciprocal lattice vector G. The other way to show things would be to fold everything back into a single brillouin zone, in which case we realize that, for example, G = 2pi/a is actually the same point in reciprocal space as G=0. This may seem strange, but there are now many eigenstates at this G, one of them with energy 0 and the other with energy hbar^2 (2 pi/a)^2/(2m), for example. S 
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