I'm having a bit of trouble trying to work out how to construct the secular equation for a 2-D k state, such at that of question 7 in the B3 exam from 2011. If the potential has 2 fourier components, does that mean that the initial k state has two possible states that it can be scattered into (one at every initial k plus each reciprocal lattice vector)? If that's the case I would expect there to be a 3x3 matrix for the amplitudes of the three k possible k states. However, later on the secular determinant is a 4x4 matrix, does that mean we have to account for a k state obtained by adding both reciprocal lattice vectors?
While thinking about that, I was also a bit confused about the Brillouin zones and it's boundaries. The definition of the Brillouin zone is ANY unit cell in the reciprocal lattice, but that means we can shift any Brillouin zone and change it's boundaries around as we please provided we keep the same shape of the cells. That means the k states on the boundaries will change so that if the centre of our BZ was at (0,0), then it's the states along (x, pi/a) and (x, -pi/a) (for example) that have the degeneracy in energies, but if we moves our BZ to match the one they give in the 2011 paper (i.e. centred at (pi/a, pi/a)), then the boundaries are different and the degenerate energies are also different (Now the boundaries are along (x,0) and (x, 2*pi/a). How are these scenarios consistent? I understand that the energies are only in terms of crystal momentum, but the crystal momentum in the case of (pi/a) and (2*pi/a) are different, surely?
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