Posted by Steve Simon on June 10, 2014, 5:11 pm, in reply to "2011 Question 7 (Nearly Free Electron Model)"
Hi MRH, Let me take your last question first. While we did not do a nearly free electron question in 2D, the syllabus has not changed since that time (and the homeworks are roughly unchanged too) so I think one should expect that if it occurred once it could occur again. Note also that the 2011 exam has some solutions posted on the web here https://www.physics.ox.ac.uk/expapers/ The phrasing of the question is confusing though. The actual potential must be a real number so if you state there is a nonzero V_G you must assume V_{G} is nonzero as well so as to make a real result. Yes, it would have been more clear if this were stated explicitly, but perhaps this was put in just to see if students would figure it out? Or maybe it was just sloppy. And yes, I think you have the right idea as to how you should handle this 2d problem (and i think I mentioned this in lecture as well). In general you have to write a trial wavefunction that includes all of the fourier components of psi which can be connected together by the scattering potential and which have the same kinetic energy (i..e, have the same magnitude of k). So when one is considering the point (pi/a,0) you need only take a superposition of (pi/a,0) and (pi/a,0). However, when you are considering (pi/a,pi/a) you need to superpose all four corner points (+pi/a, +pi/a) so your trial wavefunction is the sum of four different waves. Note that you are not actually asked to derive the 4x4 secular determinant in the problem! You are given the secular determinant and you are just asked to do some maths on it. 
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