Posted by Steve Simon on April 19, 2014, 11:47 am, in reply to "Tight Binding Chain "Done Right""
Edited by board administrator April 19, 2014, 11:55 am
I didn't quite get the same answer, although very close (I got the same denominator, but not numerator  note that H_{nm} is defined to be ( n H m ) even though n> and m> are not necessarily orthogonal  as a result I don't get the A's and B's upstairs). The eigenvalue equation to solve is H_{nm} phi_m = E S_{nm} \phi_m with sums over m implied. This does not answer your question about complex quantities though. Note that your expression is manifestly real. Be^{ik} is the complex conjugate of B*e^{ik}, so their sum is real (and A is necessarily real). If you want to make it into sins and cosines, I would write B = B e^{i z} for some real z. Then terms like B e^{i k} + B^* e^{i k} become B 2 cos(k + z) 
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