Here is my summary of what we did last class with regard to finding the eigenvectors for an infinite 1D lattice. The eigenvectors are indexed by q and the oscillation frequency depends on q. The relationship, between frequency and q, is called a dispersion relation. One has to limit the range of q to avoid duplicate eigenvectors. Can someone delineate how that works and what the range of q should be before class today? Please comment here.
It seems to me that you might have made a mistake in writing out the omega squared term. I think it should be 1-cos(qa) not cos(qa)-1.
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