A good question was asked today after lecture, and I thought it might be useful to post the answer.
Q: If k is a wavevector, we usually think of 2 pi/k as being the wavelentgth. What does it then mean that k is the same as k + 2pi/a as we found for the vibrational waves and the tight binding chain?
A: Note that the wave is only defined on discrete positions corresponding to, say, the positions of the atoms in the vibrational wave, or the position of the nucleus for the tight binding chain. Look now at the following figure
A wave with k=k+2pi/a has the same value at all of these points, and the wave is not physically meaningful in between these points.
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