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I have been trying to prove that two Bloch states with wavector k and k + (2pi/a) are identical, as the book says:

enter image description here

can anyone help me?

I have look at some more sources in the solid-state physics, and all of them say that that the block state is periodic in reciprocal space but they never prove it. enter image description here

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Maths reason

In order to compare the phase in two unit cells we need to pick two points where we know that $u_k$ is the same. The easiest way of doing this is to pick a point, and the same point in the next unit cell (at a distance of one lattice constant, $a$).

The phase difference from the exponential part between these locations is:

$e^{k.a}$,

if we say $k' = k + N \times 2\pi/a$

$e^{k'.a} = e^{k.a} e^{N2\pi} = e^{k.a}$,

(because $e^{N2\pi} = 1$).

IE. Adding or subtracting 2pi makes no mathematical difference, because exp(2pi) = 1, so adding 2pi to the exponent is the same as multiplying by 1.

Physics Reason

Why did we decide that we had to compare points one whole lattice constant apart again? It was because of the $u_k$, we needed it the same at each point to pick out just the effect of the exponential out the front. But that $u_k$ can include a phase gradient of its own (it generally will). What we have really shown is that we CAN (if we choose) take all of the spatial phase gradients that are steeper than $2\pi$ per unit cell and role them into the Bloch part of the wave (the $u_k$). This is using the "reduced Brillouin zone" and is the standard convention.

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  • $\begingroup$ I think I moreless get now. Just to confirm, to prove that f_k (r)=f_k+2pi/a (r), I need to consider that I am also moving in real space: from 0 to a. In other words, if I move in the reciprocal space I am simultaneously moving in real space? $\endgroup$ – Who Oct 27 '20 at 17:25
  • $\begingroup$ also what do you mean by role them into the Bloch part of the wave? $\endgroup$ – Who Oct 27 '20 at 23:00
  • $\begingroup$ The $u_k$ is called the Bloch part of the wave, the exponential out the front is the other part. I mean that the $u_k$ can have all the steep phases put into it, and taken out of the exponential. You do NOT need to move in real space when you move in reciprocal space. Instead move in reciprocal space and then ask what (observable) thing in the real world will actually have changed, (a phase is not an observable thing, but a phase difference between two locations is an observable thing). $\endgroup$ – Dast Oct 28 '20 at 10:35

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