Periodic Boundary Conditions proof

[Wminus]

Hi! When we model bloch-waves in a solid we assume that there exist some kind of periodic boundary conditions such that the wave function is periodic. In 1D, $\psi(x)$ repeats itself for every $L$, $\psi(x) = \psi(x+L)$, such as here:

OK, fine, we get pretty wave solutions if we assume the existence of the PBC. But what $L$? As far as I know the only repeating unit in a crystal is the Wieger Seitz cell, which is sized on the atomic scale.

Is $L$ just the wavelength?

 

[DrDu]

No, L is the size of your crystal, so e.g. 1cm. In the end, this size doesn't matter to much and you often take $L \to \infty$.

 

[Wminus]

No, L is the size of your crystal, so e.g. 1cm. In the end, this size doesn't matter to much and you often take $L \to \infty$.

How can that be? if that is true, it means $\psi$ is a standing wave... But bloch waves are traveling!

 

[DrDu]

That's why you use periodic boundary conditions, the waves then move on a circle and you can have left and right moving waves.

 

[Wminus]

this makes no sense physically. if you have a crystal cube of 1cm, you are assuming that its left side is connected to its right?

 

[DrDu]

No, it doesn't, but there is a theorem by Wigner that the influence of the boundary conditions on the states vanishes like 1/N, where N is the number of particles (or elementary cells). So if you are interested in the bulk states only, you can pick the boundary conditions as seem convenient.

 

[Wminus]

No, it doesn't, but there is a theorem by Wigner that the influence of the boundary conditions on the states vanishes like 1/N, where N is the number of particles (or elementary cells). So if you are interested in the bulk states only, you can pick the boundary conditions as seem convenient.

Ah, OK, I see. So you can pick whatever PBC you like? What about if L = length of unit cell?

 

[DrDu]

No, I said that the boundary conditions become unimportant when L is much larger than the elementary cell.

 

[Wminus]

No, I said that the boundary conditions become unimportant when L is much larger than the elementary cell.

Yes you did, my apologies. And thanks for the help!