In the Debye model, we introduced periodic boundary conditions, even though our material is not ‘periodic’ (it has a finite length L). This results in k_x = 2π/L n_x.
In footnote 2, it is explained that this should have the same results as fixed boundary conditions. The problem is that this second result is different: k_x = π/L n_x.
Why is this not a problem?
And would this be a problem if our final answer depended on L?
(I know that this doens’t seem to matter for the Debye model, because the final answers don’t depend on L, but I would like to know the general case)
if you calculate the density of states for both cases, you will find that they give the same result and same dependence on L
So, if I understand correctly: even though the periodic boudary condition is wrong, we don’t care because it gives the same results (?)
The idea is more that the system size L is so large that boundary effects don’t play a role. This becomes different if the system is so small that it can only host a few states. Then the details of the boundary start to matter.
To elaborate on that: with fixed boundary conditions we obtain solutions that are \sim \sin \pi n x/L, compared to \exp 2 \pi i n x / L. In the first expression n \geq 0 because n < 0 corresponds to the same solutions. This, however, gives the same total number of states because the distance between neighboring k-points is twice larger in the 2nd expression.