# Solid State: one-dimensional crystal

1. Apr 12, 2015

### skrat

1. The problem statement, all variables and given/known data
Investigate electronic bands in one-dimensional crystal in weak potential $$V(x)=\sum _n V_0(H(x-na)-H(x-na-a/3))$$ if $H(x)$ is Heaviside Theta function.

a) Determine primitive cell, reciprocal mesh and first Brillouin zone.
b) Calculate the difference between energy levels for first four electron bands.

2. Relevant equations

3. The attempt at a solution

a) No idea what to do?

If those wore Delta functions, than it would be easy, but now I have no idea what to do here?

b) Hmmm, my idea is very long and maybe even wrong. Since I am not sure about it, I would like to firstly describe in a few words before I start writing everything:

1. I would have to solve Schrodinger equation for both parts ($V=0$ and $V=V_0$).
2. Use correct boundary condition on $x=0$ and $x=a/3$
3. Define periodic wave functions and than rewrite the boundary conditions.
4. Have no idea, I hope this leads somewhere???