# Tight binding method for a 1D crystal with a diatomic basis

1. Jan 6, 2012

Hi, I'd be most grateful for any help regarding the following problem:

Consider a 1D crystal with 2 atoms in a primitive cell (let's call them atoms A and B). Each atom has only one valence orbital denoted as $\left|\psi_A(n)\right>$ and $\left|\psi_B(n)\right>$ respectively.

Show that the superposition
$\left|k\right> = \sum_{n = -\infty}^{\infty}e^{ikna}\left[ c_A\left|\psi_A(n)\right> + c_B\left|\psi_B(n)\right>\right]$
satisfies the Bloch's theorem ($c_A, c_B$ are constant)
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I know that the theorem is usually formulated as follows:
There is a $\vec{k}$ from the reciprocal lattice for each wavefunction so that
$\psi\left( \vec{r}+\vec{R}\right) = e^{i\vec{k}\vec{R}}\psi\left( \vec{r}\right)$
$\vec{R}$ is a vector from the direct lattice. But I still don't know what to do.

I am probably not very good with quantum mechanics yet (we had the solid state and QM simultaneously this semester) and I realize this may be very easy. But I have been stuck on this and will be grateful for any help.