Transport in 1-D crystals

The electron dispersion relation for any one-dimensional potential can be easily determined numerically. The velocity of the $k$ states can also be determined numerically,

\begin{equation} \vec{v}_{k}= \frac{1}{\hbar}\frac{dE(k)}{dk}. \end{equation}

This can be used to calculate the transport properties of one-dimensional crystals.