Past exam of the general relativity and cosmology and condensed matter physics paper of Part B of the physics course of the University of Oxford / 2011 / Section VI / 7 / i / Solution

Let $u=V_{10}<0$ andAt $X=(\pi /a,0)$ the two degenerate plane waves have momenta $(\pi /a,0)$ and $(-\pi /a,0)$ and coupling $u$. ThusTheir symmetric and antisymmetric combinations are the lower and upper states respectively. The local band gap is $2|u|$.