Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 34B / a / Solution

$E_0(k)={\hslash }^2{k}^2/(2m)$. Periodic boundary conditions give $k=2\pi n/L$. Orthogonality of plane waves gives $⟨k|U|k\prime ⟩=U_l$ when $k\prime -k=2\pi l/a$, and zero otherwise.