 Solution

Normalize the plane waves in a periodic crystal of area or volume $\mathcal{V}$. The Fourier coefficients of the periodic potential obeyTranslation of the integral by any lattice vector $\mathbf{R}$ leaves it invariant but multiplies it by $e^{-i(\mathbf{k}-{\mathbf{k}}^{\prime })\cdot \mathbf{R}}$. A nonzero result therefore requires $\mathbf{k}-{\mathbf{k}}^{\prime }$ to be a reciprocal lattice vector.

Near an isolated Bragg scattering degeneracy, ordinary nondegenerate perturbation theory fails because the free energies coincide. Retain the resonant plane waves $|\mathbf{k}⟩$, $|\mathbf{k}+\mathbf{G}⟩$ and apply degenerate perturbation theory. Absorb the constant $V_0$ into the energy origin. Since a real potential has $V_{-\mathbf{G}}=V_{\mathbf{G}}^{\ast }$, the projected Schrödinger equation has coefficient matrixA nonzero eigenstate requiresOff-resonant plane waves give higher-order energy shifts omitted in this leading nearly-free electron model.