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Nearly Free Electron Model

An approximation where electrons are treated as nearly free, allowing for straightforward calculation of energy band structure in simple periodic potentials.
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The statement of the theorem

Consider the Hamiltonian H=H0+V(r)H = H_0 + V(\mathbf{r}), where H0=p22mH_0 = \frac{\mathbf{p}^2}{2m} is the free electron Hamiltonian and V(r)V(\mathbf{r}) is the periodic potential. Using the plane wave basis set, ψk(r)=∑GcGeiG⋅r\psi_k(\mathbf{r}) = \sum_{\mathbf{G}} c_{\mathbf{G}} e^{i \mathbf{G} \cdot \mathbf{r}}, the energy eigenvalues EE and coefficients cGc_{\mathbf{G}} are determined by solving the secular equation derived from the matrix representation of HH: \ndet⁡((E−ℏ22mk⋅k)δG,G′−VG−G′)=0\det \left( \left(E - \frac{\hbar^2}{2m} \mathbf{k} \cdot \mathbf{k} \right) \delta_{\mathbf{G}, \mathbf{G}'} - V_{\mathbf{G}-\mathbf{G}'} \right) = 0 \nwhere G\mathbf{G} and G′\mathbf{G}' are reciprocal lattice vectors, and VG−G′V_{\mathbf{G}-\mathbf{G}'} are the Fourier components of the potential V(r)V(\mathbf{r}).
Source: Wikipedia