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Bloch Theorem

Electrons in a periodic potential occupy energy bands separated by gaps. The Bloch theorem states that electron wavefunctions have a specific form, resembling plane waves, within each band.
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The statement of the theorem

Let V(r)V(\mathbf{r}) be a periodic potential with lattice constant R\mathbf{R}, such that V(r)=V(r+R)V(\mathbf{r}) = V(\mathbf{r} + \mathbf{R}). For the time-independent Schrödinger equation, (−ℏ22m∇2+V(r))ψ(r)=Eψ(r)\left(-\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r})\right) \psi(\mathbf{r}) = E \psi(\mathbf{r}), the solutions ψk(r)\psi_k(\mathbf{r}) must take the form:\nψk(r)=eik⋅ruk(r)\psi_k(\mathbf{r}) = e^{i \mathbf{k} \cdot \mathbf{r}} u_k(\mathbf{r})\nwhere k\mathbf{k} is the crystal momentum (or wave vector), and uk(r)u_k(\mathbf{r}) is a function that has the same periodicity as the lattice: uk(r)=uk(r+R)u_k(\mathbf{r}) = u_k(\mathbf{r} + \mathbf{R}). This implies that the allowed energies EE are functions of k\mathbf{k}, E(k)E(\mathbf{k}).
Source: Wikipedia