Electrons in a crystal move through a potential with lattice periodicity: $$V(\mathbf r+\mathbf R)=V(\mathbf r)$$ for every lattice translation $\mathbf R$. Bloch's theorem states that energy eigenfunctions can be chosen in the form $$\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r),$$ where $$u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r).$$ Thus a crystal eigenstate is a plane-wave phase multiplied by a function with the periodicity of the lattice.

Under translation by $\mathbf R$, $$\psi_{n\mathbf k}(\mathbf r+\mathbf R)=e^{i\mathbf k\cdot\mathbf R}\psi_{n\mathbf k}(\mathbf r).$$ The wavevector $\mathbf k$ is therefore a conserved label associated with discrete translational symmetry, often called crystal momentum. States whose wavevectors differ by a reciprocal-lattice vector describe equivalent translational phases, so $\mathbf k$ can be restricted to the first Brillouin zone.

As $\mathbf k$ varies, each band index $n$ defines a dispersion relation $E_n(\mathbf k)$. Gaps can open at Brillouin-zone boundaries because the periodic potential couples plane waves whose wavevectors differ by reciprocal-lattice vectors.

Bloch's theorem is the conceptual bridge from isolated-atom quantum mechanics to electronic band structure. It explains why electrons in a periodic solid are neither simply localized on individual atoms nor completely free plane waves.