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Bloch's theorem states that the eigenstates of an electron in a periodic potential V(r) = V(r + R) for all lattice vectors R take the form psi_{nk}(r) = eik·r u_{nk}(r), where u_{nk}(r) has the periodicity of the lattice. The quantum number k (crystal momentum) lives in the first Brillouin zone, and n is the band index. This theorem is the foundation of electronic band theory: it reduces the problem of an electron in an infinite crystal to solving for u_{nk} within a single unit cell, and it explains why electronic states organize into continuous energy bands separated by gaps.
Bloch's theorem is the single most important result in the quantum theory of solids. It answers the question: what do electron wavefunctions look like in a crystal, where the potential repeats periodically? The answer is elegant — they are Bloch waves of the form psi_{nk}(r) = eik·r u_{nk}(r), where the exponential is a plane-wave envelope and u_{nk}(r) is a function with the full periodicity of the lattice. The theorem follows directly from the commutation of the Hamiltonian with lattice translation operators: since [H, T_R] = 0 for any lattice vector R, energy eigenstates can be chosen as simultaneous eigenstates of all T_R, and the eigenvalues of T_R must be phases eik·R.
The quantum number k is called the crystal momentum (up to a factor of hbar) and lives in the first Brillouin zone. It labels how the wavefunction's phase evolves from one unit cell to the next. For each k, the Schrodinger equation becomes an eigenvalue problem for u_{nk} within a single unit cell with periodic boundary conditions, yielding a discrete set of eigenvalues E_n(k) indexed by the band index n. As k varies continuously across the Brillouin zone, each E_n(k) traces out an energy band. The collection of all bands E_n(k) is the band structure of the crystal — the central object of solid-state physics.
Two features of Bloch's theorem have far-reaching consequences. First, k is defined only modulo reciprocal lattice vectors G, meaning psi_{n,k+G} and psi_{nk} describe the same physics. This is why the first Brillouin zone suffices. Second, a Bloch electron in a perfect periodic potential experiences no scattering — the wavefunction is a stationary state that propagates indefinitely. Electrical resistance comes entirely from departures from perfect periodicity: phonons, impurities, surfaces, and defects. This insight, which seems counterintuitive (how can an electron move freely through a dense array of atoms?), is the starting point for understanding metallic conduction.
The practical power of Bloch's theorem is that it reduces a many-body problem in infinite space to a tractable eigenvalue problem in a single unit cell, parameterized by k. Modern band structure calculations — whether using nearly free electron models, tight-binding, or density functional theory — all begin with this reduction. The resulting band structure E_n(k) determines whether a material is a metal, semiconductor, or insulator, governs optical absorption, dictates transport properties, and is the foundation on which all of condensed matter physics is built.