A quantum-mechanical statement that in a spatially periodic potential the eigenstates of a single electron can be written as a plane wave multiplied by a lattice-periodic function, ψ_k(r) = e^{i k·r} u_k(r), where u_k(r + R) = u_k(r) for every lattice vector R; this structure underlies band formation and the concept of crystal momentum.

Principle

Lattice translational symmetry constrains single-electron eigenfunctions to Bloch form, which implies that allowed energies form bands parameterized by crystal momentum k and that crystal momentum is conserved modulo reciprocal lattice vectors in processes respecting the periodic potential.

Demonstration

Illustrative scenario: An electron in an ideal one‑dimensional infinite lattice with potential V(x + a) = V(x). Recognition: the Hamiltonian commutes with lattice translations. Action: apply translation symmetry to the eigenvalue problem to find solutions of the form ψ_k(x)=e^{ikx}u_k(x) with u_k periodic. Consequence: the dispersion E(k) is periodic in k with band structure and gaps where no eigenvalue satisfies the Bloch condition.

Misapplication

Interpreting crystal momentum k as the same as free‑particle momentum or assuming Bloch states apply when the periodic symmetry is strongly broken by disorder, surfaces, or finite size; the semantic error is conflating crystal momentum conservation with true momentum conservation in all interactions.

Consequence

Bloch's theorem provides the formal basis for band theory, effective mass, Brillouin zone concepts and phenomena such as Bloch oscillations in idealized conditions; it also delimits analyses that rely on reciprocal‑space quantum numbers.

Reversal

If translational symmetry is absent or strongly broken (e.g., amorphous materials, strong disorder, localization, finite isolated molecules, or reduced-dimensional systems with no long-range periodicity), Bloch's theorem does not apply and eigenstates need not have Bloch form.

Boundary

Clearly within: infinite or macroscopically periodic crystalline solids where the single-particle Hamiltonian is periodic. Boundary case: large but finite crystals with surfaces or weak disorder—bulk states approximate Bloch form while surface or localized states do not. Clearly outside: amorphous solids, isolated atoms or molecules, and strongly disordered localized regimes.

Semantic Tension

Symmetry‑based delocalization versus real‑world symmetry breaking — Bloch states assume exact translational symmetry, but practical materials present defects, surfaces and interactions that partially restore locality and invalidate pure Bloch descriptions.

Synthesis

Bloch's theorem links exact lattice symmetry to a precise wavefunction structure: it is the mathematical foundation that converts spatial periodicity into momentum‑space band structure, and it indicates when reciprocal‑space quantum numbers are the appropriate variables for electronic structure.