Bloch's theorem

Bloch's theorem states that eigenstates of a particle in a periodic potential have the form of a plane wave multiplied by a function with the potential's periodicity.

Translation symmetry: A symmetry in which a system remains unchanged when all positions are shifted by a specified displacement. A lattice translation leaves the periodic potential unchanged, allowing eigenstates to be classified by translation behavior.
Schrödinger equation: The fundamental equation governing the quantum-mechanical evolution and stationary states of a system. Bloch's theorem constrains the stationary states when the Hamiltonian contains a periodic potential.
Band structure: The relationship between allowed energy levels and wavevectors in a periodic material. It is computed by solving for Bloch eigenstates across wavevector space.
Wannier function: A localized orbital-like function formed by combining Bloch states across a band. It represents the same band states in a localized basis rather than as extended Bloch waves.
Lattice translation operator: An operator that shifts a quantum state by a lattice vector. The Hamiltonian commutes with these operators, so their eigenstates can also be energy eigenstates.
Periodic potential: A potential energy function unchanged by translations through a repeating lattice. Its discrete translational symmetry is the condition that makes Bloch-form eigenstates possible.
Effective mass: A measure of how a particle responds to forces within a material, derived from band curvature. The curvature of Bloch energy bands determines carrier effective masses.
Anderson localization: The suppression of wave transport caused by coherent scattering from disorder. Disorder breaks the exact lattice periodicity underlying Bloch's theorem and can localize states.
Crystal momentum: A wavevector-based quantum label defined modulo reciprocal lattice vectors in a periodic crystal. The phase acquired under a lattice translation defines the wavevector in Bloch's theorem.
Bravais lattice: An infinite array of points generated by integer combinations of primitive translation vectors. Its translation vectors define the periodicity required by the theorem.

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