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: The result that the Eigenfunction s can be written in this form for a periodic system is called Bloch's theorem. The plane wave wavevector (or ''Bloch wavevector'') k (multiplied by the reduced Planck's Constant , this is the particle's ''crystal momentum'') is unique only up to a Reciprocal Lattice Vector , so one only needs to consider the wavevectors inside the first Brillouin Zone . For a given wavevector and potential, there are a number of solutions, indexed by ''n'', to Schrödinger's Equation for a Bloch electron. These solutions, called bands, are separated in energy by a finite spacing at each k; if there is a separation that extends over all wavevectors, it is called a (complete) Band Gap . The Band Structure is the collection of energy eigenstates within the First Brillouin Zone . All the properties of electrons in a periodic potential can be calculated from this band structure and the associated wavefunctions, at least within the Independent Electron Approximation . More generally, a Bloch-wave description applies to any wave-like phenomenon in a periodic medium. For example, a periodic Dielectric in Electromagnetism leads to Photonic Crystal s, and a periodic acoustic medium leads to Phononic Crystal s. A corollary of this result is that the Bloch wavevector k is a conserved quantity in a crystalline system (modulo addition of reciprocal lattice vectors), and hence the Group Velocity of the wave is conserved. This means that electrons can propagate without scattering through a crystalline material, almost like free particles, and that Electrical Resistance in a crystalline conductor only results from things like imperfections that break the periodicity. It can be shown that the eigenfunctions of a particle in a periodic potential can always be chosen this form by proving that Translation operators (by Lattice Vectors ) commute with the Hamiltonian . More generally, the consequences of symmetry on the eigenfunctions are described by Representation Theory . The concept of the Bloch state was developed by , it is called Floquet Theory (or occasionally the ''Lyapunov-Floquet theorem''), and the one-dimensional periodic wave equation is sometimes called Hill's Equation . REFERENCES
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