Crystal momentum

In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors of this lattice, according to (where is the reduced Planck's constant). Frequently, crystal momentum is conserved like mechanical momentum, making it useful to physicists and materials scientists as an analytical tool. A common method of modeling crystal structure and behavior is to view electrons as quantum mechanical particles traveling through a fixed infinite periodic potential such that where is an arbitrary lattice vector. Such a model is sensible because crystal ions that form the lattice structure are typically on the order of tens of thousands of times more massive than electrons, making it safe to replace them with a fixed potential structure, and the macroscopic dimensions of a crystal are typically far greater than a single lattice spacing, making edge effects negligible. A consequence of this potential energy function is that it is possible to shift the initial position of an electron by any lattice vector without changing any aspect of the problem, thereby defining a discrete symmetry. Technically, an infinite periodic potential implies that the lattice translation operator commutes with the Hamiltonian, assuming a simple kinetic-plus-potential form. These conditions imply Bloch's theorem, which states or that an electron in a lattice, which can be modeled as a single particle wave function , finds its stationary state solutions in the form of a plane wave multiplied by a periodic function . The theorem arises as a direct consequence of the aforementioned fact that the lattice symmetry translation operator commutes with the system's Hamiltonian. One of the notable aspects of Bloch's theorem is that it shows directly that steady state solutions may be identified with a wave vector , meaning that this quantum number remains a constant of motion.