Bloch state

A Bloch state is an energy eigenstate of a lattice-periodic Hamiltonian that is simultaneously an eigenstate of every lattice translation. It can be written as ${\psi }_k(x)=e^{ik\cdot x}{u}_k(x)$ with lattice-periodic $u_k$.

Crystal momentum is the translation quantum number $\hslash k$ of a Bloch state. Since $k$ and $k+G$ give the same translation character whenever $G$ belongs to the reciprocal lattice, crystal momentum is defined modulo reciprocal-lattice vectors.