Trying to Understand Bloch's Theorem "Oxford Dictionary"

A theorem relating to the quantum mechanics of crystals stating that the wave function ψ for an electron in a periodic potential has the form ψ(r) = exp(ik · r)U(r), where k is the wave vector, r is a position vector, and U(r) is a periodic function that satisfies U(r+R) = U(r), for all vectors R of the Bravais lattice of the crystal. Bloch's theorem is interpreted to mean that the wave function for an electron in a periodic potential is a plane wave modulated by a periodic function. This explains why a free-electron model has some success in describing the properties of certain metals although it is inadequate to give a quantitative description of the properties of most metals. Bloch's theorem was formulated by the German-born US physicist Felix Bloch (1905–83) in 1928.