Bloch oscillations in one-dimensional solids and solitary wave packets

Abstract

We derive the exact solution of a single-band time-dependent Schrödinger equation for an electron in an idealized one-dimensional periodic solid in the presence of a constant uniform electric field. We show that all wave functions are necessarily periodic in time. This result is the fully quantum-mechanical analog of the well-known Bloch oscillations predicted by quasiclassical dynamics. Our method of solution consists of mapping the electron Schrödinger equation to the exactly solvable problem of a quantum planar rotor in the eikonal limit subject to an arbitrary angular and time-dependent external potential. The time periodicity of the electron wave functions is due to the fact that all of the rotor wave functions have the form of solitary wave packets.