Equations of Motion of Bloch Electrons Beyond the Semiclassical Approach

A quantum mechanical description based on a Hamiltonian approach for the equations of motion of electrons in periodic systems subjected to perturbing electromagnetic fields is presented. By assuming a well localized wave packet in a Brillouin zone of reciprocal space, projected onto a subset of bands, the quantum mechanical equations of motion for the gauge invariant crystal momentum and wave packet position are obtained. These are found to contain Berry curvature effects and coupling between the bands, which can also have interesting consequences for the associated Boltzmann transport description of currents. These situations are examined in more detail in a two-band model. With this approach, we show the ease with which the equations of motion arise from quantum mechanical principles.