Electric and Magnetic Translation Group

A set of translation operators is defined which commute with the combination of operators occurring in the time-dependent Schr\"odinger equation for an electron in potentials periodic in time and space, with uniform applied electric and magnetic fields in arbitrary directions. It is shown that the operators form a group. The group is made finite by imposing periodic boundary conditions, and restrictions on the electric and magnetic fields are obtained. All irreducible representations of the group, and corresponding basis functions are generated. The limit of these functions is found as the distance between boundaries becomes infinite and the restrictions on the fields disappear.