Dynamics of Bloch Electrons in a Magnetic Field

A class of representations for Bloch electrons in a magnetic field is obtained by using only the translational properties of the Hamiltonian. The condition that these basis functions reduce to Bloch functions for zero magnetic field is included to obtain a set of magnetic Bloch functions. Since no approximation is made at any stage, these results may be carried over to a many-particle formulation. This representation is used to derive the well-known theorems for describing the motion of Bloch electrons in a magnetic field. In all cases the previous results are shown to be modified in the same manner, i. e., the wave vector $\overrightarrow{\mathrm{k}}$ is to be replaced by the operator $\overrightarrow{\mathrm{K}}$ symmetrically. Thus the appropriate space for describing the motion of Bloch electrons in a magnetic field is an operator space and not the wave-vector space.