Crystal electrons in magnetic fields: General reduction of the dimensionality and properties of the wave functions

The motion of Bloch electrons in homogeneous magnetic fields is reduced without approximations to, at most, two dimensions in the general three-dimensional case, i.e., for arbitrary crystal potential, arbitrary field-lattice geometry, and all rational fields. This is done by fully exploiting a canonical transformation and by constructing with the aid of ray-group projection operators generalized k-q functions, which separate off one degree of freedom. Previous ad hoc reductions to one dimension for essentially two-dimensional situations are recovered and explained. The solutions of the resulting lower-dimensional effective Schr\"odinger equations are functions of generalized coordinates. They are converted into the real-space wave functions by means of a contact transformation; their local and global properties are investigated. The results presented allow first-principles calculations of diamagnetic band structures and wave functions to realistic systems.