Bloch Electrons in External Electric and Magnetic Fields

The Schrödinger equation for a Bloch electron in external electric and magnetic fields in rewritten with the help of redundant variables, the wave function being obtained when a limiting process is carried out. The connection with the standard effective-mass approximation is discussed, and an application to the case of crossed electric and magnetic fields is carried out for the simple case of two parabolic bands in order to examine the singularity $\frac{1}{2}{m}^{*}{\left(\frac{E}{B}\right)}^2$ that appears in the energy in the effective-mass approximation when $B\to 0+$. We conclude that below a certain value of $\frac{E}{B}$ the character of the solution changes and the effective-mass formula is then not applicable.