Mathieu's Functions and Electrons in a Periodic Lattice

The motion of an electron in a one-dimensional sinus-oidally varying potential is considered as an illustration of Mathieu's equation. The correspondence between the special periodic Mathieu functions and the electron wave functions at band center and band edge and the symmetries of these special functions are useful in displaying the atomic and lattice periodicities of the solid state problem. This treatment has some advantages over other “model” band calculations because one can see the continuous transition from deep lying bound atomiclike states to weakly bound conduction band states as the periodic perturbation is varied.