On the General Theory of Surface States and Scattering of Electrons in Solids

In discussing the existence of surface states of electrons in solids or the scattering at a plane of discontinuity, it is necessary to match the wave function at the boundary plane, taken as the plane x = 0, using an infinite set of solutions of the Schrödinger equation in the solid. The solutions must all have the same energy E and the same ky and kz, but will have different values of kx, which may be complex. The formal structure of the energy is therefore considered, when viewed as a complex function E of the complex variable kx, and in particular the form of the lines along which E becomes real. This leads to some discussion about setting up approximate wave functions for surface states, differences between one, two and three dimensions, the energy of surface states relative to that of the Bloch bands, and the correction of a conclusion drawn by Goodwin in 1939.