The W. K. B. technique for solving the one-dimensional wave equation is extended to the case when the potential field includes a rapidly varying periodic term as well as a slowly varying term. A pair of auxiliary functions are introduced which are identical to the wave function and its derivative respectively at the edges of the periodic cells, but which have a simple exponential behaviour within the cells. The auxiliary functions satisfy a pair of auxiliary (related) differential equations, with slowly varying coefficients, which are valid for all energy values. Solution of the auxiliary equations by the well-known W. K. B. technique yields approximations to the wave function. These approximations break down in the neighbourhood of the band edges, which are the turning points of the problem. Connexion formulae are established across the band edges and employed to calculate the interband tunnelling probability. In the immediate neighbourhood of a band edge the analysis yields an effective-mass wave equation and a closed form for the wave function. The auxiliary functions are closely related to the effective-mass modulating wave function and the results of this paper may be regarded as an extension of effective-mass theory for the one-diinensional case, throughout the whole of the energy ranges of allowed bands and forbidden gaps.
One-dimensional Néel walls under applied external fields