Reflections in One-Dimensional Wave Mechanics

A method is presented for solving the one-dimensional Schroedinger equation. The method gives the wave function in the form of an infinite series. The first term of the series is the WBK approximation. Each of the higher terms in turn represents the reflections generated by the preceding term. The expansion is obtained by approximating the potential by a "staircase" potential, which is constant over short regions and changes its value discontinuously between these regions. The method is applied to periodic potentials. By taking the WBK approximation and the first reflection term into account, the zones of the periodic potential can be found in terms of a coefficient which gives the probability that a particle will be reflected in passing through one cell of the periodic potential.