NOTES AND DISCUSSIONS Kronig–Penney model with the tail-cancellation method

The Kronig–Penney model serves to illustrate the formation of energy bands in a periodic solid and appears as a pedagogical example in many textbooks in elementary solid state physics. The model is generally solved either by matching the boundary conditions for the wave functions at the cell boundaries, by a plane-wave expansion of the wave function in the reciprocal lattice space, or even by the somewhat more involved T-matrix method. In this note, we point out that perhaps the simplest way of solving the problem is by using the tail-cancellation condition, which has been used extensively in the solution of the band structure problem in realistic solids. The solution therefore serves as a simple illustration of the tail-cancellation method as well. The Schrödinger equation for a one-dimensional solid is