Numerical solution of the Schröbinger equation for an electron in a potential well of an arbitrary form

A numerical model is proposed for the solution of the Schröbinger equation. The model is based on the scattering matrix of the solution region. The scattering matrix is formed from matrices of partial regions of a simple shape. The density of the self-emission current for a metal/vacuum potential barrier of an arbitrary form is calculated. The results of test calculation of electron transmission through a rectangular potential barrier with a relative error less than 0.001% as compared to the analytical solution are presented.