Approximate and numerical solutions of the Schrödinger equation

Abstract

In practically all realistic physical situations, the hamiltonian operator is such that it is not possible to specify the potential as a single function. In this case, it is not possible to calculate the exact eigensolutions of the Schrödinger equation. Even when the potential is specified as a functional form, it is possible to solve the Schrödinger equation only for a very limited number of artificial potentials. Many analytical and numerical approximation methods are available for the solution of the equation, and in this chapter some of the main techniques will be described which will be of relevance to device modelling. These will include the WKB approximation, both time independent and time dependent perturbation theory, the variational method, and numerical methods.