In the literature a multitude of finite differences methods for solving partial differential equations can be found, see for example [Smi85,HS96] and references therein. I do not attempt to even only try to give a complete overview of this field here; rather I want to present two typical finite difference approaches to the problem of solving the SCHRÖDINGER equation, and I discuss some of the limitations of approaches of this kind. The first method presented here relies on a discretization of time only, whereas for the second both the time and position variables are being discretized.