Numerical solution of the the time-dependent Schrödinger equation.

Abstract

On the atomic scale the time-dependent Schrödinger equation is used but for systems with three or more particles like the helium atom it is not possible to solve the Schrödinger equation exactly. The motivation here is to calculate a numerical solution for the helium atom in the ground state. To this end a numerical method must be used and the choice is the summations by part method with simultaneous approximation terms at any boundaries. We analysed and ran numerical verification tests after coding in Matlab on the following PDE's. The transport equation, the heat equation and the Schrödinger equation for particle in a box. The results for the transport equation and the heat equation seems to verify the code but for the particle in a box we seem to very quickly hit the roundoff limit and this made the verification difficult for higher order. We also tested a little on the hydrogen atom before calculating the ground state of helium.