Quantum scattering via the discretisation of Schrödinger's equation

We obtain the scattering matrix for particles that encounter a quantum potential by discretising Schrödinger's time independent differential equation without the need to resort to the manipulation of the eigenfunctions directly. The singularities that arise in some solutions by other methods are handled with ease including the effects of resonances while convergence is excellent in all limits with only a small number of orders required to give accurate results. Our method compares the tunnelling probability with that of the WKB theory, exact numerical solutions and the modified Airy function method.