Non-standard FDTD implementation of the Schr\"odinger equation

In this work, we apply the Cole's non-standard form of the FDTD to solve the time dependent Schr\"odinger equation. We deduce the equations for the non-standard FDTD considering an electronic wave function in the presence of potentials which can be higher or lower in comparison with the energy of the electron. The non-standard term is found to be almost the same, except for a sine functions which is transformed to a hyperbolic sine function,as the argument is imaginary when the potential has higher energy than the electron. Perfectly Matched Layers using this methodology are also presented.