  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.
