Exponential fitting method for the time-dependent Schrödinger equation

Our purpose is to increase the accuracy of the numerical solution of the time-dependent Schrödinger equation. In particular, a modification of the standard Crank–Nicolson method by an exponential fitting Numerov formula leading to a higher order in the approximation of the second order spatial derivative along with a better description of oscillating or exponential behavior and different artificial boundary conditions aimed to reduce the reflections of the wave packet at the numerical boundaries are presented. The procedures are illustrated for the deep-tunneling case of proton emission.