"Electron ping-pong" on a one-dimensional lattice: Wave packet motion up to the first reflection

We consider the time evolution of the electron wave function on a one-dimensional lattice consisting of $N$ identical sites and a single impurity site that differs from the rest of the lattice by the on-site energy and the hopping integral. At the initial instant, the wave function is entirely localized at the impurity site. With time, a localized wave packet forms, which moves over the lattice and reflects from its end. The process of reflection from the ends repeats many times and is called "electron ping-pong". We obtain analytic expressions for the propagation of the wave packet front at different values of the problem parameters. The analytic expressions agree perfectly with numerical simulation results.