"Electron ping-pong" on a one-dimensional lattice: Multiple reflections of the wave packet and capture of the wave function by an acceptor

Considering the problem of multiple reflections of a wave function from the ends of a lattice, we observe an interesting phenomenon: the wave function amplitude is concentrated on the impurity center after reflections. The solution obtained by expanding the total wave function $a(t)$ on the impurity center in the partial amplitudes $a_k(t)$, whose contributions become essential only after the $k$th reflection from the lattice end, seems to agree very well with the results of numerical modeling. We solve the problem of the capture of the wave function by an acceptor. The obtained results can be used to explain experimental data on charge transfer in artificial oligonucleotides and polypeptides. We find expressions for the electron capture probability in some limit cases, which can be considered estimates of the quantum output of the charge transport.