Charge Transfer in the Lattice with an Impurity Site. Reflection and Transmission of the Wave Packet

The quantum dynamics of the propagation of the charge wave function in a uniform lattice containing a single impurity site is considered. A nonstationary problem is solved in the tight-binding approximation. The initial state is the wave function fully localized at one of the lattice sites. The coefficients of transmission of the wave packet through the impurity site and reflections from it are calculated as a function of the parameters of the problem, that is, the additional energy on the impurity and the distance between the impurity and the initial position of the charge. The problem is solved for two types of boundary conditions: an infinite and a semi-infinite lattice. Good agreement with numerical simulation is obtained.