Response Function of A Finite Electron-gas

The full response matrix of an idealized metal with two plane parallel surfaces is calculated in the RPA approximation using the equation of motion for the retarded density-density Green's function. The surfaces are approximated by infinite barriers and the induced charge density is related to the perturbing charge instead of the Hartree field. It is shown that the Hamiltonian of the system splits into two parts: one due to bulk interactions and the other to the presence of the surfaces. The response matrix is then formally calculated and surface plasmons are shown to be described by the poles of a function related to the response matrix.