The use of an independent boson model to study the dynamic effects of electron excitations by a particle colliding with a metal surface

We derive an algorithm which permits the calculation of our independent boson Hamiltonian for a jellium with a surface, which gives, when solved exactly, the same results as the random phase approximation. Applications to films and spheres are discussed in detail. The independent boson scheme is applied to the computation of the influence of the electron-hole pair excitations on the trajectory of a charged particle scattered by the metal surface. We show that the classical equation of motion of the particle, coupled to the quantized boson fields, is stochastic and must contain a peculiar force which depends on the final state of the quantum fields and on the past and the future of the trajectory. If the bosons are held at constant temperature, the motion of the particle is given by a quantum Langevin equation, which is derived here. The equation of motion derived here can be applied to both phonon-or electron-hole pair excitations.