Quantum multimode analysis of the free-electron laser

In this paper we discuss a longitudinal multimode quantum theory of the free-electron laser (FEL) within the framework of a nonrelativistic Hamiltonian. Furthermore, we study the evolution of the radiation field starting from the vacuum state and present the analytical expression in first order in the electron recoil of the probability amplitude of finding $l_j$ photons in the longitudinal jth mode. In conclusion we show the possibility of a self-consistent formalism adequate for numerical calculations to evaluate the evolution of the FEL radiation field from the quantum to the classical regime.