The realization of a Free Electron Laser (FEL) operating in the quantum regime could provide a compact and monochromatic X-ray source. In this thesis we de- veloped a three dimensional (3D) quantum theory for a FEL with a laser wiggler, based on a discrete Wigner function formalism taking into account the longitudi- nal momentum quantization. Starting from the exact quantum treatment, a mo- tion equation for the Wigner function coupled with self-consistent radiation field is derived in the realistic limit in which the normalized electron beam emittance is much larger than the Compton wavelength quantum limit. The model describes the 3D spatial and temporal evolution of the electron and radiation beams, including diffraction, propagation, laser wiggler, emittance and quantum recoil effects. It can be solved numerically and reduces to the 3D Maxwell-Vlasov model in the classical limit. We discuss the experimental requirements for a Quantum X-ray FEL with a laser wiggler, presenting preliminary numerical results and parameters for a possible future experiment. Moreover we present a systematic statistical analysis of the 1D SASE which confirm the "Quantum purification" phenomenon.

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