Unification of Free Electron Laser Theories

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Abstract
The quantum mechanical equations of motion of the electron in a standing wave pattern are discussed in connection with the free electron laser. The reference frame where quantization is performed is the frame in which an electron with zero initial momentum does not cause, on the average, gain or loss of the laser field. The quantum features of the electron motion are retained, while quantum fluctuations of the fields are neglected. The validity of the fully classical approach is also discussed. In the pure momentum representation the density matrix is found to be characterized by a pair of integers, (μ, v). The classical, single particle motion is recovered if one considers the index μ as a continuous parameter, i.e. in the limit of little momentum transfer, ħk→0. If, in addition one retains only the terms with v=0, ±1, the classical equations of motion for the quasi Bloch vector, previously derived by the Arizona group, can be recovered.