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A theoretical analysis is proposed to investigate the Compton scattering as well as the inverse Compton scattering. In this analysis, the incident and scattered electromagnetic (EM) waves are classically described using Maxwell's equations, while the interacting electron is represented quantum mechanically by a wavepacket with a finite spreading length. The density matrix formalism is introduced to analyze the interaction between the electron wave and the EM field in a quantum mechanical manner. The interaction mechanism is caused by the effective coupling between EM and electron waves satisfying the energy and momentum conservation rules. The effects of the electron relaxation time due to Coulomb collisions among electrons, as well as the finite spreading size of an electron, are newly considered. Expressions for the radiated power of the scattered wave are derived for Compton and inverse Compton scattering.