Summary: Inelastic scattering of a photon by a bound electron of a hydrogen-like atom is considered. An expression for the cross section of this process, which can take place both without and with a change in the energy of the photon due to atomic transitions, is obtained. Within the framework of the standard technique of Feynman diagrams with a free electron propagator, general expressions for the amplitude and cross section of the process have been obtained. Arguments in favor of the validity of using this representation of the propagator in the calculation of the amplitude in the field of a nucleus are presented. As an accompanying result, an expression for the density matrix of an electron in the field of a nucleus is found in the leading approximation in the small “atomic” expansion parameter \((Z\alpha) \ll 1, \alpha = e^2 / \hbar c\). It is shown that in a real situation at temperatures \(T\ll m_e\) of the equilibrium radiation field this process can be neglected in comparison with spontaneous emission of radiation by a hydrogen-like atom despite the lower power of the parameter \((Z\alpha)\) in its amplitude. As far as is known, this quite important question, framed in such a way, has not been discussed in the literature.