Variational principle for the elements of the $S$ matrix

A variational principle is formulated that enables one to calculate approximately the elements of the $S$ matrix in quantum electrodynamics corresponding to Feynman diagrams with one fermion and any number of external photon lines. As an example, the differential cross section is calculated for the process of elastic scattering of a photon on the hydrogen atom, and also for the shift of the energy level of a bound electron in an external electric field in the second order of perturbation theory in the external field.