  In the beginning, the synchrotron radiation (SR) was studied by classical
methods using the Li\'{e}nard-Wiechert potentials of electric currents.
Subsequently, quantum corrections to the obtained classical formulas were
studied, considering the emission of photons arising from electronic
transitions between spectral levels, described in terms of the Dirac equation.
In this paper, we consider an intermediate approach, in which electric currents
generating the radiation are considered classically, whereas the quantum nature
of the radiation is taken into account exactly. Such an approximate approach
may be helpful in some cases, it allows one to study the one-photon and
multi-photon radiation without complicating calculations using corresponding
solutions of the Dirac equation. We construct exact quantum states of the
electromagnetic field interacting with classical currents and study their
properties. By their help, we calculate a probability of photon emission by
classical currents and obtain relatively simple formulas for the one-photon and
multi-photon radiation. Using the specific circular electric current, we
calculate the corresponding SR. We discuss a relation of obtained results with
known before, for example, with the Schott formula, with the Schwinger
calculations, with one-photon radiation of scalar particles due to transitions
between Landau levels, and with some previous results of calculating the
two-photon SR.
