Synchrotron radiation was originally studied by classical methods using the Liénard–Wiechert potentials of electric currents. Subsequently, quantum corrections to the 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, an intermediate approach is considered, in which electric currents generating the radiation are considered classically while 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 one-photon and multi-photon radiation without complicating calculations using corresponding solutions of the Dirac equation. Here, exact quantum states of an electromagnetic field interacting with classical currents are constructed and their properties studied. With their help, the probability of photon emission by classical currents is calculated and relatively simple formulas for one-photon and multi-photon radiation are obtained. Using the specific circular electric current, the corresponding synchrotron radiation is calculated. The relationship between the obtained results and those known before are discussed, for example with the Schott formula, with Schwinger calculations, with one-photon radiation of scalar particles due to transitions between Landau levels, and with some previous results of calculating two-photon synchrotron radiation.