Energy loss of charged particles colliding with an oscillator

Energy loss of fast charged particles colliding with an oscillator is considered in the dipole approximation. In this approximation, the problem is solved exactly and the energy loss of the oscillator from the initial state $|m\rangle=|0\rangle$ is found in the form of the sum of single integrals. It is shown that passing to the limit, the Bethe theory for an atom with small perturbations can be obtained, and in the case of strong fields, the correction to the Bethe theory, analogous to the Bloch correction, can be calculated; in addition, a classical limit coinciding with the Bohr formula is possible.