  The semi-classical heuristic emission formula of Baier-Katkov [Sov. Phys.
JETP \textbf{26}, 854 (1968)] is well-known to describe radiation of an
ultrarelativistic electron in strong external fields employing the electron's
classical trajectory. To find the limitations of the Baier-Katkov approach, we
investigate electron radiation in a strong rotating electric field quantum
mechanically using the Wentzel-Kramers-Brillouin approximation. Except for an
ultrarelativistic velocity, it is shown that an additional condition is
required in order to recover the widely used semi-classical result. A violation
of this condition leads to two consequences. First, it gives rise to
qualitative discrepancy in harmonic spectra between the two approaches. Second,
the quantum harmonic spectra are determined not only by the classical
trajectory but also by the dispersion relation of the effective photons of the
external field.
