Semi-classical limitations for photon emission in strong external fields

TL;DR

This paper examines the limitations of the semi-classical Baier-Katkov emission formula for ultrarelativistic electrons in strong fields by comparing it with quantum mechanical calculations, revealing conditions where the semi-classical approach fails.

Contribution

It identifies additional conditions necessary for the semi-classical formula to be valid and highlights discrepancies and quantum effects not captured by the classical trajectory-based approach.

Findings

Semi-classical formula requires an extra condition beyond ultrarelativistic velocity.

Qualitative differences appear in harmonic spectra when the condition is violated.

Quantum spectra depend on the dispersion relation of effective photons, not just classical trajectories.

Abstract

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.