The complete quantum mechanical description was first performed by Bethe and Heitler. They assumed plane waves for electrons which scatter at the nucleus of an atom, and derived a cross section which relates the complete geometry of that process to the frequency of the emitted photon. The quadruply differential cross section, which shows a quantum mechanical symmetry to pair production, is
where  is the atomic number,  the fine-structure constant,  the reduced Planck's constant and  the speed of light. The kinetic energy  of the electron in the initial and final state is connected to its total energy  or its momenta  via
where  is the mass of an electron. Conservation of energy gives
where  is the photon energy. The directions of the emitted photon and the scattered electron are given by
where  is the momentum of the photon.
The differentials are given as
The absolute value of the virtual photon between the nucleus and electron is
The range of validity is given by the Born approximation
where this relation has to be fulfilled for the velocity  of the electron in the initial and final state.
For practical applications (e.g. in Monte Carlo codes) it can be interesting to focus on the relation between the frequency  of the emitted photon and the angle between this photon and the incident electron. Köhn and Ebert integrated the quadruply differential cross section by Bethe and Heitler over  and  and obtained:
with
and
However, a much simpler expression for the same integral can be found in  (Eq. 2BN) and in  (Eq. 4.1).
An analysis of the doubly differential cross section above shows that electrons whose kinetic energy is larger than the rest energy (511 keV) emit photons in forward direction while electrons with a small energy emit photons isotropically.