Comparison of classical and quantum bremsstrahlung

On May 28, 2021 we wrote about classical bremsstrahlung. If we imagine that the Planck constant goes to zero, we expect quantum bremsstrahlung to approach the corresponding calculation in classical electrodynamics.

Astrophysicists calculate bremsstrahlung with the classical approximation, cutting off impact parameters b where b is smaller than the de Broglie wavelength of the electron.

They motivate their cutoff approximation by claiming that for smaller b, "quantum effects" spoil the classical approximation.

Their motivation is very ad hoc. Why should we ignore the classical contribution of small b? Classically, most of bremsstrahlung energy is emitted with very small b.

Let us try to find a sensible motivation for the cutoff.

The radiated energy in the classical approximation is very small for b > the Compton wavelength

In the earlier blog post we calculated that classically, a "moderately relativistic" electron (1 MeV) which passes a proton at the distance 2.4 * 10^-12 m (the Compton wavelength) will only radiate ~ 10^-7 times of its kinetic energy away, roughly 0.05 eV. Classically, this energy is evenly spread over photons whose energy is up to 500 keV.

If we take the cutoff of astrophysicists literally, we have to assume that through some mysterious mechanism, a classical electron can produce a 500 keV photon even though the electron passes a proton far away, at a distance > 2.4 * 10^-12 m. This does not make much sense.

                                     500 keV photon

                      ~~~~~~~~~~~~~~~~~~~~~

                    /

      e- ------------------------------------------------

                          |

                          | virtual photon

                          |

      p -------------------------------------------------

Let us then think about the Feynman diagram of the process. The momentum transfer in the virtual photon is very large.

Semiclassically, for the moderately relativistic electron to lose a lot of its spatial momentum to the proton, it has to pass the proton at a distance < 2.8 * 10^-15 m (the classical radius of the electron).

The cross section of such a very close encounter is something like

       ~ 10^-7

times the cross section of an encounter at a distance < 2.4 * 10^-12 m.

We see that very close encounters can explain the flux of 500 keV photons which in the astrophysical cutoff approximation were the result of encounters roughly 2.4 * 10^-12 m away.

Now we have a semiclassical approximation which qualitatively explains why the astrophysical cutoff approximation might work for large-energy photons.

What about lower energy photons? A similar argument works for them.

Is there some deep underlying reason why the Feynman diagram method (qualitatively?) reproduces the cutoff approximation of astrophysicists?