stylistic:tone, structural:reformatting
Simplified quantum-mechanical description

The full quantum-mechanical treatment of bremsstrahlung is very involved. The "vacuum case" of the interaction of one electron, one ion, and one photon, using the pure Coulomb potential, has an exact solution that was probably first published by A. Sommerfeld in 1931. This analytical solution involves complicated mathematics, and several numerical calculations have been published, such as by Karzas and Latter. Other approximate formulas have been presented, such as in recent work by Weinberg and Pradler and Semmelrock.

This section gives a quantum-mechanical analog of the prior section, but with some simplifications to illustrate the important physics. We give a non-relativistic treatment of the special case of an electron of mass m e {\displaystyle m_{\text{e}}}, charge − e {\displaystyle -e}, and initial speed v {\displaystyle v} decelerating in the Coulomb field of a gas of heavy ions of charge Z e {\displaystyle Ze} and number density n i {\displaystyle n_{i}}. The emitted radiation is a photon of frequency ν = c / λ {\displaystyle \nu =c/\lambda } and energy h ν {\displaystyle h\nu }. We wish to find the emissivity j (v, ν) {\displaystyle j(v,\nu)} which is the power emitted per (solid angle in photon velocity space * photon frequency), summed over both transverse photon polarizations. We express it as an approximate classical result times the free−free emission Gaunt factor g accounting for quantum and other corrections: j (v, ν) = 8 π 3 3 Z 2 e ¯ 6 n i c 3 m e 2 v g f f (v, ν) {\displaystyle j(v,\nu)={8\pi \over 3{\sqrt {3}}}{Z^{2}{\bar {e}}^{6}n_{i} \over c^{3}m_{\text{e}}^{2}v}g_{\rm {ff}}(v,\nu)} j (ν, v) = 0 {\displaystyle j(\nu,v)=0} if h ν > m v 2 / 2 {\displaystyle h\nu >mv^{2}/2}, that is, the electron does not have enough kinetic energy to emit the photon. A general, quantum-mechanical formula for g f f {\displaystyle g_{\rm {ff}}} exists but is very complicated, and usually is found by numerical calculations. We present some approximate results with the following additional assumptions:

With these assumptions, two unitless parameters characterize the process: η Z ≡ Z e ¯ 2 / ℏ v {\displaystyle \eta _{Z}\equiv Z{\bar {e}}^{2}/\hbar v}, which measures the strength of the electron-ion Coulomb interaction, and η ν ≡ h ν / 2 m e v 2 {\displaystyle \eta _{\nu }\equiv h\nu /2m_{\text{e}}v^{2}}, which measures the photon "softness" and we assume is always small (the choice of the factor 2 is for later convenience). In the limit η Z ≪ 1 {\displaystyle \eta _{Z}\ll 1}, the quantum-mechanical Born approximation gives: g ff,Born = 3 π ln 1 η ν {\displaystyle g_{\text{ff,Born}}={{\sqrt {3}} \over \pi }\ln {1 \over \eta _{\nu }}}

In the opposite limit η Z ≫ 1 {\displaystyle \eta _{Z}\gg 1}, the full quantum-mechanical result reduces to the purely classical result g ff,class = 3 π {\displaystyle g_{\text{ff,class}}={{\sqrt {3}} \over \pi }\left} where γ ≈ 0.577 {\displaystyle \gamma \approx 0.577} is the Euler–Mascheroni constant. Note that 1 / η Z η ν = m e v 3 / π Z e ¯ 2 ν {\displaystyle 1/\eta _{Z}\eta _{\nu }=m_{\text{e}}v^{3}/\pi Z{\bar {e}}^{2}\nu } which is a purely classical expression without the Planck constant h {\displaystyle h}.

A semi-classical, heuristic way to understand the Gaunt factor is to write it as g ff ≈ ln (b max / b min) {\displaystyle g_{\text{ff}}\approx \ln(b_{\text{max}}/b_{\text{min}})} where b max {\displaystyle b_{\max }} and b min {\displaystyle b_{\min }} are maximum and minimum "impact parameters" for the electron-ion collision, in the presence of the photon electric field. With our assumptions, b m a x = v / ν {\displaystyle b_{\rm {max}}=v/\nu }: for larger impact parameters, the sinusoidal oscillation of the photon field provides "phase mixing" that strongly reduces the interaction. b m i n {\displaystyle b_{\rm {min}}} is the larger of the quantum-mechanical de Broglie wavelength ≈ h / m e v {\displaystyle \approx h/m_{\text{e}}v} and the classical distance of closest approach ≈ e ¯ 2 / m e v 2 {\displaystyle \approx {\bar {e}}^{2}/m_{\text{e}}v^{2}} where the electron-ion Coulomb potential energy is comparable to the electron's initial kinetic energy.

The above approximations generally apply as long as the argument of the logarithm is large, and break down when it is less than unity. Namely, these forms for the Gaunt factor become negative, which is unphysical. A rough approximation to the full calculations, with the appropriate Born and classical limits, is g ff ≈ max ] {\displaystyle g_{\text{ff}}\approx \max \left\right]}