The 1934 bremsstrahlung paper by Bethe and Heitler is freely readable at
Bethe and Heitler seem to use the principle later known as Fermi's Golden Rule:
where the "density" of possible end states is counted.
incoming distorted outgoing
e- | | | | | / / / / | | |
~~~~~~~~~~~
photon
● Z+ nucleus
Bethe and Heitler assume a plane wave of the incoming electron, which is perturbed (distorted) for a short time dt by the Coulomb potential of the nucleus Z+.
Then they take the projection of the distorted wave function to an off-shell electron plane wave. An off-shell electron plane wave is one where the energy-momentum relation
E² = p² + m²
does not hold.
They further calculate the transition probability of this off-shell wave to an on-shell outgoing electron wave under a perturbation of the field of the emitted photon. If we look backward in time, it is the emitted photon which causes a state transition of the outgoing electron to an off-shell electron.
Besides the process described above, there is another - less intuitive - process where the electron emits the photon before the electron meets the nucleus.
real photon
~~~~~~~~~~
/
e- ---------------------------------------------
| virtual
| photon
| k
Z+ ---------------------------------------------
In the corresponding Feynman diagram, the electron absorbs a virtual photon which is sent by the nucleus Z+ and carries 4-momentum k. The electron goes off-shell. The electron then emits a real photon and returns to on-shell.
The probability of the electron receiving a virtual photon whose 4-momentum is k, is governed by the photon propagator. Richard Feynman in his famous paper derives the propagator from the Fourier decomposition of the Coulomb potential 1 / r. Another way to derive it is from the Green's function of the Klein-Gordon equation.
An aside: why is the Fourier decomposition of the Coulomb potential similar to the Green's function? Our "sharp hammer" hypothesis in this blog is a manifestation of this relationship.
Which processes depend on the value of the Planck constant h?
Coulomb scattering is equivalent in classical physics and quantum field theory. Cross sections for it do not depend on the value of h.
Cross sections in Compton scattering depend on the classical radius of the electron and on the mass-energy of the photon. They do not depend directly on the value of h.
But in bremsstrahlung, the cross section for emitting most of the kinetic energy in a photon is inversely proportional to h.
What is the difference in these processes? An obvious difference is that in bremsstrahlung the number of particles grows by one. Bethe and Heitler count the number of possible final states in a unit volume.
The ratio of the number of final states and the number of input states probably depends on h. If we make h smaller, the ratio will probably grow. This may explain why in bremsstrahlung making h smaller will increase the cross section.
In Coulomb scattering and Compton scattering the ratio probably does not depend on h because the process is symmetric.
Hypothesis. QED processes which conserve the number of particles happen like in classical physics - if one takes into account that the photon is a particle. Their cross sections do not depend on the Planck constant h. But if the number of particles grows in the process, then the cross section is reduced by quantum physics, and a smaller h increases the cross section.
Our hypothesis claims that the "mechanics" of a process is classical. The value of the Planck constant enters the stage when we calculate the number of possible states in a unit volume.
The hypothesis explains the coincidence that Coulomb scattering is equivalent in classical and quantum physics.
Our hypothesis smashes the hopes in our previous two blog posts: the fine structure constant is not determined by the geometry of the electron static electric field.
This is because calculating the number of possible states in a unit volume is not connected to the geometry of the electron electric field.
Or could it be? Are there "resonances" in vibrations of the static electric field of the electron? If the electron orbits a proton, could it be that these resonances enforce the Bohr orbits? We need to study this.
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