Bremsstrahlung: why a large value of the Planck constant makes it weak?

Let us continue our study of bremsstrahlung.

We think we have found the explanation why a big Planck constant h makes the cross section small for the case of the electron losing most of its kinetic energy in the emission of a large photon.

If we smoothen the 1 / r potential at the origin so that there is no deep well, then the Fourier decomposition of the new potential does not contain large momenta k. This shows that it is the deep well which contributes the large momentum k exchanges with the electron. This is just like in classical physics: the electron has to go very close to the nucleus for the electron to gain a large change in the momentum.

We see that it is the close encounters which produce large photons in the QED framework.

A close encounter happens quickly. The distortion in the electron wave function lasts for a very short time.

          __          __        

        /     \___/     \___/   photon

          __

        /     \   distortion

If the Planck constant h is small enough that the distortion "fits" in half a cycle of a large photon, then the "projection" of the distortion on the photon wave is large. There is a large possibility that the photon gets emitted in the process. Having a small h makes the process look like the classical limit. In the classical limit all 1 MeV electrons which come within 10^-14 m from the nucleus (a proton) will lose most of their kinetic energy in the encounter.

       ___________

     /                       \____________/  photon

                __

              /     \    distortion

But if the value of h is big, then the large photon has a long wavelength and a long cycle. The projection of the distortion on the photon wave is small: there is a low probability that a large photon gets emitted.

The wavelength of a 250 keV photon is 5 * 10^-12 m. To emit such a photon, the 1 MeV electron has to come within 10^-14 m of the proton. We conclude that in our universe the value of the Planck constant h is "big", and few large photons are emitted in bremsstrahlung.

This means that most close encounters are elastic or almost elastic. In a classical universe no almost elastic close encounters would happen.

The classical limit and what happens if we increase the value of the Planck constant

A way to analyze bremsstrahlung is first to look at it as a classical process.

If we set h to 1 / 100 of its usual value, then the de Broglie wavelength of a 1 MeV electron is roughly 10^-14 m. A close encounter to within 10^-14 m of a proton is "almost classical".

Increasing h back to its normal value suppresses the emission probability of a large photon by a factor 1 / 100.

What about our "rubber plate" model of the electron electric field? We explained the vertex correction with the classical model. Does our explanation work if we increase h?

It is hard to produce a large real photon of energy E in a small volume in a short time. That is what our analysis showed.

But there is no problem in exchanging a similar amount of momentum k in that small volume and short time. Coulomb scattering is not affected if we raise the value of h. As if pushing or pulling is not affected by a large h, but producing a vibration (a real photon) is hampered by a large h.

The effect of the rubber plate in the vertex correction is to push and pull on the electron. We conjecture that the rubber plate model works in the classical way even if h has a large value.

We need to check if the vertex function, or correction, in QED depends on the value of h.

The length scale problem is solved!

We have spent a lot of time wondering how the very sharp turn which the 1 MeV electron makes close to the proton can produce a relatively long wave photon, the wavelength ~ 5 * 10^-12 m and energy ~ 250 keV.

Now we have the solution. The sharp turn in rare (1 / 100) cases is projected into a long-wave photon. In other cases, the encounter is elastic or almost elastic.

If the Planck constant would be 100 times smaller, then the encounter would in most cases produce a short wavelength photon, ~ 5 * 10^-14 m. The energy of the photon would still be ~ 250 keV.

A long wavelength photon has problems going through a small hole in spacetime

This setup is clearly related to the analysis above. The bremsstrahlung photon whose wavelength is ~ 5 * 10^-12 m is "born" in the area of encounter, and that area is only ~ 10^-14 m in size and ~ 10^-14 m / c in time. The would-be photon has to make it through a small hole in spacetime, in order to break out into the external world.