Photon Emission: Purcell's Textbook Presentation

[exmarine]

A textbook presentation is given by Purcell: "Electricity and Magnetism", Appendix B, "Radiation by an Accelerated Charge". He carefully shows how the changes in the Coulomb field of a quickly decelerated electron propagate outward with velocity c. Since the field was moving past the observer, presumably at some rather small relative velocity for a long time before the deceleration started, and the charge is now at rest after the deceleration is over, and Gauss’ law must be obeyed throughout this event, then a kink or wave in the field is created. He says that outgoing [spherical] wave is a photon. I have more questions about that than I can list. Here are a few:

1. The only way I know for an electron to decelerate is for it to encounter another electron. So if electron #1 emits a photon, does electron #2 also have to emit a photon? Is there any empirical evidence that photons are emitted in pairs (at the very least)?
2. Would it be more accurate to say that the outgoing kink or wave in the field describes a sort of envelope or probability distribution for the directions in which photons might be emitted?
3. A harder question to articulate accurately: is this picture consistent with SRT, specifically the feature that the velocity of light is c for all observers? I see the electron come to rest, but I may not be the only observer. Others could see it merely slow down, others could see it stop and rebound, etc.
4. If the collision of two electrons emits photons, then they carry away some energy that was originally from the kinetic energies before the impact. So as they rebound away from each other, they cannot achieve their original KE - the collision is inelastic. So are the Coulomb fields repelling each other somehow diminished during the retreat compared to the approach? It seems the dynamic / magnetic effects also repel during both approach and retreat.
5. What if the rate of deceleration is lower (or higher)? Do the electrons emit more than one photon each, or longer (shorter) wavelength photons, or many photons, etc.?

Thanks for any help with this.

 

[bhobba]

The only way I know for an electron to decelerate is for it to encounter another electron.

Can you explain your reason for believing this.

Electric fields would seem to be the agency - not other electrons.

Thanks
Bill

 

[PhilDSP]

Hello exmarine,

You're mostly asking questions that deal with classical E&M.

1. As bhobba implies, anytime the electric field fluctuates an electron will accelerate and decelerate (depending on the geometric conditions).

2. Classically, as Purcell describes, the changing field represents either force or energy. The photon is the focus point of the traveling wave created when the electron radiates. Arnold Neumeier has interpreted the entire wave as the photon in the context of QM. But usually the classical EM wave is not thought to correspond exactly with the wave described by the Schrödinger or Dirac equations.

3. This classical picture is consistent with SR. SR applies to light traveling through a vacuum. Whenever EM waves approach any charged particle the effective speed reduces to less than c because of dispersion – the field interacts with the charged particle, moving it which produces additional field fluctuations. Multi-body interactions (more than 2) is a different issue. How they are dealt with within SR is a question you might ask in the other forum or here in the context of QFT or QED.

4. Because of conservation of charge, the coulomb fields for the electrons should not change after photon emission.

5. Faster decelerations produce harder x-rays – photons with higher frequency and higher energy.