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Question:
I am providing help to a high school student in an Advanced Placement physics class who asked me a question I cannot answer. He notes that the concept of acceleration of electrons is critical to the classical electrodynamics description of radiation from moving charge. But he asks why it seems that the concept of acceleration does not enter into the quantum electrodynamics description of photon emission from moving electrons. He points to a simple Feynman diagram for an electron emitting a photon and does not recognize how electron acceleration enters the picture. Neither can I.



Replies:
Feynman diagrams do not show how acceleration enters the picture, or any other details of how electrodynamics works. Their purpose is to show you which interactions between particles have to be accounted for, not to do the actual accounting.

Many of the expressions used in quantum mechanics look much like their counterparts in classical physics, but in place of a quantity like acceleration, you will have the expectation value of an operator that represents that quantity, integrated over a complete set of the states the system under study can be in. You do not very often see the actual expressions because they are long and involved, and specific to a particular system of particles. Authors feel they will not add enough to a student's understanding to justify the space they'll take up in a book. But some introductory quantum mechanics texts do have examples of the full expressions.

-- Tim Mooney
Beamline Controls & Data Acquisition Group
Advanced Photon Source, Argonne National Lab.


Well, that is because it does not. Classical mechanics and electrodynamics is concerned with trajectories, velocity, acceleration, and such; the differential equations characterizing classical mechanics are founded on time-derivatives of position. Quantum mechanics is not. The differential equations governing quantum mechanics are constructed around time and space derivatives of a wave function, which is related to a particle's position but not quite the same thing. Energy, momentum, and position are meaningful quantities to quantum mechanics, but velocity, acceleration, and force are not, really. (You can get them in an average sense as derivatives of expectation values of observables, but they are not fundamental.) Emission of a photon is a strictly quantum process.

I realize that this is not a complete answer, but I do not have a very clear way to explain interaction of matter with light. Since your student is already studying QED, he probably is aware of the basic ideas behind it. I am just pointing out that there is no reason to expect a role for acceleration in a quantum theory.

Richard Barrans
Department of Physics and Astronomy
University of Wyoming


Didn't the diagram show the electron moving through space, then seems to veer off at a different angle? This dog-leg, this quick bend from one angle to another of the electron's "world line", its flight through time and space, is acceleration. Sometimes right at this point people draw a little scriggly line, showing the resulting photon.

By the way, are you aware of a great set of Internet steaming videos on Feynman's work? Internet search on Feynman QED lectures, he gave 4 in a series, in New Zealand, in the 1980's. You can watch them on your computer. They follow his very nice little book QED, about a 100 page book with no math, easy to read, and costs only about $6 or so. It is a good book to give a high school student, though perhaps they will not appreciate it completely.

Steve Ross


Well, the acceleration happens as a result of net force applied to the electron, specifically magnetic force, which is a subset of electromagnetism, so I suspect it would be denoted as a multiplicity of lower-energy photons colliding with the electron. Followed soon by one high-energy photon emitting. Somebody please tell me if I am right. I am also not clear how an electron gets to be a non-linear element, up-converting multiple photon low-frequency input to higher-frequency single-photon output.

Jim Swenson



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