Accelerated Charge, Photons, Feynman Diagrams
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.
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
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.
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.
Well, the acceleration happens as a result of net force applied to the
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
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Update: June 2012