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Momentum, Particle-Antiparticle Collisions
Name: Margaret
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
This is about particle-antiparticle annihilation. What is seen in a
frame of reference in which the total momentum of the annihilating pair is not
zero? (e.g. moving e+ hits stationary e- in the lab frame). Can the 2 gammas
still be the same frequency but not in opposite directions? Or will there be
two unequal gammas? If so will they be in opposite directions? Hope this is
intelligible.
Replies:
Dear Margaret,
Yes, if the total momentum of the annihilating pair is not zero, in general the
momenta of the two gammas will be different. That means, of course, that the two
gammas will in general have different frequencies.
Overall momentum (and energy) is, of course, always conserved (unchanged).
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
Margaret,
Photons of light have both energy and momentum. The directions and wavelengths
will be such that both momentum and energy balance. Remember that energy includes
mass energy (mc^2). One way to find the values is to adjust the reference frame
with relativity so that total momentum equals zero. Determine the resulting
photons, and then shift back to the original reference frame.
Dr. Ken Mellendorf
Math/Science/Engineering
Illinois Central College
Momentum will be conserved in the annihilation. In other words, in any inertial
reference frame from which you observe the interaction, the total momentum of the
photons after the annihilation will be equal to the total momentum of the electron
and positron before the annihilation.
If you look at the interaction in the center-of-mass reference frame, the total
momentum is zero throughout: the electron and positron have opposite velocities
(equal speed but opposite direction), and the produced photons have equal
frequencies and move in opposite directions.
You are correct to suppose that things are different in another reference frame.
If the electron and positron do not have exactly opposite velocities, the net
momentum of the system will not be zero. The photons produced will preserve this
non-zero net momentum. For this to happen, they must have different frequencies,
they must move in directions that are not exactly opposite to each other, or both.
The restrictions on the system are:
1. The total momentum of the two photons after the annihilation equals the total
momentum of the electron + positron after the annihilation.
2. The total energy of the two photons after the annihilation equals the total
energy (rest energy + kinetic energy) of the electron + positron.
3. The photons both move at the speed of light.
Richard Barrans, Ph.D., M.Ed.
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Update: June 2012
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