Annihilation of Electron by Positron to Produce at least 2 Gamma Rays

  1. 1. The problem statement, all variables and given/known data

    The problem is listed as follows: Show that conservation of energy and momentum require at least two gamma rays to e emitted in the annihilation of an electron by a positron.


    2. Relevant equations

    p(initial) = p(final)
    E(initial) = E(final)
    Total rest mass = 1.0218 MeV/c^2
    p=hf/c
    E=hf

    3. The attempt at a solution

    I'm checking to see whether my answer to this question is sufficient. The book (Wong) introduces the idea of a proton and anti-proton annihilation at rest, so I assumed that this process could occur at rest as well. Is that ok?

    I first tried the production of a single gamma ray. However, since initial p=0, final p=0 as well. However, since initial E=/=0, and initial E = final E, then final E =/=0. These two are in contradiction to each other, since p=hf/c and E=hf. Combining, this gives p=E/c. Since c=/= 0, this equation is impossible.

    I then explained that if the rays were travelling in opposite directions, this would allow for final p=0, fulfilling initial p=final p.

    Do you guys believe that this is a sufficient answer? It is based upon the presumption that such an interaction can occur at rest.

    Let me know, thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Just came up with a new thought, I could create a situation such that the electron and positron are approaching each other in such a manner that initial p=0. This would allow the dismissal of the "at rest" presumption. Thoughts?
     
  4. dextercioby

    dextercioby 12,314
    Science Advisor
    Homework Helper

    This is the simplest exercise in quantum field theory, showing that the vertex of QED can't have all 3 particles on the mass shell.

    You only need the energy-momentum equation describing the mass shell.

    [tex] p^2 = m^2 \, \mbox{for the electron/positron and} \, p^2=0 \, \mbox{for the photon} [/tex]
     
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