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(SP. RELATIVITY) Annihiliation of Electrons into Photons

  1. Oct 12, 2016 #1
    1. The problem statement, all variables and given/known data

    An electron e- and positron e+ annihilate to produce two photons.

    a_ Why are two photons produced rather than one?

    b_ Assume that the e- and e+ are at rest just before they annihilate. In their rest frame, what are the energies and momenta of the photons? Define the +x axis to be the direction of motion of one of the photons.

    c_ What are the energies and momenta of the photons in another frame that movies with (v/c)^2 = 4/5 with respect to the rest frame?

    2. Relevant equations

    p = y m0 u
    E total = y m0 c^2
    E rest = m0 c^2
    E kinetic = (1-y)(m0 c^2)

    3. The attempt at a solution

    a _ I think this is a simple answer that reads "momentum should be conserved after annihilation so there must be two photons created with equal and opposite momentum.

    b _ Because the electron and positron are at rest before annihilation, there is no velocity transformation or Lorentz transformation required so should the momentum of the photon be "p = m0 c" and energy be "E =m0 c^2" ?

    c _ This time the frame with photons is moving with (v/c)^2 = .8. I am guessing the electron and positron are still at rest before annihilation so there is no velocity transformation necessary. So should the momentum of the photon apply just the Lorentz transformation, making "p = y m0 c" and energy be "E = y m0 c^2" where y is the Lorentz factor with (v/c)^2 = .8 ?

    Thanks!
     
  2. jcsd
  3. Oct 12, 2016 #2

    Simon Bridge

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    (a) you appear to have assumed that the electron and positron are at rest when the annihilation has occurred - total momentum after annihilation must be zero, but photons cannot be at rest. That how you are thinking?
    What about the case where they have equal and opposite velocity in the x direction, but equal positive velocity in the z direction?
    Wouldn't energy and momentum be satisfied with a single photon travelling in the +z direction?
    Would it be possible, even in the "annihilation at rest" case, to satisfy conservation of momentum with 3 or 4 photons?

    (b) The assumption of "at rest" is explicitly stated here - so is that a good assumption for part (a)? However, your reasoning is good for this part - but why does the question tell you to consider the +x axis as the direction of one photon? (hint: momentum is a vector...)

    (c) That looks good - a way to check your understanding here is to consider what happens to the photons in this moving frame compared with the rest frame of the particles. (hint: the photons do not go any faster or slower, so what changes, and in what way?) Is your answer. above, consistent with this?
    Aside: I see the direction of the relative velocity is not given above - and the photons are in all inertial frames - so there is a slight ambiguity.
     
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