What is the case when a positron electron annihilation gives two photons and one of the photons has as small frequency as possible? I guess it is when the electron and positron are at rest before they annihilate. But what if doppler effect or conservation of momentum can cause smaller frequency for one of the photons?
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The annihilation of an electron positron pair most commonly creates either two or three photons in the final state (depending on the original angular momentum of the pair).
In each case both energy and momentum must be conserved so there are kinematic limits on the energies of the photons as measured in the center of momentum frame of the original pair.
You are presumably being asked to examine these two cases and find the highest energy that any single photon can have in that frame.
