Calculating the minimum energy

Assuming e+ and e- have no momentum…

and since ,

Which gives us this expression for the energy of the g-ray photon

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If the electron and positron have no momentum, then p=0 so the final energy is simply (2E)^2 = (2mc^2)^2. Then equating this to the expression to the initial energy (because the quantity (E^2 - (pc)^2) is a relativistic invariant), we derive the expression for the gamma-ray photon as shown.