Why is the minimum amount of energy a photon has is equal to the rest energy of one of the particles?

Question

Why is the minimum amount of energy a photon has is equal to the rest energy of one of the particle ? I know that the the minimum possible energy a photon may have is only the rest energy of particles is converted. But if that's the case then the particle and antiparticle pair must be stationary , which is impossible as they need to collide first to produce photons

Answer 1

You are correct that the total energy after the annihilation has to be equal to the total energy before the annihilation, and that has to include the kinetic energy of the electron and positron. So if the electron and positron are approaching each other at some speed $v$ we need to add the KE on to the rest mass energy.

However we can in principle consider the limit of $v \to 0$ even though we can never actually get there since the experiment would take an infinite time to perform. Then the limiting value of the photon energy is $m_ec^2$ and it seems reasonable to describe this as the minimum photon energy.

Though in fact the photons from annihilation can be slightly less energetic than this. Suppose we start with a para-positronium atom, then the mass of the positronium atom is $2m_e$ minus the binding energy of $6.8$ eV. This means the energies of the two photons produced will be about $3.4$ eV less than the photons produced by annihilation of an unbound electron and photon.

Answer 2

You are forgetting quantum mechanics rules for electron and positron. That is how one has the positronium,

it is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. The system is unstable: the two particles annihilate each other to predominantly produce two or three gamma-rays, depending on the relative spin states.

For quantum entities there is no meaning of being stationary in their bound state. Only the quantum mechanical probabilities rule.