Annihilation

We know that particles and their antiparticles annihilate each other. For example, the electron and the positron in positronium can annihilate into two photons or three photons in the final state, depending on the total angular momentum of the positronium. Why would they do such a violent action? Hint: why does any event occur in nature? We know that the rate of any quantum mechanical event, by Fermi’s Golden Rule, is proportional to the probability for the event times the density of final states. Is this statement all we need to say?

Answer

Fermi’s Golden Rule hints that we should consider the phase space available for the final particles, and this phase space is related to the entropy of the final particles. If the entropy of the final state is greater than the entropy for the initial state, the process occurs. In the simpler case, when an electron at rest and its antiparticle, the positron at rest, annihilate each other, two photons are produced to conserve quantum numbers as well as energy and momentum. The entropy of the products is greater than the reactants. Why? Because there is much freedom in the direction of the photon polarizations. The interacting particle and antiparticle begin with their spins opposite but along a specific direction, thereby having a total spin of zero. In the final state with two identical photons emerging in opposite directions in order to conserve energy and linear momentum, the photon spins are opposite that is, both spin +1 or both spin –1 with respect to their momentum directions but the polarization vectors can be in any direction in the plane perpendicular to the momentum directions.

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