I was looking at the solution to an exercise, and I came over this expression:
$$P_{i\to f} = \sum \limits_{f} {2 \pi \over \hbar }\; |\langle f |\hat V | i \rangle |^2 \delta(E_{fi}-E),$$
where I simplified this but it is in effect what the expression is a sum over the Fermi golden rule first order for the transition rate, summing over the final states. I would like to know why this sum is showing up and what it could mean in this context.
Answer
This is the total transition rate for a transition to occur. For instance, in a scattering experiment, there is a probability that an incoming particle will or will not be scattered by the target. If the quantity in which one is interested is the probability that a particle is scattered, then the quantity you have given above would give you the answer. This is because it gives the probability of scattering out of the initial state into all final states. This is not always the quantity that one is always interested in in a scattering experiment, however.
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