We were deriving Fermi's golden rule from first applying a weak time dependant (periodic) perturbation and then looking at first order changes to the Hamiltonian.
We found that the probability of finding the system in state $f$, from an initial state $i$ turns out to be a sinc function.
In the notes, the following statement was made:
“There is a high probability of causing a transition from initial state i to final state f only if the frequency of the perturbation matches the energy difference between the states.”
My issue is understanding how its possible to achieve large probabilities of transition (close to 1) when the initial assumption was that the perturbation was weak? If the perturbation is weak doesn’t that imply it can only result in small changes to our wave function? I.e it can only make small changes to the probabilities associated with the wave function being in a particular eigenstate.