Probabilities of transitions given by Fermi's golden rule

Question

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.

Answer 1

High probability here refers to different things: the book/professor might be not very clear about it.

• The perturbation is weak and the transition rate is small - these are among the underlying assumptions of the derivation. Fermi Golden rule certainly fails when probabilities are close to $1$ - in this case it is more appropriate to discuss Rabi oscillations.
• The sinc function is however sharply peaked, and the probability of transitions at the frequencies close to the resonant frequency (the level separation) are much higher than at other frequencies.

Remarks:

• Fermi Golden rule is typically used in calculation involving large collections of atoms or other situations where many identical transitions are possible. So the overall effect is large, as it is proportional to the transition probability times the number of atoms in the system (e.g., we could be easily dealing with an Avogadro number of atoms $~10^{24}$.
• The general rule in physics is that nothing can be high/big or low/small in absolute terms, but only in comparison to something else (particularly obvious for dimensional quantities: high/low means being much bigger/smaller than $1$ - obviously, no dimensional quantity can be considered as high/low, but only dimensionless ratios of such quantities).