What does the transition spectrum look like on short time scales? Instead of taking the limit t → ∞, determine the probability |〈f ψ(t)〉|², and use a program like Mathematica to plot it as a function of final state energy E_f, for various times t. You should see that at early times the spectrum oscillates rapidly, but that it converges to a Lorentzian on time scales long compared to h̄/Γ.