# In Fermi's Golden Rule, does the transition probability increase linearly with time or quadratically with time?

When deriving Fermi's Golden rule, we get that the probability of a quantum system transitioning from an initial state $$|i\rangle$$ to a final state $$|f\rangle$$ is $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\right)^2}.\tag{1}$$ If we wish to examine the behaviour of this probability when the resonance condition $$\omega_{fi}=\omega$$ is met, we can do the following $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)^2}\cdot t^2\\ \Rightarrow P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}\text{sinc}^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)\cdot t^2$$ Now since $$sinc(0)=1$$, we have that when $$\omega_{fi}=\omega$$, the probability of transition increases quadratically with time: $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|^2}{\hbar^2}t^2 \tag{2}$$ This quadratic increase in time is actually not unique to the resonance condition. It occurs for all the secondary peaks in the $$P_{i\rightarrow f}(t)$$ function. So seemingly, the probability for a transition should increase quadratically with time. However, all the books I have read (McIntyre, Shankar, Griffiths etc) as well as all the responses on this site pertaining to Fermi's Golden Rule that I have seen then go on to say that $$\lim_{t\rightarrow \infty}P_{i\rightarrow f}(t)=\lim_{t\rightarrow \infty}\frac{|V_{fi}|^2}{\hbar^2}\frac{\sin^2\left(\frac{\omega_{fi}-\omega}{2}\cdot t\right)}{\left(\frac{\omega_{fi}-\omega}{2}\right)^2}$$ $$\therefore \lim_{t\rightarrow \infty}P_{i\rightarrow f}(t)=\frac{2\pi}{\hbar^2}|V_{fi}|^2\delta(\omega_{fi}-\omega)\cdot t \tag{3}$$ This result now seems to indicate that the probability of transition increases linearly with respect to time provided we are looking at $$P_{i\rightarrow f}(t)$$ for large enought times. We no longer have the quadratic time dependence that equation (2) suggests we should. So what has happened here? Why do equation (1) and (2) imply that the probability should increase quadratically with time (regardless of how large $$t$$ is) but when we take the limit as $$t\rightarrow \infty$$, all of a sudden we now only have a linear increase with time?

I have seen the post over here (Transition probability derivation: How to prove $\lim_{\alpha\rightarrow\infty} \frac{\sin^2\alpha x}{\alpha x^2} ~=~\pi\delta(x)$?) which proves the limit but the answer does not explain how the transition probability $$P_{i\rightarrow f}(t)$$ somehow seems to be both increasing quadratically with time according to equations (1) and (2) but only linearly according to equation (3). Any help on this issue would be greatly appreciated!

• More on Fermi's golden rule. Oct 19, 2022 at 11:48
• Oct 19, 2022 at 18:17
• @Qmechanic I have looked through many of the relevant questions on stackexchange but cant find any that explain why the quadratic dependence on time turns into linear dependence on time for large $t$ values Oct 20, 2022 at 7:00
• @CosmasZachos I have read that excellent response but still cant see how a function that is quadratic with time can change into a function that is linearly dependent on time when we look at large $t$. It seems to me to be a contradiction. Oct 20, 2022 at 7:26
• Fermis Golden Rule applies only in a specific range for $t$ and within this range, the rate is approximately linear in $t$. The links in Qmechanics answer explain this, although it might be a bit hard to see on a short glance. The general expression for the probability is indeed not necessarily linear in $t$. It is only approximately so in this specific range for $t$ and Fermis Golden Rule is only valid for these values of $t$. Oct 20, 2022 at 14:43

1. OP is considering an argument $$x$$ for the $${\rm sinc}^2(x)$$ function with $$|x|\ll 1$$, where the probability $$P \propto t^2$$ is indeed quadratic in $$t$$.
2. But Fermi's golden rule applies to the region $$|x|\gtrsim 1$$, where $$P \propto t$$ is linear in $$t$$, cf. e.g. this, this and this Phys.SE posts.