Fermi's golden rule and infinite probablity?

Question

I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: $$P=\frac{2\pi t}{\hbar^2} | \langle f | \hat V | i \rangle|^2 \delta(E_f-E_i-\hbar \omega)$$ This holds for large values of $t$ but for arbitrarily large values of $t$: $$P>1$$ Which (to me anyway) makes no physical sense since there can't be a probability greater then 1 of been in the final state. If my interpretation is right then how is this problem resolved/allowable? and if it is wrong what part of what I have said is wrong?

Answer 1

This question has been discussed in the comment section, and OP has already found the answer in this wonderful post by QMechanic. In that post, we can clearly see that the magnitude of $t$ is not arbitrary: it has to be bounded both below and above, $$ \frac{2\pi}{\Delta \omega} \lesssim t \ll \frac{\hbar}{\sup_{f\in F}|V_{fi}|} $$ where the symbols are defined in QMechanic's post. This is the reason we cannot take $t\to \infty$.

It is useless to reproduce QMechanic's post here, or to try to improve it. Instead, I'll try to address my statement that, if we keep the magnitude of $t$ arbitrary, then the transition probability is an exponential.

In the usual proof of the FGR we find, at some point, the integrodifferential equation $$ c_i(t)=1-i\sum_n\int_0^tc_n(\tau)V_{ni}(\tau)\; \mathrm e^{-i\omega_{ni}\tau} $$ and take $c_n(\tau)\approx c_n(0)$ to get the first order correction (from which the FGR follows). This is the step that constrains the magnitude of $t$. If we take $c_n(\tau)\approx c_n(t)$ instead, we find (after some lengthy manipulations) $$ c_i(t)=\mathrm e^{-\Gamma t/2}\times \text{a phase} $$ where $\Gamma\propto V_{fi}$ is the decay width of the state. This formula is valid for arbitrarily large $t$. Expanding to first order ($\Gamma t\ll 1$), we get the usual formulas back. A detailed derivation (and discussion) can be found in the book by Cohen-Tannoudji Quantum Mechanics, vol 2., page 1351, "Another approximate method for solving the Schrödinger equation".