If we assume that because of factors, a quantum mechanical system needs to rearrange itself, and in doing so, it might change it's state, to another one. In our consideration, we tried to find the probability of transition between two stationary states (from $m$ to $n$), which is:
$$P_{mn}(t)=\left|\int_0^t e^{i\frac{(E_n-E_m)t'}{\hbar}}\langle n|v(t')|m\rangle\right|^2$$
where $v(t)$ is the disturbance added to the hamiltonian, to give us $H=H_0+v(t)$.
By making the following assumptions: $v(t)=v\Theta(t)$ and $t\rightarrow \infty$ (I don't understand what we mean with this. Do we want to find the probability after a lot of time passes?) we have:
$$P_{mn}(t)= t \frac{2\pi}{\hbar}\delta(E_n-E_m)|\langle n|v|m\rangle|^2$$
- Transition rate: $$\Gamma_{mn}=\frac{P_{mn}(t)}{t}=\frac{2\pi}{\hbar}\delta(E_n-E_m)|\langle n|v|m\rangle|^2$$ also known as Fermis Golden Rule (for stationary states). How can one interpret the transition rate? Does it show how the probability of transitioning between states, changes in value per unit of time? In other words, with the help of $\Gamma_{mn}$ we can find moments in time when the transition has higher probability of taking place and moments when it has not?
Then, if we assume transition to a group of states, whose range of energy values, belongs to an interval $\Delta E$ then we consider the total rate of transitioning:
$$\int \Gamma_{mn}dN_n=\rho(E_m)\frac{2\pi}{\hbar} |\langle n|v|m\rangle|^2.$$
Same question here. How does one interpret this? Similarly as to how I assumed above, for the case of the transition rate between two states, but with the difference that, here the probability over time, indicates how probable it is for a state transition to take place, where the state to which the system is transiting to, can be any of the states in the interval $\Delta E$ ?
I also have one additional question, which to me it looks like a contradiction:
We were able to derive the expression for the total rate of the state transition with the two assumptions made earlier, which were: $v(t)=v\Theta(t)$ and $t\rightarrow \infty$. For this part, important is to focus on the claim that :$t\rightarrow \infty$. Following this derivation, we make some calculations, to find the case when, Fermis Golden rule can be used, and that case is:
$$\frac{4\pi \hbar}{\Delta E}\ll t \ll \frac{4\pi \hbar}{\delta E},$$
where $\delta E$ is the energy range of an arbitrary state, in which the system can transition to, while as $\Delta E$ is the energy interval, where all the states in which the system can transition to, reside.
As one can easily notice here, initially we make the claim: $t\rightarrow \infty$ and derive Fermis Golden Rule, and then we say that we can use this rule when $$\frac{4\pi \hbar}{\Delta E}\ll t \ll \frac{4\pi \hbar}{\delta E}$$ Isn't this a contradiction?
