In the area of the time-dependent perturbation theory, if we have $$\hat{H}(t) = \hat{H}_0 + \hat{V}(t).$$ The Schrodinger equation is $$i \hbar \frac{d | \Psi(t) \rangle}{dt} = (\hat{H}_0 + \hat{V}(t))| \Psi(t) \rangle$$ where in the Shrodinger picture we have $$|\Psi(t) \rangle = \sum_n c_n(t)e^{\frac{-iE_n}{\hbar}}| \psi_n \rangle$$ In the interaction picture we have $$i \hbar \frac{d | \Psi(t) \rangle_{I}}{dt} = \hat{V}_{I}(t) | \Psi(t) \rangle_{I}$$ where $$| \Psi(t) \rangle_{I} = \hat{U}(t,t_i)| \Psi(t_i) \rangle_I$$ we therefore get $$i \hbar \frac{d \hat{U}(t, t_i)}{dt} = \hat{V}_i(t)\hat{U}_{I}(t,t_i).$$
Question: Short question, why does it follow that the transition probability corresponding to a transition from an initial unperturbed state $| \psi_i \rangle$ to another unperturbed state $| \psi_f \rangle$ is $$P_{if}(t) = |\langle \psi_f | \hat{U}_{I}(t,t_i)| \psi_i \rangle |^2$$ also why is the transition probability in terms of the expansion coefficients given by $$P_{if}(t) = |c_{f}^{0}+c_{f}^{1} +...|^2?$$ Are these postulates?
Thanks.