Time-dependant perturbation theory & "transitions" I'm studying approximation methods, and something is really bothering me about the standard treatment of time-dependant perturbation theory. In lecture, the prof introduced time-dependant perturbation theory with the following motivation: suppose we have a system with an initially time-independent Hamiltonian, initially in some eigenstate |i> of said Hamiltonian. At some time time t, a perturbation is turned on, which may be static or time-dependant itself, e.g. some fixed jump in the potential at time t, or a classical EM field applied at time t0. The question we ask is: what is the probability of finding the system at time t in another eigenstate |f> of the initial, unperturbed time-independant Hamiltonian? In other words, what is the probability of the system undergoing a transition from i to f? He then goes on the develop the standard iterative method of determining said probability, involving the mag-squared inner product of |f> with the new, time-dependent state psi(t): My problem is with the very question being asked. It seems to me ill-posed. Once the Hamiltonian is changed by the perturbation, unless the perturbation is again zero after some time, then "the probability of finding the system in state |f>", where |f> is an eigenstate of the original Hamiltonian, is (in general) zero. My reasoning is: once the Hamiltonian is perturbed, its eigenstates change with time, and the state |f> is generally no longer an eigenstate. So a measurement of energy at time t would not collapse the system into state |f>, but rather some eigenstate of the new Hamiltonian at time t. Further, even if by chance |f> is an eigenstate at some time t, the measured energy eigenvalue could be different, if for example the perturbation at that time t was a nonzero constant. I have consulted three standard texts: Shankar, Sakurai and Desai, but each of them states the question in essentially the same way. Finally, after turning to a great (but for some reason non-standard) book by Ballentine, I find some discussion of this: (Leslie E. Ballentine, Quantum Mechanics: A Modern Development (1998), pp. 351-352 This basically confirms my suspicions. It's neat to find out that the distinction has been made experimentally. However, I still have a problem: Ballentine addresses the case where the perturbation is zero outside a closed interval in time. In other texts, and in my course, we are also looking at the case where the perturbation remains in place for all time after t0. Certainly the system may then evolve to have a nonzero component along the |f> subspace, but the fact remains that a measurement of the energy will generally collapse the system to |f>, nor will it return an energy eigenvalue the same as |f>'s, in the context of H0. Sorry if that was long or hard to follow. Let me know if I need to clarify. Any thoughts?