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## Main Question or Discussion Point

When I read about the derivation of this formula from Cohen Tannoudji or similar books, I couldn't understand one thing. He starts off assuming that the unperturbed Hamiltonian has a non-degenerate discrete spectrum, writes down the formulas for the evolution of an initial state using the resolution of the identity corresponding to this spectrum, and then assumes that the initial state is an eigenvector of the unperturbed Hamiltonian. After all of this he calculates the probability amplitude (at first order) that the evolved state at time t is found in another eigenstate of the unperturbed Hamiltonian (different from the original one he started from). So far everything's ok. But the he says that he wants to calculate the probability amplitude for the initial state to be found in a

Long story short, why doesn't one use Dyson series and then (as it is done in presence of a simple non-degenerate discrete spectrum) insert in this series of operators a lot of resolutions of the identity that account for the case in which there's a continuous spectrum too? That would be the most general formula, IMHO...

__continuum__(I hope my expression is correct) of__eigenstates__of the original hamiltonian and uses exactly the same formulas for the evolution of the initial state! That is, the resolution of the identity he uses is the same as before! Then my question is: if the unperturbed Hamiltonian had a continuous and discrete spectrum, shouldn't both of them enter in the resolution fo the identity? The formulas would be different this way when I calculate the evolution of an arbitrary initial state.Long story short, why doesn't one use Dyson series and then (as it is done in presence of a simple non-degenerate discrete spectrum) insert in this series of operators a lot of resolutions of the identity that account for the case in which there's a continuous spectrum too? That would be the most general formula, IMHO...