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I have a problem with the very nature of time dependant perturbation theory (TDPT). In TDPT, we consider a system of Hamiltonian H(t) = H_0 (for t<0), H(t)=H_0 + kW(t) (for t>0) [where k<<1], where H_0 is, for simplicity, discrete, non-degenerate and time-independent, and given that at t=0, the state of the system is |phi_i> (an eigenstate of H_0), we are interested in calculating P_if(t), the probability of finding the system in another eigenstate of H_0, |phi_f>, at time t.

But this does not make sense because at t, the hamiltonian is no longer H_0, so it will, in general, not have |phi_f> as an eigenstates. But we know that the result of a measurement will project the wave-function into one of the eigenfunction. So as soon as |phi_f> is not an eigenstate of H(t), the probability of finding the system in |phi_f> will be 0.

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# Time dependant perturbation

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