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In time-independent perturbation theory, the objective was to perturb the Hamiltonian and find out corrections in energy eigenvalues and eigenfunctions. In most of the textbooks, they find up to second-order correction to both.

However, why is it the case that when we go to time-dependent perturbation theory, somehow the goal changes to finding transition probability from one state to another? Why are we not interested in corrections to eigenvalues and eigenfunctions anymore?

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    $\begingroup$ The reason we care about the eigenstates of the Hamiltonian is that they correspond to the possible outcomes of energy measurements. In practice, we don't usually measure the energy of the interacting Hamiltonian, but rather some observable of the final state after the interaction is turned off again. For example, if a system is driven from the ground to excited state and the interaction is turned off, then we can measure it to be there, and we don't care about the eigenstates of the Hamiltonian at every time t. See also the answer to this question $\endgroup$
    – fulis
    Commented Mar 25, 2020 at 20:06
  • $\begingroup$ @fulis, that was a good explanation. I also read in the link below that a time-dependent Hamiltonian implies no conservation of energy, which is why it does not make sense to look for energy corrections. galileo.phys.virginia.edu/classes/752.mf1i.spring03/… $\endgroup$
    – time12
    Commented Mar 27, 2020 at 13:08

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