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Consider a system whose Hamiltonian can be written $H(t) = H_0 + H_1(t).$ Here, $$H_0$$ is again a simple time-independent Hamiltonian whose eigenvalues and eigenstates are known exactly. However, $$H_1$$ now represents a small time-dependent external perturbation. Let the eigenstates of $$H_0$$ take the form $H_0\,\psi_m = E_m\,\psi_m.$ We know (see Section [sstat]) that if the system is in one of these eigenstates then, in the absence of an external perturbation, it remains in this state for ever. However, the presence of a small time-dependent perturbation can, in principle, give rise to a finite probability that if the system is initially in some eigenstate $$\psi_n$$ of the unperturbed Hamiltonian then it is found in some other eigenstate at a subsequent time (because $$\psi_n$$ is no longer an exact eigenstate of the total Hamiltonian). In other words, a time-dependent perturbation allows the system to make transitions between its unperturbed energy eigenstates. Let us investigate such transitions.