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8: Time-Dependent Perturbation Theory

Suppose that the Hamiltonian of the system under consideration can be written

\[H = H_0 + H_1(t) \tag{739}\]

where \(H_0\) does not contain time explicitly, and \(H_1\) is a small time-dependent perturbation. It is assumed that we are able to calculate the eigenkets of the unperturbed Hamiltonian:

\[H_0 \,\vert n\rangle = E_n \,\vert n\rangle. \tag{740}\]

We know that if the system is in one of the eigenstates of \(H_0\) then, in the absence of the 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 a system initially in some eigenstate  \(\vert i\rangle\) of the unperturbed Hamiltonian is found in some other eigenstate at a subsequent time (because \( \vert i\rangle\) 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.