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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.