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Consider a system whose Hamiltonian can be written
Here, is again a simple time-independent Hamiltonian whose eigenvalues and eigenstates are known exactly. However,
now represents a small time-dependent external perturbation.
Let the eigenstates of take the form
We know (see Sect. 4.12) 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 of the unperturbed
Hamiltonian then it is found in some other eigenstate at a subsequent time
(since is no longer an exact eigenstate of the total
Hamiltonian). In other words, a time-dependent perturbation can cause
the system to make transitions between its unperturbed energy eigenstates.
Let us investigate this effect.