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Consider a system whose Hamiltonian can be written

(1017) 
Here, is again a simple timeindependent Hamiltonian whose eigenvalues and eigenstates are known exactly. However,
now represents a small timedependent external perturbation.
Let the eigenstates of take the form

(1018) 
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 timedependent
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 timedependent perturbation can cause
the system to make transitions between its unperturbed energy eigenstates.
Let us investigate this effect.
Richard Fitzpatrick
20100720