Next: Harmonic Perturbations
Up: TimeDependent Perturbation Theory
Previous: Spin Magnetic Resonance
Let us recall the analysis of Sect. 13.2. The
are the stationary orthonormal eigenstates of the timeindependent
unperturbed Hamiltonian, . Thus,
,
where the are the unperturbed energy levels, and
. Now, in the presence of a small
timedependent perturbation to the Hamiltonian, , the wavefunction
of the system takes the form

(1057) 
where
. The amplitudes satisfy

(1058) 
where
and
. Finally, the probability of finding the system in the th eigenstate
at time is simply

(1059) 
(assuing that, initially,
).
Suppose that at the system is in some initial energy eigenstate labeled . Equation (1058) is, thus, subject to the initial condition

(1060) 
Let us attempt a perturbative solution of Eq. (1058) using
the ratio of to (or to
, to be more exact) as our expansion parameter.
Now, according to (1058), the are constant in time in the
absence of the perturbation. Hence, the zerothorder solution is simply

(1061) 
The firstorder solution is obtained, via iteration, by substituting the zerothorder
solution into the righthand side of Eq. (1058). Thus, we obtain

(1062) 
subject to the boundary condition
. The solution to
the above equation is

(1063) 
It follows that, up to firstorder in our perturbation expansion,

(1064) 
Hence, the probability of finding the system in some final energy
eigenstate labeled at time , given that it is definitely in a different initial energy eigenstate labeled at time , is

(1065) 
Note, finally, that our perturbative solution is clearly only valid provided

(1066) 
Next: Harmonic Perturbations
Up: TimeDependent Perturbation Theory
Previous: Spin Magnetic Resonance
Richard Fitzpatrick
20100720