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Let us now try to find approximate solutions of Eq. (740) for a general
system. It is convenient to work in terms of the time evolution
operator, , which is defined

(768) 
Here,
is the state ket of the
system at time , given that the state ket at the initial
time is . It is easily seen that the time evolution operator
satisfies the differential equation

(769) 
subject to the boundary condition

(770) 
In the absence of the external perturbation, the time evolution operator
reduces to

(771) 
Let us switch on the perturbation and look for a solution of the
form

(772) 
It is readily demonstrated that satisfies the differential
equation

(773) 
where

(774) 
subject to the boundary condition

(775) 
Note that specifies that component of the time evolution operator
which is due to the timedependent perturbation. Thus, we would expect
to contain all of the information regarding transitions between different
eigenstates of caused by the perturbation.
Suppose that the system starts off at time in the eigenstate of
the unperturbed Hamiltonian. The subsequent evolution of
the state ket is given by Eq. (735),

(776) 
However, we also have

(777) 
It follows that

(778) 
where use has been made of
.
Thus, the probability that the system is found in state at time
, given that it is definitely in state at time ,
is simply

(779) 
This quantity is usually termed the transition probability
between states and .
Note that the differential equation (773), plus the boundary condition
(775), are equivalent to the following integral equation,

(780) 
We can obtain an approximate solution to this equation by iteration:
This expansion is known as the Dyson series.
Let

(782) 
where the superscript refers to a firstorder term in the expansion,
etc. It follows from Eqs. (778) and (781) that
These expressions simplify to
where

(789) 
and

(790) 
The transition probability between states and is
simply

(791) 
According to the above analysis, there is no chance of a
transition between states and ()
to zerothorder (i.e., in the absence of the perturbation). To
firstorder, the transition probability is proportional to
the time integral of the matrix element
,
weighted by some oscillatory phasefactor. Thus, if the matrix
element is zero, then there is no chance of a firstorder transition between
states and . However, to secondorder,
a transition between states and is possible
even when the
matrix element
is zero.
Next: Constant perturbations
Up: Approximation methods
Previous: Spin magnetic resonance
Richard Fitzpatrick
20060216