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Preliminary Analysis
Suppose that at the state of the system is represented by

(1019) 
where the are complex numbers. Thus, the initial state is some
linear superposition of the unperturbed energy eigenstates. In the
absence of the timedependent perturbation, the time evolution
of the system is simply (see Sect. 4.12)

(1020) 
Now, the probability of finding the system in state at time is

(1021) 
since the unperturbed eigenstates are assummed to be orthonormal:
i.e.,

(1022) 
Clearly, with , the probability of finding the system in state
at time is exactly the same as the probability of finding the
system in this state at the initial time, . However, with ,
we expect and, hence, to vary with time. Thus, we can write

(1023) 
where
. Here, we have carefully separated the
fast phase oscillation of the eigenstates, which depends on the
unperturbed Hamiltonian, from the slow variation of the
amplitudes , which depends entirely on the perturbation
(i.e., is constant in time if ). Note that in Eq. (1023) the eigenstates are timeindependent (they are actually the eigenstates of evaluated at the initial time, ).
The timedependent Schrödinger equation [see Eq. (199)]
yields

(1024) 
Now, it follows from Eq. (1023) that

(1025) 
We also have

(1026) 
since the are timeindependent. According to
Eq. (1024), we can equate the righthand sides of the previous
two equations to obtain

(1027) 
Projecting out the component of the above equation which is proportional
to , using Eq. (1022), we obtain

(1028) 
where

(1029) 
and

(1030) 
Suppose that there are linearly independent eigenstates of
the unperturbed Hamiltonian. According to Eqs. (1028),
the timedependence of the set of coefficients , which specify the
probabilities of finding the system in these eigenstates at time , is determined
by coupled firstorder differential equations. Note that Eqs. (1028)
are exactwe have made no approximations at this stage. Unfortunately,
we cannot generally find exact solutions to these equations. Instead, we
have to obtain approximate solutions via suitable
expansions in small quantities. However, for the particuilarly
simple case of a twostate system (i.e., ), it is actually
possible to solve Eqs. (1028) without approximation. This solution
is of great practical importance.
Next: TwoState System
Up: TimeDependent Perturbation Theory
Previous: Introduction
Richard Fitzpatrick
20100720