Time-dependent perturbation theory 539= + + + ...
t 0
t
t 0
t
t 0
t
t’
t 0
t
t’’
t’
t 0
t
t’’’
t’’
t’
Fig. 14.3 Pictorial representation of the perturbation expansion ofthe time-dependent prop-
agator.
undergo some undetermined dynamical process betweent 0 andt. On the right side of
the equal sign, the process is broken down in terms of the action ofthe perturbationHˆ 1
at specific intermediate times (which must be integrated over), indicated in the figure
by the dots. At thekth order, the perturbation HamiltonianHˆ 1 acts on the system
atkinstances in time. The limits of integration indicate that these time instances are
ordered chronologically.
The specific ordering of the times at whichHˆ 1 acts on the unperturbed system
raises an important point. In each term in the expansion forUˆI(t;t 0 ), the order in
which the operatorsHˆI(t′),HˆI(t′′),... are multiplied is critical. The reason for this
is that the commutator [HˆI(t),HˆI(t′)] does not vanish ift 6 =t′. Thus, to remove
any ambiguity when specifying the order of the operatorsHˆI(t′),HˆI(t′′),... in a time
series, we introduce thetime-ordering operator,T. The action ofT on a product
of time-dependent operatorsAˆ(t 1 )Bˆ(t 2 )Cˆ(t 3 )···Dˆ(tn) reorders the operators in the
product chronologically in time from earliest to latest proceeding from right to left in
the product. This means that operators earliest in time act beforeoperators at later
times. For example, the action ofTon two operatorsAˆ(t 1 ) andBˆ(t 2 ) is
T
[
Aˆ(t 1 )Bˆ(t 2 )]
=
Aˆ(t 1 )Bˆ(t 2 ) t 2 < t 1Bˆ(t 2 )Aˆ(t 1 ) t 1 < t 2(14.2.25)
Let us now apply the time-ordering operatorTto the second-order term. We first
write the double integral
I 2 (t 0 ,t) =∫tt 0dt′∫t′t 0dt′′HˆI(t′)HˆI(t′′) (14.2.26)as a sum of two terms generated by interchanging the dummy variablest′andt′′:
I 2 (t 0 ,t) =1
2
[∫
tt 0dt′∫t′t 0dt′′HˆI(t′)HˆI(t′′) +∫tt 0dt′′∫t′′t 0dt′HˆI(t′′)HˆI(t′)