1549380323-Statistical Mechanics Theory and Molecular Simulation

540 Quantum time-dependent statistical mechanics

t’

t’’

t 0 t

t 0

t

t’

t’’

t 0 t

t 0

t

(a) (b)

Fig. 14.4The two integration regions in eqn. (14.2.27): (a) The regiont′′∈[t 0 ,t],t′∈[t 0 ,t′′].
(b) The regiont′∈[t 0 ,t],t′′∈[t′,t].

Fig. 14.4(a) illustrates the integration regiont′′∈[t 0 ,t],t′∈[t 0 ,t′′] in thet′-t′′plane,
which is covered in the second term on the right side of eqn. (14.2.27). The same region
can be covered by choosingt′∈[t 0 ,t] andt′′∈[t′,t], as illustrated Fig. 14.4(b). With
this choice, eqn. (14.2.27) becomes

I(t 0 ,t) =

1

2

[∫

t

t 0

dt′

∫t′

t 0

dt′′HˆI(t′)HˆI(t′′) +

∫t

t 0

dt′

∫t

t′

dt′′HˆI(t′′)HˆI(t′)

]

. (14.2.28)

In the first term on the right side of eqn. (14.2.28),t′′< t′andHI(t′′) acts first,
followed byHI(t′). In the second term,t′ < t′′andHˆI(t′) acts first, followed by
HˆI(t′′). The two terms can thus be combined with botht′andt′′lying in the interval
[t 0 ,t] by introducing the time-ordering operator:

∫t

t 0

dt′

∫t′

t 0

dt′′HˆI(t′)HˆI(t′′) =

1

2

∫t

t 0

dt′

∫t

t 0

dt′′T

[

HˆI(t′)HˆI(t′′)

]

. (14.2.29)

The same analysis can be applied to each order in eqn. (14.2.17), recognizing that a
product ofkoperators can be ordered ink! ways by the time-ordering operator. Eqn.
(14.2.17) can then be rewritten in terms of a time-ordered product

UˆI(t;t 0 ) =

∑∞

k=0

(

−

i

̄h

)k

1

k!

×

∫t

t 0

dt 1

∫t

t 0

dt 2 ···

∫t

t 0

dtkT

[

HˆI(t 1 )HˆI(t 2 )···HˆI(tk)

]

. (14.2.30)

The sum in eqn. (14.2.30) resembles the power-series expansion of an exponential, and
consequently we can write the sum symbolically as