1549380323-Statistical Mechanics Theory and Molecular Simulation

Time-dependent perturbation theory 537

=Iˆ−

i

̄h

∫t

t 0

dt′HˆI(t′) +

(

i

̄h

) 2 ∫t

t 0

dt′

∫t′

t 0

dt′′HˆI(t′)HˆI(t′′), (14.2.14)

and so forth. Thus we see that thekth-order solution is generated from the (k−1)th-
order solution according to the recursion formula:

UˆI(k)(t;t 0 ) =Iˆ−i

̄h

∫t

t 0

dt′HˆI(t′)UˆI(k−1)(t′;t 0 ). (14.2.15)

Settingk= 3 in eqn. (14.2.15), for example, yields the third-order solution

UˆI(3)(t;t 0 ) =Iˆ−i

̄h

∫t

t 0

dt′HˆI(t′) +

(

i

̄h

) 2 ∫t

t 0

dt′

∫t′

t 0

dt′′HˆI(t′)HˆI(t′′)

−

(

i

̄h

) 3 ∫t

t 0

dt′

∫t′

t 0

dt′′

∫t′′

t 0

dt′′′HˆI(t′)HˆI(t′′)HˆI(t′′′). (14.2.16)

By taking the limitk→ ∞in eqn. (14.2.15) and summing over all orders, we obtain
the exact solution forUˆI(t;t 0 ) as a series:

UˆI(t;t 0 ) =

∑∞

k=0

(

−

i

̄h

)k∫t

t 0

dt(1)···

∫t(k−1)

t 0

dt(k)HˆI(t(1))···HˆI(t(k)). (14.2.17)

The propagatorUˆI(t;t 0 ), approximated at any order in perturbation theory, can
be used to approximate the time evolution of the state vector|Φ(t)〉in the interaction
picture. In general, this evolution is

|Φ(t)〉=UˆI(t;t 0 )|Φ(t 0 )〉, (14.2.18)

and from this expression, the time evolution of the original state vector|Ψ(t)〉in the
Schr ̈odinger picture can be determined as

|Ψ(t)〉=e−i

Hˆ 0 (t−t 0 )/ ̄h

|Φ(t)〉

=e−i

Hˆ 0 (t−t 0 )/ ̄hˆ

UI(t;t 0 )|Φ(t 0 )〉

=e−i

Hˆ 0 (t−t 0 )/ ̄hˆ

UI(t;t 0 )|Ψ(t 0 )〉

≡Uˆ(t;t 0 )|Ψ(t 0 )〉. (14.2.19)

Here, we have used the fact that|Φ(t 0 )〉=|Ψ(t 0 )〉. In the last line, the full propagator
in the Schr ̈odinger picture is identified as

Uˆ(t;t 0 ) =e−iHˆ^0 (t−t^0 )/ ̄hUˆI(t;t 0 ) =Uˆ 0 (t;t 0 )UˆI(t;t 0 ). (14.2.20)