1549380323-Statistical Mechanics Theory and Molecular Simulation

538 Quantum time-dependent statistical mechanics

From eqn. (14.2.20), the structure of the full propagator is revealed. Let us use eqn.
(14.2.20) to generate the first few orders in the propagator. Substituting eqn. (14.2.12)
into eqn. (14.2.20) yields the lowest order contribution toUˆ(t;t 0 ):

Uˆ(0)(t;t 0 ) =e−iHˆ^0 (t−t^0 )/ ̄h=Uˆ 0 (t;t 0 ). (14.2.21)

Thus, at zeroth order, eqn. (14.2.21) implies that the system is propagated using the
unperturbed propagatorUˆ 0 (t;t 0 ) as though the perturbation did not exist. At first
order, we obtain

Uˆ(1)(t;t 0 ) =e−iHˆ^0 (t−t^0 )/ ̄h−i

̄h

e−i

Hˆ 0 (t−t 0 )/ ̄h

∫t

t 0

dt′HˆI(t′)

=e−i

Hˆ 0 (t−t 0 )/ ̄h

−

i

̄h

e−i

Hˆ 0 (t−t 0 )/ ̄h

∫t

t 0

dt′ei

Hˆ 0 (t′−t 0 )/ ̄hˆ

H 1 (t′)e−i

Hˆ 0 (t′−t 0 )/ ̄h

=e−i

Hˆ 0 (t−t 0 )/ ̄h

−

i

̄h

∫t

t 0

dt′e−i

Hˆ 0 (t−t′)/ ̄hˆ

H 1 (t′)e−i

Hˆ 0 (t′−t 0 )/ ̄h

=Uˆ 0 (t;t 0 )−

i

̄h

∫t

t 0

dt′Uˆ 0 (t;t′)Hˆ 1 (t′)Uˆ 0 (t′;t 0 ), (14.2.22)

where in the second line, the definition

HˆI(t) = eiHˆ^0 (t−t^0 )/ ̄hHˆ 1 (t)e−iHˆ^0 (t−t^0 )/ ̄h (14.2.23)

has been used. Eqn. (14.2.22) indicates that at first order, the propagator is composed
of two terms. The first term is simply the unperturbed propagationfromt 0 tot. In
the second term, the system undergoes unperturbed propagation fromt 0 tot′. Att′,
the perturbationHˆ 1 (t′) acts, and finally, fromt′tot, the propagation is unperturbed.
In addition, we must integrate over all possible intermediate timest′ at which the
perturbation is applied.
In a similar manner, it can be shown that up to second order, the fullpropagator
is given by

Uˆ(2)(t;t 0 ) =Uˆ 0 (t;t 0 )−i

̄h

∫t

t 0

dt′Uˆ 0 (t;t′)Hˆ 1 (t′)Uˆ 0 (t′;t 0 )

+

(

i

̄h

) 2 ∫t

t 0

dt′

∫t′

t 0

dt′′Uˆ 0 (t;t′)Hˆ 1 (t′)Uˆ 0 (t′;t′′)Hˆ 1 (t′′)Uˆ 0 (t′′;t 0 ). (14.2.24)

At second order, the new (last) term involves unperturbed propagation fromt 0 to
t′′, action ofHˆ 1 (t′′) att′′, unperturbed propagation fromt′′tot′, action ofHˆ 1 (t′)
att′, and finally unperturbed propagation fromt′tot. Again, we must integrate
over intermediate timest′andt′′at which the perturbation is applied. A pictorial
representation of the full propagator is given in Fig. 14.3. The picture on the left side
of the equal sign in Fig. 14.3 indicates that the perturbation causesthe system to