1549380323-Statistical Mechanics Theory and Molecular Simulation

556 Quantum time-dependent statistical mechanics

Substituting the definitions ofR(ω) andR(−ω) from eqns. (14.3.7) and (14.3.15) into
this expression forQ(ω) yields

Q(ω) = ̄hω|F(ω)|^2

1

̄h^2

∫∞

−∞

dte−iωt

〈

ˆV(0)Vˆ(t)−Vˆ(t)Vˆ(0)

〉

=−ω|F(ω)|^2

1

̄h

∫∞

−∞

dte−iωt

〈[

Vˆ(t),Vˆ(0)

]〉

=iω|F(ω)|^2

∫∞

−∞

dte−iωtΦVV(t). (14.5.23)

Next, we divide the time integration into an integration xfrom−∞to 0 and from 0
to∞so that

Q(ω) =iω|F(ω)|^2

[∫ 0

−∞

dte−iωtΦVV(t) +

∫∞

0

dte−iωtΦVV(t)

]

=iω|F(ω)|^2

[∫∞

0

dteiωtΦVV(−t) +

∫∞

0

dte−iωtΦVV(t)

]

, (14.5.24)

where in the first term the transformationt→−thas been made. In order to proceed,
we need to analyze the time-reversal properties of the after-effect function.
Consider a general after-effect function ΦAB(t):

ΦAB(t) =

i

̄h

〈[

ei

Hˆ 0 t/ ̄hˆ

Ae−i

Hˆ 0 t/ ̄h

,Bˆ

]〉

. (14.5.25)

Substituting−tinto eqn. (14.5.25) yields

ΦAB(−t) =

i

̄h

〈

e−i

Hˆ 0 t/ ̄hˆ

Aei

Hˆ 0 t/ ̄hˆ

B−Bˆe−i

Hˆ 0 t/h ̄ˆ

Aei

Hˆ 0 t/h ̄〉

=

i

̄h

[

Tr

(

ρˆ 0 e−i

Hˆ 0 t/ ̄hˆ

Aei

Hˆ 0 t/ ̄hˆ

B

)

−Tr

(

ˆρ 0 Bˆe−i

Hˆ 0 t/ ̄hˆ

Aei

Hˆ 0 t/ ̄h)]

. (14.5.26)

Because the trace is invariant under cyclic permutations of the operators and ˆρ 0 com-
mutes with the propagators exp(±iHˆ 0 t/ ̄h), we can express eqn. (14.5.26) as

ΦAB(−t) =

i

̄h

[

Tr

(

ρˆ 0 e−i

Hˆ 0 t/ ̄hˆ

Aei

Hˆ 0 t/ ̄hˆ

B

)

−Tr

(

ρˆ 0 Bˆe−i

Hˆ 0 t/ ̄hˆ

Aei

Hˆ 0 t/ ̄h)]

=

i

̄h

[

Tr

(

e−i

Hˆ 0 t/h ̄

ρˆ 0 Aˆei

Hˆ 0 t/ ̄hˆ

B

)

−Tr

(

Bˆe−iHˆ^0 t/ ̄hAˆeiHˆ^0 t/ ̄hˆρ 0

)]

=

i

̄h

[

Tr

(

ρˆ 0 Aˆei

Hˆ 0 t/ ̄hˆ

Be−i

Hˆ 0 t/ ̄h)

−Tr

(

Bˆe−iHˆ^0 t/ ̄hAˆρˆ 0 eiHˆ^0 t/ ̄h

)]

=

i

̄h

[

Tr

(

ρˆ 0 Aˆei

Hˆ 0 t/ ̄hˆ

Be−i

Hˆ 0 t/ ̄h)

−Tr

(

ρˆ 0 ei

Hˆ 0 t/ ̄hˆ

Be−i

Hˆ 0 t/ ̄hˆ

A

)]