1549380323-Statistical Mechanics Theory and Molecular Simulation

Quantum linear response theory 553

where〈Aˆ〉is the equilibrium ensemble average ofAˆ. When eqn. (14.5.6) is substituted
into eqn. (14.5.7), we obtain

〈Aˆ〉t=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

{

Aˆe−iHˆ^0 (t−s)/ ̄h

[

Vˆ,ˆρ 0

]

ei

Hˆ 0 (t−s)/ ̄h}

Fe(s)

=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

{

ei

Hˆ 0 (t−s)/ ̄hˆ

Ae−i

Hˆ 0 (t−s)/ ̄h

[

Vˆ,ρˆ 0

]}

Fe(s)

=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

{

Aˆ(t−s)

[

ˆV(0),ρˆ 0

]}

Fe(s). (14.5.8)

In the second line, we have used the fact that the trace is invariantto cyclic permu-
tations of the operators, Tr(AˆBˆCˆ) = Tr(CˆAˆBˆ) = Tr(BˆCˆAˆ). In the last line,Aˆ(t−s)

denotes the operatorAˆin the interaction picture at timet−s, andˆV(0) denotes an
operator in this picture att= 0. Using the cyclic property of the trace again, the
expression in eqn. (14.5.8) can be further simplified. Expanding the commutator in
the last line of eqn. (14.5.8) yields

〈Aˆ〉t=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

[

Aˆ(t−s)Vˆ(0)ˆρ 0 −Aˆ(t−s)ˆρ 0 Vˆ(0)

]

Fe(s)

=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

[

ρˆ 0 Aˆ(t−s)ˆV(0)−ρˆ 0 Vˆ(0)Aˆ(t−s)

]

Fe(s)

=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

[

ρˆ 0

(

Aˆ(t−s)Vˆ(0)−ˆV(0)Aˆ(t−s)

)]

Fe(s)

=〈Aˆ〉−

1

i ̄h

∫t

t 0

dsTr

{

ρˆ 0

[

Aˆ(t−s),Vˆ(0)

]}

Fe(s). (14.5.9)

Eqn. (14.5.9) is the quantum analog of the classical linear response formula given in
eqn. (13.2.27) and is hence the starting point for the development of quantum Green-
Kubo expressions for transport properties. Since these expressions are very similar
to their classical counterparts, we will not repeat the derivationsof the Green-Kubo
formulae here.
The time correlation function appearing in eqn. (14.5.9) is referred to as theafter-
effect functionΦAV(t), which is defined as

ΦAV(t) =

i

̄h

〈[Aˆ(t),Vˆ(0)]〉. (14.5.10)

Note that although the operator combination [Aˆ(t),Vˆ(0)] is anti-Hermitian,^3 thei
prefactor in eqn. (14.5.10) fixes this: the operatori[Aˆ(t),Vˆ(0)] is Hermitian. In order

(^3) An anti-HermitianBˆsatisfiesBˆ†=−Bˆ.