1549380323-Statistical Mechanics Theory and Molecular Simulation

552 Quantum time-dependent statistical mechanics

to the eigenstates ofHˆ 0. This approach, known asquantum linear response theory, is
the quantum version of the classical linear response theory described in Section 13.2,
and it also the basis for the calculation of quantum transport properties. Since the
eigenstate approach to linear spectroscopy derived in Section 14.3employed first-order
perturbation theory (Fermi’s Golden Rule), we expect to use a linearization of the
quantum Liouville equation, as was done in Section 13.2, in order to establish the
connection between the eigenstate and density-matrix approaches.
Recall that the quantum Liouville equation is

∂ˆρ(t)

∂t

=

1

i ̄h

[

Hˆ(t),ρˆ(t)

]

. (14.5.1)

In order to keep the discussion as general as possible, we will consider the solution of
eqn. (14.5.1) for a general class of Hamiltonians of the form

Hˆ(t) =Hˆ 0 −ˆVFe(t), (14.5.2)

whereVˆis a Hermitian operator andFe(t) is an arbitrary function of time.
As in the classical case, we take an ansatz for ˆρ(t) of the form

ρˆ(t) = ˆρ 0 (Hˆ 0 ) + ∆ˆρ(t), (14.5.3)

where ˆρ 0 (Hˆ 0 ) is the equilibrium density matrix for a system described by the unper-
turbed HamiltonianHˆ 0 and which therefore satisfies an equilibrium Liouville equation

[

Hˆ 0 ,ρˆ 0

]

= 0,

∂ρˆ 0

∂t

= 0. (14.5.4)

Assuming that the system is in equilibrium before the perturbation is applied, the
initial condition on the Liouville equation is ˆρ(t 0 ) = ˆρ 0 (Hˆ 0 ), ∆ˆρ(t 0 ) = 0. When

eqn. (14.5.3) is substituted into eqn. (14.5.1) and terms involving bothVˆand ∆ˆρare
dropped, we obtain the following equation of motion for ∆ˆρ(t):

∂∆ˆρ(t)

∂t

=

1

i ̄h

[

Hˆ 0 ,∆ˆρ(t)

]

−

1

i ̄h

[

ˆV,ρˆ 0

]

Fe(t). (14.5.5)

Since this equation is a first-order inhomogeneous linear differentialequation for an
operator and involves a commutator, left and right integrating factors in the form of
Uˆ 0 †(t) = exp(iHˆ 0 t/ ̄h) andUˆ 0 (t) = exp(−iHˆ 0 t/ ̄h), respectively, are needed. With these
integrating factors, the solution for ∆ˆρ(t) becomes

∆ˆρ(t) =−

1

i ̄h

∫t

t 0

dse−i

Hˆ 0 (t−s)/ ̄h[ˆ

V,ρˆ 0

]

ei

Hˆ 0 (t−s)/ ̄h

Fe(s). (14.5.6)

The ensemble average of an operatorAˆin the time-dependent quantum ensemble
is given by

〈Aˆ〉t= Tr

[

ρˆ(t)Aˆ

]

= Tr

[

ˆρ 0 Aˆ

]

+ Tr

[

∆ˆρ(t)Aˆ

]

=〈Aˆ〉+ Tr

[

∆ˆρ(t)Aˆ

]

, (14.5.7)