1549380323-Statistical Mechanics Theory and Molecular Simulation

546 Quantum time-dependent statistical mechanics

Although quantum time correlation functions possess many of the same properties as
their classical counterparts, we point out one crucial difference at this juncture. The
operatorsVˆ(0) andVˆ(t) are individually Hermitian, but since [Vˆ(0),Vˆ(t)] 6 = 0, the
autocorrelation function in eqn. (14.3.7) is an expectation value of anon-Hermitian
operator productVˆ(0)Vˆ(t). Such a non-Hermitian expectation value suggests that
something fundamental is missing from the above analysis.
A little reflection reveals that the problem lies with our choice ofω >0 in eqn.
(14.2.39). A complete analysis requires that we examineω <0 as well, in which case
the only first term on the right side of eqn. (14.2.40) is retained. Thisis tantamount
to substituting−ωforωin eqn. (14.3.3), which yields

R(−ω) =

2 π

̄h

|F(ω)|^2

∑

i,f

wi

∣

∣

∣〈Ef|Vˆ|Ei〉

∣

∣

∣

2

δ(Ef−Ei+ ̄hω). (14.3.9)

Unlike eqn. (14.3.3), which refers to an absorption process withEf=Ei+ ̄hω, eqn.
(14.3.9) describes a process for whichEf=Ei− ̄hω, orEf< Ei, which is anemission
process. The system starts in a state with energyEiand releases an amount of energy
̄hωas it decays to a state with lower energyEf. We will now show that eqn. (14.3.9)

can be expressed in terms of the correlation functionˆV(t)Vˆ(0), which when added to
eqn. (14.3.7) leads to the Hermitian combinationVˆ(t)Vˆ(0) +ˆV(0)Vˆ(t).
We begin by interchanging the summation indicesiandfin eqn. (14.3.9), which
gives

R(−ω) =

2 π

̄h

|F(ω)|^2

∑

i,f

wf

∣

∣

∣〈Ei|Vˆ|Ef〉

∣

∣

∣

2

δ(Ei−Ef+ ̄hω), (14.3.10)

where

wf=

e−βEf

Q(N,V,T)

=

e−βEf

Tr

[

e−βHˆ^0

]. (14.3.11)

The interchange of summation indices in eqn. (14.3.10) causes theδ-function condition
in eqn. (14.3.10) to revert to that contained in eqn. (14.3.9), namely,Ef=Ei+ ̄hω.
Substituting this condition into the expression forwfgives

wf=

e−β(Ei+ ̄hω)

Q(N,V,T)

=wie−β ̄hω. (14.3.12)

Sinceδ(x) =δ(−x), eqn. (14.3.10) can be expressed as

R(−ω) =

2 π

̄h

|F(ω)|^2 e−β ̄hω

∑

i,f

wi

∣

∣

∣〈Ei|Vˆ|Ef〉

∣

∣

∣

2

δ(Ef−Ei+ ̄hω). (14.3.13)

Comparing eqn. (14.3.13) with eqn. (14.3.3) reveals that

R(−ω) = e−β ̄hωR(ω), (14.3.14)

which is known as the condition ofdetailed balance. According to this condition, the
probability per unit time of an emission event is smaller than that of an absorption