1549380323-Statistical Mechanics Theory and Molecular Simulation

544 Quantum time-dependent statistical mechanics

ofRfion the frequencyω. When the integral in the third line of eqn. (14.2.42) is
replaced by theδ-function, the remaining integral becomes simplyTsinceωfi=ω,
and thisTcancels theTin the denominator. For this reason, the division byTin eqn.
(14.2.38) is equivalent to expressing the rate as a proper derivativelimT→∞dPfi/dt.
The expression for the rate now becomes

Rfi(ω) =

2 π

̄h

|F(ω)|^2

∣

∣

∣〈Ef|Vˆ|Ei〉

∣

∣

∣

2

δ(Ef−Ei− ̄hω), (14.2.43)

which is known asFermi’s Golden Rule. The rule states that, in first-order perturbation
theory, the transition rate depends only on the square of the matrix element of the
operatorVˆbetween initial and final states and explicitly requires energy conservation
via theδ-function. Fermi’s Golden Rule predicts the rate of transitions fromaspecific
initial state|Ei〉to a final state|Ef〉, both of which are eigenstates ofHˆ 0 and which
are connected via the energy conservation conditionEf=Ei+ ̄hω.

14.3 Time correlation functions and frequency spectra

In this section, the Fermi Golden Rule expression will be used to analyze the output
of an experiment in which a monochromatic field is applied to an ensembleof systems.
If we wish to calculate the transition rate for the ensemble, we mustremember that
the systems in the ensemble arenotin a single initial state|Ei〉. Rather, there is a
distribution of initial states prescribed by the equilibrium density matrixρ(Hˆ 0 ), which
satisfies the equilibrium Liouville equation [Hˆ 0 ,ρ(Hˆ 0 )] = 0. Thus, in the canonical
ensemble, the probability that a given ensemble member is in an eigenstate ofHˆ 0
with energyEiis the density matrix eigenvalue

wi=

e−βEi

Q(N,V,T)

=

e−βEi

Tr

(

e−βHˆ^0

). (14.3.1)

The rate we seek is the ensemble average ofRfi(ω) over initial states, denotedR(ω),
which is given by
R(ω) =〈Rfi(ω)〉=

∑

i,f

Rfi(ω)wi. (14.3.2)

Although both initial and final states are summed in eqn. (14.3.2), weknow that the
sum over final states is not independent, since the only permissible final states are those
connected to initial states by energy conservation. Eqn. (14.3.2)indicates that the
contribution from each possible initial state|Ei〉to the average rate is the probability
withat a given member of the ensemble is initially in that state. Ultimately,we sum
over those final states that can be reached from the initial statewithout violating
energy conservation to obtain the average transition rate.
When we substitute eqn. (14.2.43) forRfi(ω) into eqn. (14.3.2), the average rate
becomes

R(ω) =

2 π

̄h

|F(ω)|^2

∑

i,f

wi

∣

∣

∣〈Ef|Vˆ|Ei〉

∣

∣

∣

2

δ(Ef−Ei− ̄hω). (14.3.3)