1549380323-Statistical Mechanics Theory and Molecular Simulation

542 Quantum time-dependent statistical mechanics

=−

i

̄h

e−iEft/ ̄heiEit^0 / ̄h

∫t

t 0

dt′ei(Ef−Ei)t

′/ ̄h

〈Ef|Hˆ 1 (t′)|Ei〉. (14.2.35)

We now define atransition frequencyωfiasωfi= (Ef−Ei)/ ̄h. Taking the absolute
square of the last line of eqn. (14.2.35), we obtain the probability at first order as

Pfi(1)(t) =

1

̄h^2

∣

∣

∣

∣

∫t

t 0

dt′eiωfit

′

〈Ef|Hˆ 1 (t′)|Ei〉

∣

∣

∣

∣

2

. (14.2.36)

At first order, the probability depends on the matrix element of theperturbation
between the initial and final eigenstates.
Thus far, the formalism we have derived is valid for any perturbationHˆ 1 (t). The
specific choice of this perturbation determines the manifold of eigenstates ofHˆ 0 it
probes, as we will demonstrate in the next subsection.

14.2.3 Fermi’s Golden Rule

In Section 14.1, we formulated the Hamiltonian of a material system coupled to an
external electromagnetic field, and we derived expressions for the electromagnetic field
in the absence of sources or physical boundaries as solutions of the free-field wave
equation. For the remainder of this chapter, we will focus on weak fields, so that the
term in eqn. (14.1.11) proportional toA^2 can be neglected. We will also focus on
a class of experiments for which the wavelength of the electromagnetic radiation is
long compared to the size of the sample under investigation. Under this condition, the
spatial dependence of the electromagnetic field can also be neglected, since cos(k·r−
ωt+φ 0 ) = Re[exp(ik·r−iωt+φ 0 )], and exp(ik·r)≈1 in the long-wavelength limit
where|k|= 2π/λ≈0. Thus,Hˆ 1 (t) reduces to the specific form

Hˆ 1 (t) =− 2 ˆVF(ω) cos(ωt) =−VˆF(ω)

[

eiωt+e−iωt

]

, (14.2.37)

whereVˆis a Hermitian operator.
As noted above, we are interested in the probability that the system will undergo
a transition from an eigenstate|Ei〉to|Ef〉ofHˆ 0 under the perturbation of eqn.
(14.2.37). However, since the perturbation is periodic in time, the problem must be
stated a little differently. If we apply the perturbation over a long time, what is the
mean probability per unit time of a transition ormean transition rate? In order to
simplify the calculation of this rate, let us consider a time intervalT and choose
t 0 =−T/2 andt=T/2. The mean transition rate can be expressed as

R(1)fi(T) =

Pfi(1)(T)

T

=

|F(ω)|^2

T ̄h^2

∣

∣

∣

∣

∣

∫T/ 2

−T/ 2

dt

[

ei(ωfi+ω)t+ ei(ωfi−ω)t

]

∣

∣

∣

∣

∣

(^2) ∣
∣
∣〈Ef|Vˆ|Ei〉

∣

∣

∣

2

. (14.2.38)

For finiteT, the time integral can be carried out explicitly yielding