548 Quantum time-dependent statistical mechanics
Finally, substituting eqn. (14.3.21) into eqn. (14.3.18) and the resultinto eqn. (14.3.16),
the energy spectrum becomes
Q(ω) =
2 ω
̄h|F(ω)|^2 tanh(β ̄hω/2)∫∞
−∞dte−iωt〈
1
2
[
Vˆ(0),Vˆ(t)]
+〉
. (14.3.22)
Eqn. (14.3.22) demonstrates that the energy spectrumQ(ω) can be expressed in terms
of the ensemble average of a Hermitian operator combination [Vˆ(0),Vˆ(t)]+. In partic-
ular,Q(ω) is directly related to the Fourier transform of a symmetric quantum time
correlation function〈[Vˆ(0),Vˆ(t)]+〉.
It is instructive to examine the classical limit of the quantum spectrum in eqn.
(14.3.22). In this limit, the operatorsVˆ(0) andVˆ(t) revert to classical phase space
functions so thatVˆ(0)Vˆ(t) =ˆV(t)ˆV(0) and [Vˆ(0),Vˆ(t)]+−→ 2 V(0)V(t). Also, as ̄h−→
0, tanh(β ̄hω/2)−→β ̄hω/2. Combining these results, we find that the classical limit
of the quantum spectrum is just
Qcl(ω) =ω^2
kT|F(ω)|^2∫∞
−∞dte−iωt〈V(0)V(t)〉cl, (14.3.23)where the notation〈V(0)V(t)〉clserves to remind us that the time correlation function
is aclassicalone.
14.4 Examples of frequency spectra
From eqn. (14.3.22), it is clear that in order to calculate a spectrum,we must be able
to calculate a quantum time correlation function. Unfortunately, numerical evaluation
of these correlation functions is an extremely difficult computational problem, an issue
we will explore in more detail in Section 14.6, where we will also describe approaches
for approximating quantum time correlation functions from path-integral molecular
dynamics. In this section, we will use a simple, analytically solvable example, the har-
monic oscillator, to illustrate the general idea of a quantum time correlation function.
As discussed in Section 10.4.1, expressions for equilibrium averages and thermody-
namic quantities for a harmonic oscillator form the basis of simple approximations for
general anharmonic systems. We will use the result we derive here for the position
autocorrelation function of a harmonic oscillator to devise a straightforward approach
to approximate absorption spectra fromclassicalmolecular dynamics trajectories.
14.4.1 Position autocorrelation function of a harmonic oscillator
We begin by considering the position autocorrelation function〈xˆ(t)ˆx(0)〉and sym-
metrized autocorrelation function〈[ˆx(t),xˆ(0)]+〉of a simple harmonic oscillator of
frequencyω 0. In order to calculate the time evolution of the position operator, we use
the fact that the Schr ̈odinger operator ˆxcan be expressed in terms of the creation and
annihilation operators (or raising and lowering operators) ˆa†and ˆa, respectively, as
xˆ=(
̄h
2 mω 0) 1 / 2
(
aˆ+ ˆa†