In the next lecture, we will solve the quantum Liouville equation
perturbatively and derive quantum linear response theory. However, the
transition rate can actually be determined directly within perturbation theory
using the Fermi Golden Rule approximation, which states that the
probability of a transition's occuring per unit time, 
, is given by
The 
-function expresses the fact that energy is conserved.
This describes the rate of transitions between specific states 
and
. The transition rate between any initial and final states can be
obtained by summing over both i and f and weighting the sum by the
probability that the system is found in the initial state :
where 
is an eigenvalue of the density matrix, which we will take to be
the canonical density matrix:
Using the expression for 
, we find
Note that
This quantity corresponds to a time-reversed analog of the
absorption process. Thus, it describes an emission event 
with 
, i.e., emission of a photon with
energy 
. If can also be expressed as a process
by recognizing that
or
Therefore
If we now interchange the summation indices, we find
where the fact that 
has been used. Comparing this expression
for 
to that for 
, we find
which is the equation of detailed balance. We see from it that
the probability of emission is less than that for absorption. The reason
for this is that it is less likely to find the system in an excited
state initially, when it is in contact with a heat bath and hence
thermally equilibrated.
However, we must remember that the microscopic laws of motion (Newton's equations
for classical systems and the Schrödinger equation for quantum systems) are
reversible. This means that
The conclusion is that, since 
, reversibility is lost when the
system is placed in contact with a heat bath, i.e., the system is being driven
irreversibly in time.
Define
then
Now using the fact that the -function can be written as
becomes
Recall that the evolution of an operator in the Heisenberg picture is given by
if the evolution is determined solely by 
. Thus, the expression for
becomes
which involves the quantum autocorrelation function 
.
In general, a quantum time correlation function in the canonical ensemble is defined by
In a similar manner, we can show that
since
in general. Also, the product B(0)B(t) is not Hermitian. However, a hermitian
combination occurs if we consider the energy difference between absorption and emission.
The energy absorbed per unit of time by the system is 
, while
the emitted into the bath by the system per unit of time is 
.
The energy difference 
is just
But since
it follows that
or
Note, however, that
where 
is known as the anticommutator: 
. The
anticommutator between two operators is, itself, hermitian. Therefore, the energy
difference is
The quantity 
is the symmetrized quantum autocorrelation function.
The classical limit is now manifest ( 
):
The classically, the energy spectrum is directly related to the
Fourier transform of a time correlation function.