[MUSIC PLAYING]
Having set up the basics
of the time-dependent
perturbation theory for simple
oscillating, or what we
sometimes call harmonic
perturbations, we're now going
to derive an extremely useful
rule in quantum mechanics.
This is generally known as
Fermi's golden rule.
This is a simple formula that
works for a very broad range
of problems and gives us a
simple answer for transition
rates, such as optical
absorption and other kinds of
interactions involving
oscillating fields, such as
interactions with vibrations
or with acoustic fields and
solid materials, as we said,
often described as phonon
scattering.
Fermi's golden rule applies
whenever we have got a dense
set of possible final states
in a system which is a very
common situation in
many materials.
So now then, we're going to
consider the case only
associated with absorption
presuming we're starting in a
lower energy state and
transitioning to a higher
energy one.
The treatment of the stimulated
emission case would
be essentially identical
with the energies
of the states reversed.
However, here we're going to
look at the absorption case.
Then we have this equation from
our previous derivation.
Leaving only the absorption term
in here, which resonates
when omega is approximately
equal to omega jm--
the separation between the j
state and the m state in
energy divided by H-bar.
So analyzing the case of a
transition between one state
and exactly one other state
using this approach turns out
to have some formal
difficulties.
The problem is as we let the
time t naught become
arbitrarily large, the sinc
squared term becomes
arbitrarily sharp in omega.
Unless the frequency is exactly
correct, then we will
get no absorption.
We can solve this problem with
a more sophisticated analysis
involving what's called density
matrices, but we're
not going to be able
to do that here.
That will take some
more setup.
But that allows "widths" to the
absorption lines, rather
than just the sinc function
becoming arbitrarily sharp.
And that gets us out of
this problem with
a singularity here.
If we were to go to the density
matrix form, we would
end up with more of a Lorentzian
function, rather
than a sinc squared function.
So the Lorentzian function
looks like this.
And this is a mathematical form
of a Lorentzian function.
And it has a width, a
half-width, for example, of
one over this parameter
T2 here.
So the total width at half
maximum would be 2 over this
T2 parameter.
There's a physical meaning to
the T2 parameter, once we do
the density matrix theory.
And that is that it's
essentially the time between
the scattering that the system
encounters, which could be,
for example, collisions
with other atoms.
That's something we've not put
into our theory so far, and so
we, therefore, don't get
rid of our singularity.
But with the density matrix
formalism, we can put in that
effect, and their singularity
disappears, and we get this
kind of Lorentzian
function instead.
However for the moment, we're
not going to make this
particular change, because there
is still something quite
useful we can do in
the current form.
We can incidentally rationalize
this Lorentzian
form here, even though
we're not going to
go through the physics.
We can rationalize why we don't
just get an arbitrarily
sharp function based on an
energy-time uncertainty
relation kind of argument.
And this is an approximate
argument, but it gives a sense
of why lines don't get
arbitrarily sharp.
If a system only exists in its
original form for some time
T2, with T2 be in the standard
way of notating this
particular phenomenon in the
density matrix theory, then we
should expect that the energy
of the transition is only
defined to some energy within
something like h-bar over T2,
or in frequency, angular
frequency here, to within
something like 1 over T2.
Incidentally, if you followed
through the precise
uncertainty principle arguments,
it would be another
factor of 2 in here, but this
is only a rough argument.
So the idea is that physically
in the way we are looking at
the problem at the moment,
we're being a little
unphysical, because we haven't
put in the fact that all sorts
of systems are going to suffer
collisions of one kind or
another, and from something like
an uncertainty principle
argument, no line is going to
get arbitrarily sharp, because
that would mean its energy was
arbitrarily well defined.
And if it has some lifetime kind
of effect associated with
it, that simply physically
won't happen.
Anyway, as I said, we're not
going to do this Lorentzian
analysis at the moment, but that
is still something useful
we can do even with the
analysis we have.
The analysis we have so far does
turn out to work quite
well, if we're thinking about
dense sets of possible
transitions.
So a major class of problems
then can be
analyzed using our approach.
Suppose, we have not one
possible transition with a
very specific energy difference
H-bar omega jm, but
a dense set of transitions
near this photon
energy h-bar omega.
So something like this here.
Here's our photon energy,
this magnitude here.
And we find that in our system
maybe we have many different
atoms, and they might have
slightly different energies
associated with their
transitions here.
And instead of having one
transition, we have a dense
set of possible transitions,
each with
slightly different energies.
But presumably, they all have
essentially identical matrix
elements, they have essentially
identical stands
for these transitions, if we
get the frequency right.
Well, this kind of situation
occurs quite routinely in
solid materials, for example.
So not just one transition
energy, but a dense set of
possible transitions, but
somewhat different energies.
We presume that this set of
possible transitions then is
very dense in frequency or
energy, and so we could talk
about a density per unit energy
near the photon energy
h-bar omega.
We could call that density g
sub g of h-bar bar mega.
That is that within some range
here of possible photon
energies, for example, so with
photon energies anywhere from
this amount all the way up to,
say, this amount here,
separated by some amount of
energy, delta E, then we would
find a total of gj of h-bar
omega times delta E. So gj of
h-bar omega is the density of
possible transitions in
energy, and then we multiply
that by a small amount of
energy to get the total number
of transitions within that
energy range.
gj of h-bar omega is sometimes
known as a joint density of
states since it refers to
transitions between states.
In distinction to just a density
of states, which would
just be density of particular
energy states, not of
transition energies that we're
talking about here.
So if we're talking about
dense sense of possible
transitions, we can add up the
probabilities for absorbing
transitions and obtain a total
probability of absorption by
this set of transitions
of this amount here.
So we're summing up over all the
possible transitions, or
what we are actually going to do
here is instead of doing a
sum, we're going to do an
integral over this density of
transitions and, of
course, a little
increment of energy here.
So it could have been a sum
with gj of h-bar omega jm
times delta E. But what we're
doing instead is an integral
with the infinitesimal
in here.
So this would be as adding up
the probabilities for all the
absorbing transitions within
some given range of energies
or photon energies, or energy
separations as a way of
looking at it.
So what we're doing is adding
up the probabilities for all
the possible transitions that
are near photon energy
frequency omega.
And we're using a density
of transitions in here.
Approximately speaking, we're
presuming that this density is
constant over small
energy ranges.
So again, this is a density of
possible transitions in the
system, a density
in energy range.
And presumably then this sinc
squared term that we have here
is not narrow in omega gm.
We're presuming that
t0 is quite large.
Hence, this density, joint
density of states here, is
approximately constant
everywhere in our integral, so
we can take it out the front.
And therefore, we're going to
obtain just a factor gj of
h-bar omega, because,
approximately speaking, omega
jm for everywhere within this
integral that we're doing
here, will be approximately
h-bar omega.
And so as long as this density
of state is approximately
constant over some range of
energies, then we can just
take it out as this constant.
So formally changing the
variable in this integral just
for mathematical purposes.
Then we can perform this
integral, and our integral
just turns into a purely
mathematical one.
And this particular integral
is one that we can perform.
The result of that particular
mathematical
integral is just pi.
And so as a result, we'll obtain
a total probability of
making some transition that
is given by the following
expression.
So 2pi times t naught-- that's
the length of time we have our
perturbation on for--
divided by h-bar, multiplied by
the modulus squared of this
matrix element between the
initial and final states.
And remember, in here and our
electromagnetic case, this was
just the electronic charge times
the magnitude of the
electric field that's
oscillating, E times position
z, which is an operator
in here.
So this is our formula for the
total probability of making
some transition as a result of
having this perturbation for
time t nought.
And it's a meaningful
kind of calculation.
We've managed to get rid of that
infinitely sharp spike
concept that was bothering
us before.
And that's because we've got a
dense set of transitions we
are working with.
Now, we can see that we are
able to calculate a total
probability of making
some transition.
It is proportional to the time
for which the perturbation has
been applied.
And so therefore, we can deduce
a transition rate.
That is a rate of absorption
here of photons.
That is basically that
probability that we just
calculated divided by the time
for which the perturbation had
been applied.
So this will give us the rate
of absorption of photons.
The probability that a photon
would be absorbed in a total
amount of time t nought, but
divided by that time t nought.
So that gives a rate of
absorption of photons, which
we call W. And here we have then
a nice simple formula for
that, whenever we've got
a dense set of possible
transitions.
So this result is called
Fermi's golden rule.
And that's because Fermi
actually popularized this.
This rule was also derived by
Dirac, when he was working on
the theory of optics and
quantum mechanics.
This turns out to be one of
the most useful results of
time-dependent perturbation
theory.
It's something that we use in
many different situations.
And for example, we can use
it to calculate optical
properties, like optical
absorption spectra of solids,
which typically have very
dense sets of possible
transitions.
And we can also apply it to many
other problems involving
simple harmonic perturbations.
For example, problems involving
photons, which are
acoustic vibrations inside
solids and which often give
scattering processes that
we see when we're
working with electrons.
We can state Fermi's golden rule
in alternative way, which
is using Dirac's
delta function.
This is me merely a formal
change compared to what we
were looking at.
And in this case, we are
using a small w here.
And this is the transition
rate between
state m and state g.
And of course, it does have this
formal difficulty that it
has a sharply-spiked function
here, which is what the delta
function is.
But this would give us formally
in a mathematical
sense the transition rates
between states m
and g of the system.
And as I said, this is the Dirac
delta function, which is
an infinitely high and
sharp spike, that
nonetheless has unit area.
So that's a mathematical concept
that helps us write
something down in a compact form
here, although you have
to be careful in understanding
what you can and cannot do
with Dirac's delta functions.
Then, if we use this formalism,
the total
transition rate involving
all the possible similar
transitions in the neighborhood
of this
transition energy we're
interested in, the photon
energy here, is then formally
the integral of this wgm,
small w, with the density
of states.
And dh bar omega jm.
This is actually the same as the
formula we were looking at
a minute ago, it's just a
different mathematical way of
writing it.
And you'll often see Fermi's
golden rule stated this way.
[MUSIC PLAYING]
