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PROFESSOR: Today we're going to
finish our discussion of atoms

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and light in interaction
and finally get

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some of those rates.

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Last time, we reviewed
the Einstein's argument.

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Einstein used his argument
to discover spontaneous--

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stimulated emission
of radiation and how

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it accompanies the process
of absorption all the time.

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There was also, of
course, this process

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of spontaneous radiation
that we found the formula

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for that transition rate
in terms of the transition

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rate for stimulated radiation.

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So let me review a couple
of things that we had.

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So we have an electric field
that interacts with the atom.

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There's a position vector
that points to the charge, q.

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And we have two levels,
level b with energy E b

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and level a with energy E a.

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We introduce a dipole
moment operator,

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which was defined as q times r.

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And it's an operator
because r is an operator.

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The position is an operator
in quantum mechanics.

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So we observe, and we've
observed several times,

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that the process is b going
to a of stimulated emission

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and the process of a
going to b of absorption

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are the same, not
equivalent, the same.

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The first order in
perturbation theory, and we

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wrote, last time,
the formula for it

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using our general formula
for harmonic transitions.

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Not firm is golden
rule, because this

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was a discrete
transition, apparently,

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between these two levels.

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So here is the formula.

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Let me remind you what it was.

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There was an electric field.

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It was of the form 2E 0 cosine
omega t times the vector n.

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OK.

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All our results from the
discussion of light and atoms

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from the previous
lecture, so here we go.

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The transition probability
as a function of time

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is given by this quantity, n
0 is half the peak amplitude

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of the electric field.

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We always have
this extra 2 here.

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d ab are the matrix
elements of the operator d

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between the states
here, a and b.

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So that's an inner product.

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This is dotted with
the vector n that

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refers to the polarization
of the electric field.

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That half number is squared.

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And then we have the sine
squared over x squared function

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that we've usually had.

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So this is our situation.

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But what's going to happen,
what we're interested

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and the situation that
Einstein was considering,

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was when you had a box,
you had thermal radiation.

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And those atoms were there.

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Here, we have a discrete
process of transition

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from one state to another state.

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Neither one is a continuum.

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When we had ionization,
we were going

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from one state bound to the
continuum of flying electrons.

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And therefore, a Fermi golden
rule arose by integration.

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Here, what are we to integrate?

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Well, there is
something to integrate.

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The fact of the
matter is that when

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you have this
black-body radiation,

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you will have a spectrum,
omega, of electric fields.

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You don't just have
one electric field

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at one specific frequency.

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You will have many components
of the electric field

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at different frequencies
at different directions.

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So in particular, when you
have a black-body over this,

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you will have, in this
spectrum, here is omega ba.

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That is the frequency
for which the transitions

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tend to be more important.

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Remember, when you're
omega, your frequency omega

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is the frequency of
the external light.

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When your frequency
omega coincides

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with the omega of
this transition,

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this stimulated emission or
the absorption get enhanced.

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So here is omega ba.

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That's one of the frequencies
in the black-body of radiation.

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But there are many,
many more nearby.

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And they are incoherent.

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That is, in the
black-body of radiation,

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each amplitude for a
different frequency

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is independent of
the other ones.

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They're, like, random.

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So here is our sum, therefore.

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We must, we know, that when we
have several inputs, not just

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one electric field but
many electric fields,

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we will have to add the
transition probability caused

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by the different
electric fields.

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And in this case,
we will have to sum

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the effects of
the electric field

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squares of the many components
of the black-body of radiation

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that lie near this omega ba.

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That is the one that would
trigger a transition.

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So this is how we're
going to obtain

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something that will look like a
Fermi golden rule as well here.

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There is a transition.

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The transition is
between discrete states.

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But the trigger
for the transition

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is a superposition of
electromagnetic waves

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that we have to integrate over.

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So one can say that the
electric field, electric field

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is a superposition of incoherent
waves with different omega i's,

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different frequencies,
different E 0 of omega i,

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different amplitudes, and
different vectors, unit

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vectors.

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So each mode of this
black-body radiation

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has a different amplitude
and a different omega

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and a different polarization.

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And we have to sum over
those that are near omega ba.

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And we know only over near
ones, because this function

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doesn't allow you to--

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or it becomes too small
when you get far away

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from that central frequency.

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So as a first step, I can
rewrite this transition

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probability to make
it a little clearer.

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I would say, here it is.

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The transition probability
for spontaneous emission

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of radiation associated
to a particular mode

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of the radiation
field would be given

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by 4E 0 of omega i
squared, h squared, d ab.

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That is some matrix element
of the dipole operator

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in the atom.

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That doesn't depend
on anything having

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to do with the radiation.

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n is possibly dependent
on the radiation.

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Sine squared of 1/2
omega ba minus omega i.

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The omega i is the omega
of the electric field.

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So I've put all these things
to represent the contribution

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from the ith mode of
the radiation field

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to the transition amplitude.

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That's what it does.

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So you could say,
oh, the ith mode

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is one in which I
would write 2E 0 of i.

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So here would be 2E 0 of
omega i cosine of omega i t

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times the vector n omega i.

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That's the ith mode
of the electric field.

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And that ith mode of
the electric field,

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that's this transition.

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And we have to sum
over all these modes

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that are nearby here.

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OK.

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This is one of the
little non-trivial things

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that we have to do now.

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But the way to think
about it is that we've

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seen, from the
Einstein arguments

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and in general black-body, that
there is a function u of omega,

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which is the energy
density per unit frequency

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so that when you
multiply by the omega,

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this is the energy
density contained

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in the part of the spectrum
with a range d omega.

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That's your energy density.

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So that's an important
quantity because it

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represents a number of photons.

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And that's how Einstein
argued that that defines or is

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a part of the ingredient
of constructing

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the rate at which transitions
would be happening.

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So we have to relate
this to energy.

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So let's try to
relate it to energy.

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So we use things from
electromagnetism,

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some basic facts from
electromagnetism.

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If you have-- so I'll go here.

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If you have a wave,
in general for a wave,

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you have an E field
and a B field.

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And you have energies u
electric and u magnetic.

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These are energy densities.

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And there's a simple
formula for them.

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The electric energy is
equal to the magnitude

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of the electric field
squared over 8 pi.

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And in our case, the magnitude
of the electric field

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is, the electric field
in our conventions,

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is 2E 0 cosine omega t.

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So what do we get here?

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4E 0 squared cosine
squared omega t over 8 pi.

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So this is 1 over 2
pi E 0 squared cosine

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squared of omega t.

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But nobody really cares about
the fluctuation of the energy.

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You need the average
energy at the volume.

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So the time dependence
must be averaged.

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So from here, the average over
time of the electric field

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energy density is the average
over time of cosine squared,

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which is 1/2.

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So we have 1 over
4 pi E 0 squared.

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So that's the average electric
field energy for a wave.

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And that's the
average over time.

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But in a electromagnetic wave,
the average electric energy

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and the average magnetic
energy are the same.

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So u M magnetic is equal to u
E. The averages are identical.

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And then the total energy
density, therefore, is--

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therefore, total u,
it's already average.

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Energy density would be u
E plus u M, which is 2u E.

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And it's, therefore, 1
over 2 pi E 0 squared.

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So our end result is
that the energy density

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u is 1 over 2 pi E 0 squared.

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OK.

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Usually the thing
here that is important

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is that there's lots
of factors of 2.

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There's factors of 2 in
defining the energy density,

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in averaging over
time, in adding

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electric and magnetic
contributions up.

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This is correct.

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It's important to get
those factors of 2 right,

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otherwise your final
formulas will be off.

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So here is-- the end result
is a nice formula that

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allows you to rewrite the
amplitude of your oscillation

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in terms of its contribution
to the energy density.

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So here it goes.

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We can now think
of this by saying,

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if I have an electric field
squared at some frequency,

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well, that's 2 pi times the
energy density contributing

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00:16:23,570 --> 00:16:30,810
at that frequency, at just
the specialization of this

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00:16:30,810 --> 00:16:36,150
to an arbitrary frequency, an
omega i and an omega i here.

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00:16:38,710 --> 00:16:51,330
So with this, we can now rewrite
this transition amplitude.

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The transition amplitude
can be now expressed

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in terms of energy densities.

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So what do we have?

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We have p b to a of t
i would be equal to 4.

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That 4 remains there.

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And then instead of E 0 squared,
we put 2 pi u of omega i

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over h squared.

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And not much changes anymore.

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The ab n omega i
squared sine squared

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of 1/2 omega ba minus
omega i t over omega ba

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minus omega i squared.