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PROFESSOR: We have now expressed
in terms of the energy density

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on this mode omega
I the contribution

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to the transition amplitude.

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00:00:12,310 --> 00:00:18,120
The fact that all these
modes act incoherently

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means that each one has a shot
in producing the transition.

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And the probabilities
now must be added.

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So the way to sum probabilities
now is the following.

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If you have a sum over I of this
frequency sum I of some energy

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00:00:39,240 --> 00:00:47,370
density u of I times
any function of omega I,

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you can replace it by
an integral d omega.

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Instead of little omega
I's that you're summing,

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you now integrate
over omega the energy

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density of your radiation field
times the function of omega.

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So instead of having
a sum of these things,

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you now integrate over
a continuous variable.

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And this represents the
energy density in the range.

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This whole thing is
the energy density

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in the range d omega,
which is in that range d

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omega, the energy range.

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The energy density is the
sum of the energy densities

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of each of the contributions.

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So that's what we're going to
do here to express and to get

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our transition amplitude.

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So we'll say that we have
the sum over I of Pab I of t

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equals 8 pi over h squared
integral u of omega d omega dab

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and omega sine squared of
1/2 omega ba minus omega t.

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So all the omega I's
have become omegas.

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That's the variable
of integration.

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And this is information
about your radiation field.

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If it's a black body radiation,
we gave the formula last time.

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So this is your
transition amplitude.

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And we have to try
to do the integral.

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This one I that remained here.

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So what do we have to do now?

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Well, is it, again, the
kind of useful story?

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We know that this factor
from Fermi's golden rule

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tends to say, basically, you
get only the contribution

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from omegas equal to omega ba.

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So whatever else is
being integrated,

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nothing varies as fast as this.

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And you can take it out of
the integral approximating

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omega for omega ba.

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So that is a little
more delicate here.

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So I'll just write
an equality here.

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So it's 8 pi over h squared.

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I'll take the u at omega ba out
of the integral, this factor.

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And I want to take
this one out as well.

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But this one is a little funny.

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As I change the omega,
the polarization

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of this electric field is going
to come in all directions.

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It's going to come at random.

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So even for a given
frequency omega,

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there might be many
modes of frequency omega

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that are coming at the particle
at different directions.

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And that's exactly
what I would expect

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for thermal radiation,
which is truly

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the case we're doing here,
incoherence superposition

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of light.

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So when I have this, and
I integrate over omegas,

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the various omegas,
even for a given omega,

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there might be many
lines that correspond

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to just a different direction.

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Because the field comes
in all directions.

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So the interpretation is
that we can take this out,

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but we must do an average
over all directions of omega.

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So this factor will
go out as dab dot n.

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But here, we will average
over all directions of n.

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And it's a square here.

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So this is important because the
field comes in all directions.

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We've taken care of this.

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Then, finally, we
have the integral,

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the omega of the sine squared
function a minus omega.

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All right.

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So a couple of things
still remain to be done.

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One thing that I don't think
we need to do explicitly

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again is this integral.

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We've done it a couple of times.

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You do a change of variables.

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The t will go out
linearly, and this

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becomes an integral of sine
squared over x squared,

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which is equal to pi.

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And so this integral has
been done a couple of times.

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Let me just write the answer to
this thing is 1/2 t times pi.

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It's linear in t.

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That we've observed.

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And it's pretty important
that it's linear in t

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because that means that
the probability-- you know,

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we've been writing this thing--
the probability of transition

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is linear in time.

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Therefore, you
can divide by time

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to get that rate of transition.

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So we're almost there.

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Let's put this together.

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We have this is the sum of P's.

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And there's this t there.

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So the transition
rate, W from b to a

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is this probability
divided by time.

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So it's going to
cancel this time.

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Then the 2 is going to give a 4.

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The pi is going to
give a pi squared.

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There's going to
be an h squared.

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There's going to be this
factor dab dot n squared.

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And there's going to
be the u of omega ba.

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So this is going to be the rate.

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We've divided by t, the previous
result. And there we have it.

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We're almost finished
with the calculation.

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The rate is here.

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The only difficulty
here is this average,

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but it is not complicated,
in fact, doing this average.

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It's actually kind of simple.

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So let's do it.

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So what is it?

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It's all the matter of
writing things properly here.

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There's the average
of dab dot n squared.

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So this is the average of dab
dot n complex conjugate times

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dab dot n.

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So it's good to do this
thing because, actually, you

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have to face as to what
these symbols really mean.

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dab is a vector with
complex components.

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Electric field is a vector,
but it has real components.

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dab is a vector.

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Why is it a vector?

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Because it has a
vector index here.

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It really came from this
operator over there d is qr.

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So it's x, y, and z.

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And each one has matrix
elements between states a and b.

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And with matrix elements
between states a

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and b that are complex
wave functions,

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this can be complex numbers.

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So in general, this dab
is a complex vector.

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And the star is necessary here.

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So, this is a number.

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This, however, is a number.

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And I took the number star
times the number and average.

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So the dot products
can be written as sums.

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So I'll put here a sum
over I and a sum over j.

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dab, the ith component
times ni, the ith component.

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Here, this should be star.

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And that's the sum over i
is the first dot product.

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dabj and j, the sum over j
is the second dot product.

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These d's are numbers, so
they don't have anything

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to do with the average
that we're doing

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over different directions.

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So we have i and j of
di ab star dj ab times

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the average of ni nj.

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This average, if you wish,
you could simply do it.

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If you don't want to
use symmetry arguments

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00:11:53,690 --> 00:11:57,560
to do an average like
that, you take a vector,

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parameterize it
with theta and phi

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and just do the integral
over solid angle.

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And divide by 4 pi.

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This should give
you the same answer.

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If you're uncomfortable with
what we're going to say now,

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00:12:12,470 --> 00:12:14,870
you should do that
because that average just

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means take the vector
n, integrate it

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00:12:18,860 --> 00:12:26,150
over all directions, all
solid angles, and average.

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So what is this ni nj?

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I claim this thing
is 1/3 delta ij.

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And it's based on
the following idea,

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that the average
between nx and ny,

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by the time you integrate
over the sphere, is 0.

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The average of off diagonal
things don't have averages.

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nx with nx, however,
would have an average

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because it's always positive.

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00:13:03,350 --> 00:13:07,010
ny with ny would have an average
because it's always positive.

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And nz with nz would
have an average.

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And each one of these
three would be the same

168
00:13:13,880 --> 00:13:16,550
because there's no real
difference between the x, y,

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00:13:16,550 --> 00:13:18,440
and z directions.

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And therefore, you
have three things.

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And at the end of the day,
this average of nx and x

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plus ny and y plus nz
and z is this average

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00:13:31,190 --> 00:13:34,170
of n squared, which
is equal to 1.

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00:13:34,170 --> 00:13:36,520
So it should be equal to 1.

175
00:13:36,520 --> 00:13:43,520
So that each one of these
nx, nx, or ny, ny, or nz, nz

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00:13:43,520 --> 00:13:45,750
must add up to a total of 1.

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00:13:45,750 --> 00:13:48,740
Therefore, this is the 1/3.

178
00:13:48,740 --> 00:13:50,900
So that's what this is.

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00:13:50,900 --> 00:13:56,350
And if you find
this a little funny,

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you should do the integral and
then think about this argument

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again.

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So anyway, that's the answer.

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And therefore, we get 1/3.

184
00:14:06,440 --> 00:14:09,830
And the delta ij
establishes a dot product

185
00:14:09,830 --> 00:14:11,150
between these two things.

186
00:14:11,150 --> 00:14:19,470
So this is dab vector star
dotted with the dab vector.

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00:14:19,470 --> 00:14:22,340
This is a vector complex
conjugated because we said

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00:14:22,340 --> 00:14:25,070
the components can be complex.

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00:14:25,070 --> 00:14:26,380
So that's the answer.

190
00:14:26,380 --> 00:14:29,860
And most people like
to just simply write it

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00:14:29,860 --> 00:14:36,620
as dab vector squared like that.

192
00:14:36,620 --> 00:14:39,410
But this is not just the
vector dotted with itself.

193
00:14:39,410 --> 00:14:44,590
It's a vector dotted with
its complex conjugate.

194
00:14:44,590 --> 00:14:48,810
So with this number, the
whole formula is now finished.

195
00:14:48,810 --> 00:14:50,010
This is the final form.

196
00:14:52,890 --> 00:14:55,680
The rate for
spontaneous transition

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00:14:55,680 --> 00:15:00,690
triggered by
electromagnetic fields,

198
00:15:00,690 --> 00:15:04,060
an incoherent superposition
of electromagnetic fields

199
00:15:04,060 --> 00:15:06,450
is 4 pi squared.

200
00:15:06,450 --> 00:15:10,570
We get the 1/3 here
over 3h squared.

201
00:15:13,700 --> 00:15:20,810
dab vector squared u omega ba.

202
00:15:30,230 --> 00:15:35,570
And it's a transition
rate per atom.

203
00:15:35,570 --> 00:15:37,800
We've considered a single atom.

204
00:15:37,800 --> 00:15:44,880
So it's transition
rate per atom.

205
00:15:48,720 --> 00:15:49,930
All right.

206
00:15:49,930 --> 00:15:56,350
Long derivation, but a
lot of physics in it.

207
00:15:56,350 --> 00:15:59,050
A transition between
discrete state

208
00:15:59,050 --> 00:16:03,490
got convoluted with an
integral of the stimuli

209
00:16:03,490 --> 00:16:08,760
as it comes in and produce
a nice result in the style

210
00:16:08,760 --> 00:16:10,360
of Fermi's golden rule.

211
00:16:12,890 --> 00:16:17,870
Griffiths called that
Fermi golden rule.

212
00:16:17,870 --> 00:16:20,420
He works with Si units.

213
00:16:20,420 --> 00:16:24,860
And there's epsilon zeros
and different numbers of pi's

214
00:16:24,860 --> 00:16:27,370
there.

215
00:16:27,370 --> 00:16:28,920
I'm sorry.

216
00:16:28,920 --> 00:16:29,770
With checked.

217
00:16:29,770 --> 00:16:35,920
This is consistent, and we
can't do much about that.