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We're going to do
a couple of results

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with this formula that connect
even a little more with what

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we're doing.

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Maybe I'll leave
that blackboard here,

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because [INAUDIBLE] simple
formulas come out of this,

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which is very nice, actually.

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So one simple formula was
that the total cross section--

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we wrote the formula for the
differential cross-section.

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But the total cross-section is
the integral of fk of theta,

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in this case the omega.

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So this is the integral
of fk of theta star

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fk of theta with omega.

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But we have fk of theta here.

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That 0 looks like
theta, isn't it?

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That's not good.

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That's a 0.

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And we can plug all this in.

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So what do we have?

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We do this because the
answer is very simple.

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We get a 4 pi over k squared.

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We have a sum over l and l prime
because we have two factors--

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2l plus 1, 2l prime plus 1.

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The phase shift for l prime
and the phase shift for l--

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so e to the minus i
delta l sine of delta l.

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So I guess I'm using l for this
one and l prime for that one.

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e to the i delta l prime
sine of delta l prime.

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But happily, all that will
not matter, because then you

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have the integral over solid
angle of yl star 0 of omega

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yl 0 of omega prime here.

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Seems like a lot
of work, but we're

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doing nothing else than
integrating this f of theta.

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Let's look at it for a second.

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It's just that formula squared.

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So basically, you
star the one y.

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You don't start the other, but
you have to integrate them.

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And there orthonormality
of our spherical harmonics

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means that this
is delta ll prime.

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So you can set l
prime equal to l.

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These phases will cancel.

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This factor will be squared.

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This square root will disappear.

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And you have a simple formula
that the cross-section

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is 4 pi over k squared sum 2l
plus 1 sine squared delta l.

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Very nice formula for
a physical observable.

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What did you get?

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That the cross-section is
the sum of contributions.

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The total cross-section
is the sum

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of contributions from
partial cross-sections

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from each partial wave.

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That's not true for the
differential cross-section.

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Because in the
differential cross-section,

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you don't integrate.

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And as long as you
don't integrate,

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you have mixing
between different l's.

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The different partial waves
interfere in the differential

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cross-section, but they
will not interfere anymore

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in the total cross-section.

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Each partial wave contributes
to the total cross-section.

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That's why it's important to
calculate this phase shift.

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So you say, oh, just a phase.

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Phases don't matter.

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No.

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The relative phases
matter a lot.

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Those are the phases
that enter here,

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and the cross-section
is expressed

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in terms of those phases.

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There's one more result
that is famous in scattering

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and has to do with
the optical theorem.

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The optical theorem
is something you

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may have seen already
in electromagnetism

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or in other fields.

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It's a statement
about probability

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of conservation of flux.

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And it's fairly non-trivial,
and it's a constraint

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on the scattering amplitude.

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Basically-- and we may
discuss it in a problem.

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It's the statement that
when you have an object,

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the thing that you detect as
the scattering cross-section

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is all these particles that
were deflected from the object.

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And they were deflected
from the object

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because the object
creates a shadow.

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At least in the
electromagnetism,

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that intuition is very clear.

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You have a sphere here,
maybe a conducting sphere,

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an observancy.

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You shine light.

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You create the shadow.

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And that is the light that
if you didn't have here,

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it would have gone through.

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But if it's here, the
shadow is responsible.

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What you lost from the shadow
is what you got scattered.

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So whatever you get--

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in the forward direction
here, you get nothing--

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you get a shadow--

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carries the information about
the wave that scattered.

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It's a little more complicated
than that mathematically

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when you do it, because
the total wave function

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in the forward direction
is a combination

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of the incoming
wave function that

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has some forward direction and
the scattered wave function

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that has a forward direction.

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So the theorem is quite
interesting to prove,

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and we prove it with
flux conservation

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that gives you enormous
insight into the physics.

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But here is the power of algebra
and the power of phase shifts.

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You don't have to be brilliant
to discover the optical theorem

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in this setup.

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The physics has already done--

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the mathematics
of the phase shift

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has already done all
the work for you.

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Let's see it happen.

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Let's figure out how
does the forward--

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the scattering cross-section
look in the forward direction.

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So you have your object here.

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You send in your waves.

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They get scattered.

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But there is something
in the forward direction.

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That is fk at theta equals
0 is the forward scattering,

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the word "scattering."

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And now this forward
scattering is--

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we can calculate it.

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For that, we need to know
that yl 0 of theta equals 0--

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well, yl 0 of theta
is actually 2l

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plus 1 over 4 pi pl,
the Legendre polynomial

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of cosine theta.

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So yl 0 at theta equals 0
is just square root of 2l

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plus 1 over 4 pi.

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00:08:41,090 --> 00:08:47,660
And the Legendre polynomials are
defined always so that pl at 1

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is equal to 1.

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All pl's at x equals
1 are always 1.

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So that's the spherical harmonic
in the forward direction.

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00:09:01,140 --> 00:09:06,320
So you have here from that
formula on the right 4 pi

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00:09:06,320 --> 00:09:10,660
over k sum from l equals 0.

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00:09:10,660 --> 00:09:13,340
This is an investigation
of what happens

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00:09:13,340 --> 00:09:15,830
to the forward
scattering amplitude.

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The yl 0 gives you another
square root of 2l plus 1

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and a square root
of 4 pi as well.

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And then you get e to
the i delta l sine delta.

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So look here, here, and
these factors simplify.

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00:09:48,670 --> 00:10:01,870
So fk of theta equals 0 is 1
over k sum of l 2l plus 1 e

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00:10:01,870 --> 00:10:04,450
to the i delta sine delta.

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00:10:07,330 --> 00:10:09,500
That's a nice formula.

143
00:10:09,500 --> 00:10:13,330
Maybe something-- it has
to do with a cross-section.

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00:10:13,330 --> 00:10:18,400
f in this forward
direction has that formula,

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and the cross-section
has this formula.

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So how do I get to relate them?

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00:10:35,910 --> 00:10:41,490
I get to relate
them if I set now--

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00:10:41,490 --> 00:10:45,450
I ask for the imaginary
part of this fk.

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00:10:45,450 --> 00:10:52,460
Because the imaginary part
of this fk at theta equals 0

150
00:10:52,460 --> 00:11:01,410
is equal to 1 over k sum
over l, this 2l plus 1.

151
00:11:01,410 --> 00:11:05,540
And the imaginary part of
that turns it into sine

152
00:11:05,540 --> 00:11:09,510
squared delta, because the
imaginary part of the e to i

153
00:11:09,510 --> 00:11:11,760
delta is sine delta.

154
00:11:11,760 --> 00:11:16,010
So I get the sine
squared delta there.

155
00:11:16,010 --> 00:11:17,690
And you say, look at this sum.

156
00:11:17,690 --> 00:11:20,930
It's identical to the
sum of the cross-section.

157
00:11:20,930 --> 00:11:26,750
You have discovered a relation
between the total cross-section

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00:11:26,750 --> 00:11:33,390
and the imaginary part of the
forward scattering amplitude.

159
00:11:33,390 --> 00:11:36,860
So what is the final formula?

160
00:11:36,860 --> 00:11:41,540
The final formula
is that sigma--

161
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maybe I can do it here.

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Sigma is equal to--

163
00:12:02,600 --> 00:12:07,010
well, it differs at
least from a 4 pi

164
00:12:07,010 --> 00:12:15,860
over k imaginary part of f of
theta equals 0, fk of theta.

165
00:12:20,790 --> 00:12:27,150
This is the optical
theorem, which

166
00:12:27,150 --> 00:12:30,210
was discovered in
the context of optics

167
00:12:30,210 --> 00:12:32,350
by thinking of the
physics of shadows.

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00:12:36,340 --> 00:12:41,740
Whatever-- basically, the
shadow contains all that kind

169
00:12:41,740 --> 00:12:43,840
of information
about all that was

170
00:12:43,840 --> 00:12:46,690
lost from the incoming beam.

171
00:12:46,690 --> 00:12:48,580
So that's a nice--

172
00:12:48,580 --> 00:12:52,300
it's not absolutely
obvious how it

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00:12:52,300 --> 00:12:54,700
works, the process
of interference

174
00:12:54,700 --> 00:12:59,120
between the incoming wave
and the scattered wave

175
00:12:59,120 --> 00:13:00,370
and what they do.

176
00:13:00,370 --> 00:13:02,540
And you have to look
carefully at this thing.

177
00:13:02,540 --> 00:13:07,420
So that explains why you
need the imaginary part

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00:13:07,420 --> 00:13:09,340
and why these factors show up.

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00:13:09,340 --> 00:13:12,901
But the intuition is
nevertheless simple.