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PROFESSOR: We are going to
recap some of the last results

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we had on scattering and
then push them to the end

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and complete our discussion
of this integral equation

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and how we approximate
scattering with it.

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It's a pretty powerful method
of thinking about scattering

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and gives you a
direct solution, which

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is in a sense more intuitive
perhaps than when you're

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dealing with partial waves.

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So let's see where
we were last time.

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So we set up our solution
of the Schrodinger equation

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in a funny way.

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We had an incoming
wave plus a term that

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still had an integral over
all of space of our Green's

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function integrated
against a scaled version

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of the potential.

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Remember that v
was defined to be

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equal to the potential up
to some constants, h bar

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squared over m.

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And we had this
integral equation

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where one could show
that phi satisfies

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the correct
Schrodinger equation.

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Now this function G plus
was our Green function

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that we also discussed.

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And it was given by
the following formula.

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It just depends on the magnitude
of this vector argument

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of the Green's
function, so 1 over 4 pi

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e to the ik, magnitude
of that divided

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by magnitude of that vector.

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So in most applications, we
look at the wave function.

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This should be phi of r.

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We look at the wave
function far away.

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And if the potential
is localized,

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this integral, since it
involves the potential,

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can only be non-zero, have
non-zero contributions

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to the integral on the region
where the potential exists.

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So r prime, whenever
you're doing this integral,

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must be small compared to r.

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This we can draw here.

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So assume we have a potential.

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And you have an
observation point, r.

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We'll call the unit vector
in this direction n.

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This is the vector r.

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And an integration point
is a point, r prime here.

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And there is this
distance, this vector,

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in between r minus r prime.

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And finally, on this graph,
we could also have a direction

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that we could call the
incident direction, perhaps

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the z-direction.

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And that's the direction
which your wave is incident.

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This is the wave that
were represented here.

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So we did a little approximation
of this Green's function

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for the case where r
prime where you're looking

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is much smaller than r.

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In that case, the
distance, r minus r prime,

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is approximately
equal to r minus--

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so that's the
distance, the vectors.

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When you don't put an
arrow, we mean the distance.

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So this is r minus
n dot r prime.

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To 0-th approximation, this
is the most important term.

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To next approximation
comes this term

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because that term is
suppressed with respect

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to the original term by a
factor of r prime over r.

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So if you factored, for
example, the r here,

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you would say 1 minus
n r prime over r.

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So you see that
this ratio enters

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here, which says that the second
term is smaller by this factor.

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We could even include
more terms here.

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But those would not
be relevant unless k

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is extremely high energy.

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So when k is very large,
it would make a difference.

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But it has to be
extremely large.

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And in general,
this approximation

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isn't enough for any k
for sufficiently far away.

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So this is really good enough.

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And the approximation is
done on the Green's function

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at two levels.

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You need to be mildly accurate
with respect to this quantity.

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But you need to be more accurate
with respect to this phase

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because the phase can
change rather fast depending

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on the wavelength of the object.

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So if you make an error in
the total distance of 1%,

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that's not big deal.

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But if you make an
error here, such that k

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dot times this distance
is comparable to 1,

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the phase could
be totally wrong.

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So the phase is always
a more delicate thing.

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And therefore, with
this approximation,

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your G plus of r minus r prime
becomes minus 1 over 4 pi e

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to the ikr over r.

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So we replace this
term by just r.

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And in this exponent,
the absolute value

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is replaced by two terms.

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The first one is e
to the ikr, is here.

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The second one is e to
the minus ikn dot r prime.

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We also have here
the incident vector.

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You can write e to the ikz
as e to the ik incident

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dotted with r if k incident
is a vector, k, times the unit

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vector in the z-direction.

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So that's a matter
of just notation.

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Might as well use a
vector for k incident

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because that is
really the vector

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momentum that is coming in.

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And it's sometimes useful to
have the flexibility, perhaps

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some moment.

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You don't necessarily
want the wave

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to come in the z-direction.

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There's another vector that
is kind of interesting.

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You're observing the wave in
the direction of the vector

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r, the direction of
the unit vector n.

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So it's interesting
to call k an incident.

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I think people write it with
just i, so ki for incident, k.

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And then, well, we're
looking at the direction r.

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So we call the
scattering momentum

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or the scattered momentum, kn.

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You're looking at that distance.

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So might as well
call it that way.

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So with these things,
we can rewrite

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the top equation making
use of all the things

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that we've learned.

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And it's going to be an
important equation, psi of r

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is equal to e to the iki dot r.

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So we have to replace
the e to the ikz by this.

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Then we have the
Green's function

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that has to be integrated.

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And you have to see,
we've decided already

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to approximate the
Green's function.

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So it's integral over r prime.

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So the only term here that has
an r prime is this exponential.

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So all of this can go out.

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So the way I want to make it go
out, we will have the minus 1

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over 4 pi here and the
integral d cubed r prime.

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And I'm going to have the rest
of the Green's function placed

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in the standard position
we've used for scattering.

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So we're still redoing the
top integral, the G plus,

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which includes some of G plus.

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We've taken care of
all of this part.

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And now we have
that other factor.

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So it's e to the minus ikn.

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I could call it
the scattering one.

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But I will still not do that.

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Then we have the u of r
prime and the psi of r prime.

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So this is our
simplified expression

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when we make use of the fact
that we're looking far enough

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and the Green's
function has simplified.

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So let's keep that
for a moment and try

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to think how we could solve
this integral equation.

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Now there's no
very simple methods

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of solving them exactly.

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You have to do
interesting things here.

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It's fairly nontrivial.

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But finding approximate
solutions of integral equations

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is something that we can do.

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And it's relatively simple.

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So we will come back to
this because in a sense,

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this is nice.

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But you must feel that somehow
the story is not complete.

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We want to know the
wave function far away.

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This has the form of this
function that when we wrote psi

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is equal to e to the ikz
plus f of theta phi e

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to the ikr over r.

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If we know this f
of theta phi, we

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know everything
about the scattering.

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But it almost looks like
this is f of theta and phi.

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But we don't know psi.

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So have we made any progress?

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Well, the Born
approximation is the way

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in which we see that
we could turn this

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into something quite simple.

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So let's do Born approximation.

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So our first step is to
rewrite once more for us

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this equation, psi
of r is equal to e

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to the ikir plus
integral d q bar prime G

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plus of r minus r prime u
of r prime psi of r prime.

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OK.

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We wrote it.

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Good.

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Now let's rewrite it
with instead of r,

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putting an r prime here.

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You'd say, why?

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Well, bear with me a second.

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Rewrite with r
replaced by r prime.

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If we have an r
prime, replace it

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with r double prime and so on.

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So let's rewrite this.

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So we have psi of r prime
is equal to e to the iki

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r prime plus
integral of d cubed r

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double prime G plus of r prime
minus r double prime u of r

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double prime psi
of r double prime.

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So I shifted.

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Whenever I had an
r, put an r prime.

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Whenever I had an r prime,
put an r double prime.

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The reason we do this
is that technically, I

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can now substitute this
quantity here into here

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and see what I get.

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And I get something
quite interesting.

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I get the beginning
of an approximation

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because I now get psi
of r is equal to e

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00:15:42,100 --> 00:15:57,030
to the ikir plus d cubed r prime
G plus of r minus r prime--

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I'm still copying the first
equation-- u of r prime.

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00:16:01,140 --> 00:16:04,110
And here is where the
first change happens.

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00:16:04,110 --> 00:16:07,430
Instead of psi of
r prime, I'm going

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to put this whole other thing.

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So I'll write two terms.

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00:16:13,540 --> 00:16:20,085
The first is just having
here, e to the iki r prime.

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00:16:23,440 --> 00:16:26,480
The second, I'll write
this another line.

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00:16:26,480 --> 00:16:32,200
So integral d cubed
r prime G plus of r

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minus r prime u of r prime.

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00:16:39,130 --> 00:16:43,680
And then the second term
would be another integral,

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00:16:43,680 --> 00:16:49,710
d cubed r double prime
G plus of r prime minus

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00:16:49,710 --> 00:16:56,480
r double prime u of r double
prime psi of r double prime.

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00:17:02,440 --> 00:17:07,079
So look at it and see
what has happened.

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00:17:07,079 --> 00:17:13,369
You have postponed the fact that
you had an integral equation

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00:17:13,369 --> 00:17:15,710
to a next term.

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00:17:15,710 --> 00:17:21,920
If I would drop
here this term, this

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would not be an integral
equation anymore.

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00:17:24,859 --> 00:17:28,510
Psi is given to this
function that we know.

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00:17:28,510 --> 00:17:29,660
Green's function we know.

218
00:17:29,660 --> 00:17:30,680
Potential we know.

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00:17:30,680 --> 00:17:32,060
Function we know.

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00:17:32,060 --> 00:17:33,980
It's a solution.

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00:17:33,980 --> 00:17:37,620
But the equation is
still not really solved.

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00:17:37,620 --> 00:17:38,960
We have this term here.

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00:17:41,760 --> 00:17:46,440
And now you could go
on with this procedure

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00:17:46,440 --> 00:17:51,600
and replace this psi
of r double prime.

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00:17:51,600 --> 00:17:56,580
You could replace it
by an e to the iki r

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00:17:56,580 --> 00:18:01,200
double prime plus
another integral d

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00:18:01,200 --> 00:18:06,690
r triple prime of G, like that.

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00:18:06,690 --> 00:18:13,080
And then here you would have
an e to the ik r double prime

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00:18:13,080 --> 00:18:18,010
and then more terms
and more terms.

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00:18:18,010 --> 00:18:22,410
So I can keep doing
this forever until I

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00:18:22,410 --> 00:18:29,340
have this integral, G
plus u incident wave.

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00:18:29,340 --> 00:18:34,950
And here I have G plus u
G plus u incident wave.

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00:18:34,950 --> 00:18:38,200
And then I will have G plus u,
G plus u, G plus u, incident

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00:18:38,200 --> 00:18:39,290
wave.

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00:18:39,290 --> 00:18:44,100
And it would go
on and on and on.

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00:18:44,100 --> 00:18:46,850
So this is called the
Born approximation

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00:18:46,850 --> 00:18:52,230
if you stop at some stage
and you ignore the last term.

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00:18:52,230 --> 00:18:59,510
So for example, we could ignore
this term, in which we still

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have an unknown psi, and
say, OK, we take it this way.

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00:19:05,180 --> 00:19:10,050
And this would be the
first Born approximation.

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00:19:10,050 --> 00:19:11,720
You just go up to here.

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00:19:11,720 --> 00:19:14,070
The second Born
approximation would

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00:19:14,070 --> 00:19:19,620
be to include this term with psi
replaced by the incident wave.

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00:19:19,620 --> 00:19:22,500
The third Born approximation
would be the next term

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00:19:22,500 --> 00:19:24,420
that would come here.

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00:19:24,420 --> 00:19:30,790
So let me write the whole Born
approximation schematically,

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00:19:30,790 --> 00:19:35,920
whole Born approximation
schematically.

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00:19:35,920 --> 00:19:39,700
I could write three terms
explicitly or four terms

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if you wish.

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00:19:40,390 --> 00:19:44,200
But I think the pattern
is more or less clear now.

251
00:19:44,200 --> 00:19:45,270
What do you have?

252
00:19:45,270 --> 00:19:52,990
Psi of r equal e
to the ikir plus--

253
00:19:52,990 --> 00:19:56,250
and now it becomes schematic--

254
00:19:56,250 --> 00:19:59,650
integral-- so I
put just integral--

255
00:19:59,650 --> 00:20:06,070
G plus u e to the ikr.

256
00:20:06,070 --> 00:20:07,980
Don't put labels or anything.

257
00:20:07,980 --> 00:20:10,720
But that's the first integral.

258
00:20:10,720 --> 00:20:13,960
And if you had to
put back the labels,

259
00:20:13,960 --> 00:20:15,520
you would know how to put them.

260
00:20:15,520 --> 00:20:18,580
You'd put an integral
over some variable.

261
00:20:18,580 --> 00:20:21,610
Cannot be r because that's
where you're looking at.

262
00:20:21,610 --> 00:20:25,450
So you do an r prime,
the G of r minus r prime,

263
00:20:25,450 --> 00:20:28,660
the u at r prime, and
this function at r prime.

264
00:20:28,660 --> 00:20:31,300
That's how this integral
would make sense.

265
00:20:31,300 --> 00:20:41,980
The next term would be integral
G u of another integral of G u

266
00:20:41,980 --> 00:20:43,000
e to the ikr.

267
00:20:47,340 --> 00:20:49,490
That's the next term.

268
00:20:49,490 --> 00:20:59,250
And the next term would be
just G u G u G u e to the ikr.

269
00:21:02,730 --> 00:21:04,390
And it goes on forever.

270
00:21:04,390 --> 00:21:06,752
That's the Born series.