PROFESSOR: We are going to recap some of the last results
we had on scattering and then push them to the end
and complete our discussion of this integral equation
and how we approximate scattering with it.
It's a pretty powerful method of thinking about scattering
and gives you a direct solution, which
is in a sense more intuitive perhaps than when you're
dealing with partial waves.
So let's see where we were last time.
So we set up our solution of the Schrodinger equation
in a funny way.
We had an incoming wave plus a term that
still had an integral over all of space of our Green's
function integrated against a scaled version
of the potential.
Remember that v was defined to be
equal to the potential up to some constants, h bar
squared over m.
And we had this integral equation
where one could show that phi satisfies
the correct Schrodinger equation.
Now this function G plus was our Green function
that we also discussed.
And it was given by the following formula.
It just depends on the magnitude of this vector argument
of the Green's function, so 1 over 4 pi
e to the ik, magnitude of that divided
by magnitude of that vector.
So in most applications, we look at the wave function.
This should be phi of r.
We look at the wave function far away.
And if the potential is localized,
this integral, since it involves the potential,
can only be non-zero, have non-zero contributions
to the integral on the region where the potential exists.
So r prime, whenever you're doing this integral,
must be small compared to r.
This we can draw here.
So assume we have a potential.
And you have an observation point, r.
We'll call the unit vector in this direction n.
This is the vector r.
And an integration point is a point, r prime here.
And there is this distance, this vector,
in between r minus r prime.
And finally, on this graph, we could also have a direction
that we could call the incident direction, perhaps
the z-direction.
And that's the direction which your wave is incident.
This is the wave that were represented here.
So we did a little approximation of this Green's function
for the case where r prime where you're looking
is much smaller than r.
In that case, the distance, r minus r prime,
is approximately equal to r minus--
so that's the distance, the vectors.
When you don't put an arrow, we mean the distance.
So this is r minus n dot r prime.
To 0-th approximation, this is the most important term.
To next approximation comes this term
because that term is suppressed with respect
to the original term by a factor of r prime over r.
So if you factored, for example, the r here,
you would say 1 minus n r prime over r.
So you see that this ratio enters
here, which says that the second term is smaller by this factor.
We could even include more terms here.
But those would not be relevant unless k
is extremely high energy.
So when k is very large, it would make a difference.
But it has to be extremely large.
And in general, this approximation
isn't enough for any k for sufficiently far away.
So this is really good enough.
And the approximation is done on the Green's function
at two levels.
You need to be mildly accurate with respect to this quantity.
But you need to be more accurate with respect to this phase
because the phase can change rather fast depending
on the wavelength of the object.
So if you make an error in the total distance of 1%,
that's not big deal.
But if you make an error here, such that k
dot times this distance is comparable to 1,
the phase could be totally wrong.
So the phase is always a more delicate thing.
And therefore, with this approximation,
your G plus of r minus r prime becomes minus 1 over 4 pi e
to the ikr over r.
So we replace this term by just r.
And in this exponent, the absolute value
is replaced by two terms.
The first one is e to the ikr, is here.
The second one is e to the minus ikn dot r prime.
We also have here the incident vector.
You can write e to the ikz as e to the ik incident
dotted with r if k incident is a vector, k, times the unit
vector in the z-direction.
So that's a matter of just notation.
Might as well use a vector for k incident
because that is really the vector
momentum that is coming in.
And it's sometimes useful to have the flexibility, perhaps
some moment.
You don't necessarily want the wave
to come in the z-direction.
There's another vector that is kind of interesting.
You're observing the wave in the direction of the vector
r, the direction of the unit vector n.
So it's interesting to call k an incident.
I think people write it with just i, so ki for incident, k.
And then, well, we're looking at the direction r.
So we call the scattering momentum
or the scattered momentum, kn.
You're looking at that distance.
So might as well call it that way.
So with these things, we can rewrite
the top equation making use of all the things
that we've learned.
And it's going to be an important equation, psi of r
is equal to e to the iki dot r.
So we have to replace the e to the ikz by this.
Then we have the Green's function
that has to be integrated.
And you have to see, we've decided already
to approximate the Green's function.
So it's integral over r prime.
So the only term here that has an r prime is this exponential.
So all of this can go out.
So the way I want to make it go out, we will have the minus 1
over 4 pi here and the integral d cubed r prime.
And I'm going to have the rest of the Green's function placed
in the standard position we've used for scattering.
So we're still redoing the top integral, the G plus,
which includes some of G plus.
We've taken care of all of this part.
And now we have that other factor.
So it's e to the minus ikn.
I could call it the scattering one.
But I will still not do that.
Then we have the u of r prime and the psi of r prime.
So this is our simplified expression
when we make use of the fact that we're looking far enough
and the Green's function has simplified.
So let's keep that for a moment and try
to think how we could solve this integral equation.
Now there's no very simple methods
of solving them exactly.
You have to do interesting things here.
It's fairly nontrivial.
But finding approximate solutions of integral equations
is something that we can do.
And it's relatively simple.
So we will come back to this because in a sense,
this is nice.
But you must feel that somehow the story is not complete.
We want to know the wave function far away.
This has the form of this function that when we wrote psi
is equal to e to the ikz plus f of theta phi e
to the ikr over r.
If we know this f of theta phi, we
know everything about the scattering.
But it almost looks like this is f of theta and phi.
But we don't know psi.
So have we made any progress?
Well, the Born approximation is the way
in which we see that we could turn this
into something quite simple.
So let's do Born approximation.
So our first step is to rewrite once more for us
this equation, psi of r is equal to e
to the ikir plus integral d q bar prime G
plus of r minus r prime u of r prime psi of r prime.
OK.
We wrote it.
Good.
Now let's rewrite it with instead of r,
putting an r prime here.
You'd say, why?
Well, bear with me a second.
Rewrite with r replaced by r prime.
If we have an r prime, replace it
with r double prime and so on.
So let's rewrite this.
So we have psi of r prime is equal to e to the iki
r prime plus integral of d cubed r
double prime G plus of r prime minus r double prime u of r
double prime psi of r double prime.
So I shifted.
Whenever I had an r, put an r prime.
Whenever I had an r prime, put an r double prime.
The reason we do this is that technically, I
can now substitute this quantity here into here
and see what I get.
And I get something quite interesting.
I get the beginning of an approximation
because I now get psi of r is equal to e
to the ikir plus d cubed r prime G plus of r minus r prime--
I'm still copying the first equation-- u of r prime.
And here is where the first change happens.
Instead of psi of r prime, I'm going
to put this whole other thing.
So I'll write two terms.
The first is just having here, e to the iki r prime.
The second, I'll write this another line.
So integral d cubed r prime G plus of r
minus r prime u of r prime.
And then the second term would be another integral,
d cubed r double prime G plus of r prime minus
r double prime u of r double prime psi of r double prime.
So look at it and see what has happened.
You have postponed the fact that you had an integral equation
to a next term.
If I would drop here this term, this
would not be an integral equation anymore.
Psi is given to this function that we know.
Green's function we know.
Potential we know.
Function we know.
It's a solution.
But the equation is still not really solved.
We have this term here.
And now you could go on with this procedure
and replace this psi of r double prime.
You could replace it by an e to the iki r
double prime plus another integral d
r triple prime of G, like that.
And then here you would have an e to the ik r double prime
and then more terms and more terms.
So I can keep doing this forever until I
have this integral, G plus u incident wave.
And here I have G plus u G plus u incident wave.
And then I will have G plus u, G plus u, G plus u, incident
wave.
And it would go on and on and on.
So this is called the Born approximation
if you stop at some stage and you ignore the last term.
So for example, we could ignore this term, in which we still
have an unknown psi, and say, OK, we take it this way.
And this would be the first Born approximation.
You just go up to here.
The second Born approximation would
be to include this term with psi replaced by the incident wave.
The third Born approximation would be the next term
that would come here.
So let me write the whole Born approximation schematically,
whole Born approximation schematically.
I could write three terms explicitly or four terms
if you wish.
But I think the pattern is more or less clear now.
What do you have?
Psi of r equal e to the ikir plus--
and now it becomes schematic--
integral-- so I put just integral--
G plus u e to the ikr.
Don't put labels or anything.
But that's the first integral.
And if you had to put back the labels,
you would know how to put them.
You'd put an integral over some variable.
Cannot be r because that's where you're looking at.
So you do an r prime, the G of r minus r prime,
the u at r prime, and this function at r prime.
That's how this integral would make sense.
The next term would be integral G u of another integral of G u
e to the ikr.
That's the next term.
And the next term would be just G u G u G u e to the ikr.
And it goes on forever.
That's the Born series.