WEBVTT
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PROFESSOR: Great.
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So I will begin
with phase shifts
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and do the introduction of how
to make sure we can really--
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so this is the
important part of this.
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Just like when we added the
reflected and transmitted wave
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we could find the
solution I'm going
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to try to explain
why with this things
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we can find
solutions in general.
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So this is the subject
of partial waves,
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and it's a nice subject,
a little technical.
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There might seem to be
a lot of formulas here,
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but the ideas are relatively
simple once one keeps in mind
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the one dimensional analogies.
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The one dimensional analogies
are very valuable here,
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and we will
emphasize them a lot.
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So we will discuss partial
waves and face shifts.
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So it's time to simplify
this matters a little bit.
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And to do that I
will assume from now
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on that the potential is central
so v of r is equal to v of r.
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That will simplify the
azimuthal dependence.
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There will be no
azimuthal dependencies.
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You see, the thing is
spherical is symmetric,
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but still you're coming from
a particular direction, the z.
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So you can expect now that
the scatter wave depends
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on the angle of the
particle with respect
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to z because it's
spherically symmetrical.
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But it shouldn't depend
on five, the angle five,
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should just depend on theta.
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So expect f of theta.
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Now, a free particle
is something
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we all know how to
solve, e to the ikx.
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Why do we bother with the
free particle in so many ways?
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Because free particle
is very important.
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Part of the solution
is free particles.
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To some degree far away it
is free particles as well.
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And we need to
understand free particles
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in spherical coordinates.
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So it's something we've done
in 805 and sometimes in 804,
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and we look at the
radial equation which
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is associated to
spherical coordinates
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for a free particle.
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So we'll consider free
particle and we'd say, well,
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that's very simple
but it's not all that
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simple in spherical
coordinates, and you'd say, OK,
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if it's not simple, it's
spherical coordinates,
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why do we bother?
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We bother because
scattering is happening
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in spherical coordinates.
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So we can't escape having
to do the free particle
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in spherical coordinates.
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It is something you have to do.
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So what are solutions in
spherical coordinates?
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We'll have solution SI of r.
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Remember the language
with coordinates
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was a U of r divided by
r and of Ylm of omega.
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That was a typical solution, a
single solution of the showing
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our equation will--
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the U only depends on l, the
m disappears, so this is r.
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This r's are r's without the
vector because you're already
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talking about the
radial equations,
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and depend on the
energy and depend
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on the value of the
l quantum number.
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So what is the
Schrodinger equation?
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The radial equation is
minus h squared over 2m,
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the second the r squared plus.
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h squared over 2m l times
l plus 1 over r squared.
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Remember the potential
centrifugal barrier
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in the effective potential,
then you would have v of r here,
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but it's free particle,
so v of r is equal to 0.
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So if nothing else,
U of El of little r
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is equal to the
energy, which is h
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squared k squared over 2m UEl.
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And that's a
parliamentary session
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of the energy in terms of
the k squared, like that.
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Well, there's lots of
h squared, k squared,
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and 2m's, so we can
get rid of them.
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Cancel the h squared over 2m.
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You get minus d second
dr squared plus l times l
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plus 1 over r.
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UEl is equal to k squared UEl.
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It's a nice equation.
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It's the equation
of the free particle
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in spherical coordinates.
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Now, this is like the
Schrodinger equation.
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And I think when you look at
that you could get puzzled
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whether or not the value
of k squared or the energy
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might end up being quantized.
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With the Schrodinger
equation many times
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quantized is the energy, but
here it shouldn't happen.
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This is a free particle.
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All values of k
should be allowed,
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so there should be
no quantization.
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This is an r squared here.
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You can see one reason,
at least analytically,
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that there is no quantization
is that you can define
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a new variable row
equal kr and then
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this whole differential
equation becomes
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minus the second the row
squared plus l times l
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plus 1 over row squared.
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Well, I can put the other
number in there as well,
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or should I not?
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No, it's not done here.
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UEl is equal to UEl, and the k
squared disappeared completely.
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That tells you that the case
will kind of get quantized.
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If there is a solution of
this differential equation
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it holds for all values of k.
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And these are going to
be like plane waves,
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and maybe that's
another reason you
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can think that k
doesn't get quantized
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because these solutions are
not normalizable anyway,
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so it shouldn't get quantized.
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So with this equation in here
we get the two main solutions.
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The solutions of this
differential equation
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are vessel functions,
spherical vessel functions.
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UEl is equal to a constant
Al times row times the vessel
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function lowercase j of row.
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There's a row times
that function.
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That's the way it shows up.
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It's kind of interesting.
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It's because in fact you
have to divide U by r,
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so that would mean
dividing U by row,
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and it means that the radial
function is just the vessel
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function without anything else.
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And then there's the
other vessel function,
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the n of l a row times
of n of l of row.
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So those are spherical
vessel functions.
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As you're familiar
from the notation
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that j is the one that
this healthy at row
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equals 0 doesn't diverge
the n is the solution
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that diverges at the origin.
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And both of them
behave nicely far away.
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So Jl of x goes like 1 over x
sine of x minus l pi over 2,
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and ADA l of x
behaves like minus 1
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over x cosine of x
minus l pi over 2.
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This is for x big, x
much greater than 1,
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you have this behavior.
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So these are our
solutions, and here is
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the thing that we have to do.
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We have to rewrite our solutions
in terms of spherical waves
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because this was the spherical
wave so we should even write
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this part as a spherical wave.
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And this is a very
interesting and in some way
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strange representation of E to
the ikz You have E to the ikz
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that you have an
intuition for it
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as a plane wave
in the z direction
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represent it as an infinite
sum of incoming and outgoing
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spherical waves.
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That's what's going to happen.
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So this is the last
thing we need do here.
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We have that e to the ikz
is a plane wave solution,
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so it's a solution
of a free particle,
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so I should be able to write the
superpositions of the solutions
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that we have found.
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So it should be a superposition
of solutions of this type.
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So it could be a sum of
coefficients al times,
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well, alm you think of
some a's times solutions.
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Remember, we're writing
a full solution,
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so a full solution
you divide by r.
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So you divide by this quantity.
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So you could have
an alm Jl of row
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plus Blm ATA l of row times Ylm.
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So this should be
a general solution,
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and that would be a sum
over l's and m's of all
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those quantities.
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But that's a lot more
than what you need.
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First, this does not
diverge near r equals 0.
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It has no divergence anywhere
and the ATAs or the n's, I
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think they're n such and not
ATAs, the n's diverge for row
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equal to 0.
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So none of this are necessary,
so I can erase those.
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l and m.
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But there is more.
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This function is
invariant and there
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are some beautiful rotations.
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If you have your axis
here, here's the z,
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and you have a point here and
you rotate that the value of z
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doesn't change.
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It's independent of
phi for a given theta,
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z just depends on
r of cosine theta.
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So there's no phi dependence
but all the Ylm's with m
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difference from 0
have phi dependent.
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So m cannot be here either.
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m must be 0.
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So you must be
down to sum over l,
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al some coefficient,
Jl of row, Yl0.
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And all of those would be
perfectly good plane wave
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solutions.
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Whatever numbers you
choose for the little al's,
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those are good solutions
because we've build them
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by taking linear combinations
of exact solutions
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of this equation.
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But to represent this
quantity the al's
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must take particular values.
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So what is that formula?
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That formula is quite
famous, and perhaps even you
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could discuss this
in recitation.
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e to the ikz, which is e
to the ikr cosine theta,
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is the sum 4 pi.
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Now you have to get all
the constants right.
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Square root of 4 pi,
sum from l equals 0
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to infinity, square
root of 2l plus 1.
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Coefficients are pretty funny.
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They get worse very fast.
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Now you have of i to the
I, i to the l, Yl0 of theta
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doesn't depend on phi, Jl of kr.
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This is the expansion
that we need.
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There's no way we can make
problems with this problem
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unless we have this expansion.
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But now if Y the intuition
that I was telling you
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of these waves
coming in and out,
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well, you have e to the
ikz, you sum an infinite sum
00:16:42.370 --> 00:16:44.620
over partial waves.
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A partial wave is a
different value of l.
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These are partial waves.
00:16:49.960 --> 00:16:54.730
As I was saying, any solution
is a sum of partial waves
00:16:54.730 --> 00:16:56.530
is a sum over l.
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And where are the waves?
00:16:58.120 --> 00:17:10.569
Well, the Jl of kr far away
is a sine, and the sine of x
00:17:10.569 --> 00:17:17.050
is an exponential ix minus
e to the minus ix over 2.
00:17:17.050 --> 00:17:21.250
So here you have
exponentials of e to the ikr
00:17:21.250 --> 00:17:24.849
and exponentials of
e to the minus ikr,
00:17:24.849 --> 00:17:31.060
which are waves that are
here like outgoing waves
00:17:31.060 --> 00:17:32.360
and incoming waves.
00:17:32.360 --> 00:17:38.080
So the E to the ikz's are sum of
ingoing and outgoing spherical
00:17:38.080 --> 00:17:38.740
waves.
00:17:38.740 --> 00:17:42.790
And that's an intuition that
we will exploit very clearly
00:17:42.790 --> 00:17:43.920
to solve this problem.
00:17:43.920 --> 00:17:47.580
So we will do that next.