WEBVTT
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OK.
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So that's our introduction
to the subject.
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Now, we have to get going.
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We have to explore how to set
up this scattering problem.
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So equations that
we need to solve.
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Well, what are the equations?
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We will have a
Hamiltonian, which
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is p squared over 2m
plus a potential v of r.
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Need not be yet a
central potential.
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As usual, we will think
of a wave function that
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depends on r and t
that will be written
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as a psi that depends on r times
e to the minus iet over h bar.
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An energy eigenstate.
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And then the equation
that you have to solve,
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the time independent the
Schrodinger equation, becomes
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minus h squared
over 2m Laplacian
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plus v of r acting on psi
of r is equal to e psi of r.
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So these are the equations
that you have seen already
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endlessly in quantum mechanics.
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These are equations we
write to get warmed up,
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and we just repeat
for ourselves that we
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have a Hamiltonian
in the picture where
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the particles are scattering
off of a central potential.
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Not a central
potential, in fact,
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just often of a potential.
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Work with energy
eigenstates, and these
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are going to be the equation
for the energy eigenstates.
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Now, let's write the first
picture of the scattering
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process as some
sort of target here
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or potential and
particles that come in
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and scatter off
of this potential.
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So we're looking for
energy eigenstates.
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And we will try to identify
our energy eigenstates,
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and what we're going to
assume is that this potential
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is finite range as well.
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Range.
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Finite range we can
deal with potentials
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that fall off relatively fast.
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Already, the coolant potential
doesn't fall off that fast,
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but potentials that fall faster
and the cooler potentials
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are potentials that
are just localized,
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which is pretty common
if you have an atom
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and you scatter
things off of it.
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If it's a neutral atom, the
potential due to the atom
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is zero outside the atom, but as
soon as you go inside the atom
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you start to experience all
the electrical forces that
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are due to the nucleus
and the electrons.
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So a finite range
potential, and we're
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going to think of
solutions that--
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OK, away from the finite
range our plane waves,
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solutions of constant
energy specified
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with perhaps some momentum,
so we will think of e as h
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squared k squared over 2m.
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This is a way of thinking of
the energy of a given energy
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eigenstate.
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So it's another
label for the energy.
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This one, it looks like
I've done something
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but I haven't done much except
to begin an intuition process
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in your head in
which somehow these
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are going to be related
to energy eigenstates that
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have some momentum as they
propagate all over space.
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So if I write that, I
could just simply put
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this on the left hand
side and get an equation
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that is kind of nicer.
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Minus h squared
over 2m Laplacian
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squared plus k squared plus
v of r psi of r equals zero.
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OK, it's not really
a matter scattering
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of solving this
equation at this moment.
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There is infinitely many
solutions of this equation,
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and most of them may
not be relevant for us.
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We're not trying to find every
solution of this equation.
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We're trying to
find solutions that
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have something to
do with physics,
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and you've done
that when you had
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potentials in one dimension.
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And that intuition is
going to prove invaluable.
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So when you have a potential in
one dimension, you didn't say,
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OK I'm going to find all
the energy eigenstates.
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You said, let's
search for things
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that are reasonable and
physically motivated,
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so you put in a wave
that was moving in
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and you said, OK, this
wave is a solution
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until it reaches this point
where it just stops being
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a solution and you need more.
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If you put in this wave, you
will generate a reflected wave
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and a transmitted wave.
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Those two waves are
going to be generated,
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and then you write an
ansatz for this wave,
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some coefficient a e to the
ikx, b e to the minus ikx, c
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e to the other, and then
you solve your equation.
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So we need to do the same thing
with this kind of equation
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and this kind of potential.
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We have to set up
some sort of situation
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where we have the physics
intuition of a wave that
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is coming in and then whatever
the system will do to that wave
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to upgrade it into
a full solution.
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So that's what we want to do
here in analogy to that thing.
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This could be called
the incoming wave,
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and this whole thing the
reflected and the transmitted
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could be called the scattered
wave, the thing that
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gets produced by the
scattering process.
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So if you had that there is lots
of solutions of that equation,
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if v was identical to zero,
if you had no potential,
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you could have
lots of solutions,
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because in fact if the
potential is not zero,
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plane waves are always
solutions without any potential.
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Particles that move as
plane waves so if equal
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zero, plane waves of the
form psi equal e to the ikx
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are solutions with k equal
the square root of k dot k.
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Or with k squared
equals k dot k.
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Plane waves are always
solutions of this equation.
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So if plane waves are solutions
of this equation for v
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equal to zero, this
is the same thing
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as saying here
that ae to the ikx
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is a solution of this
equation as long as you
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don't hit the potential.
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So here, we're going to do
something quite similar.
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We're going to say that we're
going to put in an incident
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wave function, and I will
instead of writing psi,
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I will call it 5x.
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The incident wave function 5x is
going to be just e to the ikz.
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So it's a wave function.
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I call it phi to
distinguish it from psi.
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Psi in general is
a full solution
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of the Schrodinger equation.
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That's our understanding of psi.
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So phi, it reminds you that
well, it's some wave function.
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I'm not sure it's a solution.
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In fact, it probably is
not the solution as soon
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as you have the potential
different from zero.
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So is this common, we
forget that ksv equals zero,
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and now we put an incident wave
function which is of this form.
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This solves the equation
as long as you're away
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from the potential.
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This is true, it's a solution
of the Schrodinger equation
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away from v of r.
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Away from v of r means
wherever v of r is equal zero,
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you have a solution.
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Nevertheless, so if we call
the range of the potential--
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let's call the range of the
potential finite range a--
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that is to take that if
there is an origin here
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up to a radius a, there is some
potential and beyond the radius
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a the potential vanishes.
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So this definitely works.
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It's all ks away from vr or
as long as r is greater than a
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for whatever value
of z you take.
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This is fine, so
here is our wave,
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and now this is just
the incident wave.
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This is not going to
be the whole solution.
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Just like in the one dimensional
case, there must be more.
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So what is there more?
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And our challenge to
begin with this problem
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is to set up what
else could there be.
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So looking at it,
you'd say, all right.
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So the thing comes in.
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If there is
scattering, particles
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are sometimes going to go
off in various directions.
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So the outgoing wave, here
there were outgoing waves
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reflected and transmitted.
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The outgoing wave in the three
dimensional scattering problem
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should be some sort of spherical
wave moving away from r
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equals zero, which
is the origin.
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That should be the other
wave that I would write.
00:13:05.540 --> 00:13:10.260
So my ansatz should
be that there
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is some sort of spherical
wave that is moving away.
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So to complete this with
a spherical outgoing wave.
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So while here this is
a plane wave moving
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in the direction
of the vector k,
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if I want to write the spherical
wave, I would write e to the i
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just kr.
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E to the ikr is
spherically symmetric,
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and it propagates radiantly out.
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If you remember
as usual that you
00:14:04.660 --> 00:14:12.220
have e to the minus iet over h
bar, so you have kr minus et,
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that is a wave that
propagates radially out.
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So maybe this is kind
of the scattered wave.
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This e to the ikr
moving out everywhere
00:14:29.190 --> 00:14:30.735
would be your scattered wave.
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If that is the
scattered wave, remember
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the scattered wave should
solve the Schrodinger equation.
00:14:44.250 --> 00:14:46.280
In fact, the sum
of these two should
00:14:46.280 --> 00:14:48.450
solve the scattering equation.
00:14:48.450 --> 00:14:50.390
On the other hand,
we've seen that this
00:14:50.390 --> 00:14:55.370
solves it as long as you're
away from the potential,
00:14:55.370 --> 00:14:58.100
and therefore this
should also solve it
00:14:58.100 --> 00:15:00.290
if you're away
from the potential.
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So I ask you, do you think this
solves the Schrodinger equation
00:15:07.340 --> 00:15:09.170
when you're away
from the potential?
00:15:14.140 --> 00:15:17.840
Would e to the ikr
solve the equation?
00:15:17.840 --> 00:15:21.240
Well for that, you would
need that if you're away
00:15:21.240 --> 00:15:26.250
from the potential, do
you have Laplacian of e
00:15:26.250 --> 00:15:30.680
to the ikr roughly equals zero?
00:15:34.540 --> 00:15:35.190
Is that true?
00:15:44.550 --> 00:15:46.850
Would it solve it.
00:15:46.850 --> 00:15:47.520
No.
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It doesn't solve it.
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Doesn't even come
close to solving it.
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It's pretty bad, but the reason
it's bad is physically clear.
00:15:57.990 --> 00:16:04.890
This wave as it expands out
must become weaker and weaker
00:16:04.890 --> 00:16:09.720
so that the probability
flux remains constant.
00:16:09.720 --> 00:16:11.910
You know, you don't
want an accumulation
00:16:11.910 --> 00:16:16.110
of probability between a shell
at one kilometer and a shell
00:16:16.110 --> 00:16:17.230
at two kilometers.
00:16:17.230 --> 00:16:22.190
So whatever flux is going out
from the shell in one kilometer
00:16:22.190 --> 00:16:24.930
should be going out
of the bigger shell.
00:16:24.930 --> 00:16:29.280
So therefore, it
should fall off with r.
00:16:29.280 --> 00:16:30.300
Oh, I'm sorry.
00:16:30.300 --> 00:16:34.320
I didn't write this well.
00:16:34.320 --> 00:16:39.090
So if you are going
to have this to be
00:16:39.090 --> 00:16:43.320
a solution of the Schrodinger
equation outside of vr
00:16:43.320 --> 00:16:49.250
equals zero, you should
have Laplacian k squared
00:16:49.250 --> 00:16:51.570
of this thing equal to zero.
00:16:51.570 --> 00:16:55.210
So if this is equal to zero and
the potential is equal to zero,
00:16:55.210 --> 00:16:57.480
the whole thing
is equal to zero.
00:16:57.480 --> 00:17:00.030
So we need this to hold.
00:17:00.030 --> 00:17:03.880
But even this one, of
course, is not true.
00:17:03.880 --> 00:17:06.010
It is just absolutely not true.
00:17:06.010 --> 00:17:09.599
The one that works
is the following.
00:17:12.690 --> 00:17:23.329
Laplacian plus k squared
of e to the ikr over r
00:17:23.329 --> 00:17:28.830
is equal to zero for
r different from zero.
00:17:28.830 --> 00:17:32.750
This is a computation
I think you guys have
00:17:32.750 --> 00:17:35.390
done before when
you were studying
00:17:35.390 --> 00:17:37.880
the Hermiticity of p squared.
00:17:37.880 --> 00:17:40.310
You ended up doing
this kind of things.
00:17:40.310 --> 00:17:45.860
This Laplacian produces a delta
function at r equals zero,
00:17:45.860 --> 00:17:49.620
but r equals zero is not the
place we're interested in.
00:17:49.620 --> 00:17:53.780
We're trying to find
how the waves look away
00:17:53.780 --> 00:17:57.420
from the scattering center.
00:17:57.420 --> 00:18:02.240
So we need this to
hold away from r
00:18:02.240 --> 00:18:06.860
equals zero, in fact,
bigger for r bigger than a.
00:18:06.860 --> 00:18:11.630
One way of checking
this kind of thing
00:18:11.630 --> 00:18:18.200
is to remember that the
Laplacian of a function of r
00:18:18.200 --> 00:18:27.290
is in fact one over r d
second dr squared r times f.
00:18:27.290 --> 00:18:32.330
That's a neat formula for the
Laplacian of a function that
00:18:32.330 --> 00:18:33.620
just depends on r.
00:18:33.620 --> 00:18:36.960
If it depends on theta and
phi, it's more complicated.
00:18:36.960 --> 00:18:40.730
But with this function, it
becomes a one line calculation
00:18:40.730 --> 00:18:44.450
to do this, and the r
here is just fantastic,
00:18:44.450 --> 00:18:48.830
because by the time you
multiply by r this function is
00:18:48.830 --> 00:18:49.930
just exponential.
00:18:49.930 --> 00:18:55.820
You take two derivatives,
you get minus k squared,
00:18:55.820 --> 00:18:58.940
and then the r gets
canceled as well
00:18:58.940 --> 00:19:02.490
and everything works
beautifully here.
00:19:02.490 --> 00:19:09.770
And so this equation holds OK.
00:19:09.770 --> 00:19:15.890
And then we do have a
possible scattered wave.
00:19:15.890 --> 00:19:18.410
So we're almost there.
00:19:18.410 --> 00:19:29.550
We can write the scattering
wave size scattering
00:19:29.550 --> 00:19:45.470
of x could be e to the ikr over
r and then leave it at that.
00:19:45.470 --> 00:19:48.120
But this would not
be general enough.
00:19:48.120 --> 00:19:53.470
There is no reason why
this wave would not
00:19:53.470 --> 00:19:56.270
depend also on theta and phi.
00:19:59.560 --> 00:20:03.130
Here is the z
direction, and there's
00:20:03.130 --> 00:20:07.450
points with angle theta, and if
you rotate it with some angle
00:20:07.450 --> 00:20:13.930
phi here, and therefore
this function could as well
00:20:13.930 --> 00:20:16.420
depend on theta and phi.
00:20:16.420 --> 00:20:24.590
So we'll include that
factor f of theta and phi.
00:20:31.550 --> 00:20:34.000
Now you would say,
look, you have
00:20:34.000 --> 00:20:37.090
a nice solution of the
Schrodinger equation
00:20:37.090 --> 00:20:45.110
already here, and what
should you be doing?
00:20:45.110 --> 00:20:48.020
Why do you add this factor?
00:20:48.020 --> 00:20:52.100
With this factor, it may
not be anymore a solution
00:20:52.100 --> 00:20:53.537
of the Schrodinger equation.
00:20:53.537 --> 00:20:55.745
The Schrodinger equation is
going to have a Laplacian
00:20:55.745 --> 00:20:59.040
and it's going to
be more complicated.
00:20:59.040 --> 00:21:03.140
Well, this is true and we
will see that better soon,
00:21:03.140 --> 00:21:07.040
but this will remain
an approximate solution
00:21:07.040 --> 00:21:14.690
for r much bigger than a is the
leading term of the solution,
00:21:14.690 --> 00:21:19.385
leaving term of the solution.
00:21:30.850 --> 00:21:36.660
So if you have a leading
term of a solution here,
00:21:36.660 --> 00:21:38.810
this is all you want.
00:21:38.810 --> 00:21:44.270
You are working at r much bigger
than a, and your whole wave
00:21:44.270 --> 00:21:46.880
and finally be written.
00:21:46.880 --> 00:21:53.100
So your full wave psi, or your
full energy eigenstates psi
00:21:53.100 --> 00:21:57.380
of rt, is going to be
equal, approximately
00:21:57.380 --> 00:22:00.740
equal to e to the ik--
00:22:00.740 --> 00:22:03.260
well, I'll write it this way.
00:22:03.260 --> 00:22:09.680
Phi of r, that incident
wave we wrote--
00:22:09.680 --> 00:22:12.020
I wrote if x there, I'm sorry.
00:22:12.020 --> 00:22:17.940
Plus psi scattering of--
00:22:17.940 --> 00:22:22.320
I should decide x or r.
00:22:22.320 --> 00:22:26.550
Let's call this r.
00:22:26.550 --> 00:22:27.870
And I should call this r.
00:22:33.826 --> 00:22:37.930
Psi of r plus psi
scattering of r,
00:22:37.930 --> 00:22:47.460
and it's therefore equal to e
to the ikz plus f of theta phi e
00:22:47.460 --> 00:22:56.160
to the ikr over r, and this is
only valid for r much bigger
00:22:56.160 --> 00:22:57.690
than a.
00:22:57.690 --> 00:22:59.520
Far away.
00:22:59.520 --> 00:23:05.430
This had an analog in our
problem in one dimension.
00:23:05.430 --> 00:23:12.860
In one dimension, you set up a
wave and you put here the wave
00:23:12.860 --> 00:23:17.430
and there is a reflected wave,
and there's a transmitted wave,
00:23:17.430 --> 00:23:21.180
and in setting the problem, say
this is valid far to the left,
00:23:21.180 --> 00:23:25.530
far to the left meaning at least
to the left of the barrier,
00:23:25.530 --> 00:23:28.110
this is valid to the right.
00:23:28.110 --> 00:23:33.300
And these were exact
solutions in this region here.
00:23:33.300 --> 00:23:37.470
You can't do that well, but
you can do reasonably well,
00:23:37.470 --> 00:23:44.850
you can write solutions that
are leading term accurate
00:23:44.850 --> 00:23:48.370
as long as you are far away.
00:23:48.370 --> 00:23:54.630
So this is the first step in
this whole process in which we
00:23:54.630 --> 00:23:59.830
are setting up the
wave functions that
00:23:59.830 --> 00:24:01.840
is the most important equation.
00:24:01.840 --> 00:24:05.810
This is the way we're going
to try to find solutions.
00:24:09.030 --> 00:24:11.640
Now, when you try
to find solutions
00:24:11.640 --> 00:24:13.770
at the end of the
day, you will have
00:24:13.770 --> 00:24:19.050
to work in the region
r going to zero.
00:24:19.050 --> 00:24:22.410
So sooner or later
we'll get there.
00:24:22.410 --> 00:24:25.980
But for the time being, we
have all the information
00:24:25.980 --> 00:24:30.510
about what's going on far
away, and from there we
00:24:30.510 --> 00:24:33.360
can get most of what we need.
00:24:33.360 --> 00:24:38.160
In particular, this
f of theta and phi
00:24:38.160 --> 00:24:42.510
is the quantity we
need to figure out.
00:24:42.510 --> 00:24:47.160
This f of theta
and phi is called
00:24:47.160 --> 00:24:49.770
the scattering amplitude.
00:24:49.770 --> 00:24:58.040
F of theta and phi called
scattering amplitude.