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PROFESSOR: If you have
potential transmission

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coefficient for
a potential where

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z0 is equal to 13 pi over 4.

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That's a square well
of certain depth,

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and we represent it in this way.

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Remember n must be greater
than or equal than 2z0 over pi.

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So this will be 13/2.

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And 13/2 means that we can start
with n equals 7, 8, 9, and all

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those.

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Remember, this n counts
which bound state

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of the infinite square
well you're talking about.

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And the energy that you
must use are your integers,

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are positive energy.

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So positive energies mean that
you have sufficiently large n,

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and the n that this sufficiently
large is 7 in this case.

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So you can then determine
from this formula what

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is the value of e n over v0.

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So for example E7 over v0
turns out to be 0.15976.

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E8 over v0 turns
out to be 0.514793.

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And E9 over v0 is 0.91716.

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So if you plot it, you have here
E, or capital energy, over v0.

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And you want to plug the
transmission probability.

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And it begins with 0.

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That was the question
a second ago.

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And then it may reach 1.

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And it will reach 1 at
each one of those values.

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So if, here is 1, 0.15.

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There will be 0.15,
0.51, and 0.92.

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So you get this, and here
another one, and here

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another one.

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Probability like that.

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So that's a typical graph for
the transmission probability.

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It oscillates, and it
reaches 1 at several points

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forever and ever.

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And the amplitude
become smaller,

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so it's really
overall tending to 1.

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So these two people
we're talking about,

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Ramsauer and Townsend.

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They lived from 1860s
to 1940s and '50s.

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And they did their famous
experiment in 1921.

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So their experiment
was elastic scattering

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of low energy electrons
off of rare gas atoms.

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So Ramsauer and
Townsend, in 1921,

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they scattered elastically.

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That means the particles
didn't change their identities.

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They didn't create
more particles.

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It was just electrons came
in and electrons went out.

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Electrons.

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And these are low
energy electrons,

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off of rare gas atoms.

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So these are noble gases.

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Their shells are
completely filled.

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And they're rather inert, very
unreactive, high ionization

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energies, no low
energy states you

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can scatter these atoms into.

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So basically very
unreactive atom.

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And you can imagine it as a
very beautiful spherical cloud.

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We can draw some
electrons, there's

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some protons, a nucleus
here, and an electron cloud.

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So how does this
look to an electron?

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Well, you know
from electrostatics

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that if you have total charge
0 and it's totally spherically

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symmetric, no electric
field outside.

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So the electron comes
in, feels nothing.

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And as soon as you penetrate
this, at any point here,

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the electric field points in.

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Or, well, it
actually points out,

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but the electron
will feel a force in.

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Because the charge
in the outside

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shell doesn't produce any field.

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But now, the protons
in the nucleus

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beat the effect
of the electrons.

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So there's a force in, a
force in, that goes in.

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So basically this is like a deep
square well, or spherical well,

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representing the atom.

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The atom can be some
sort of spherical well

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that attracts the electrons.

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So what these people did were
throwing these electrons.

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And they considered that
this electron scattered

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a lot when they bounced back.

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On the other hand,
if they continued,

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if the electrons pass by, they
said nothing has happened.

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So the reflection coefficient
for them, the reflection

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coefficient.

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Reflection coefficient
is a proxy,

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a good representation for
the scattering cross-section.

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So the reflection coefficient,
what they found experimentally

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was a reflection coefficient,
R, that as a function of energy

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was very high.

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And people thought
at this moment, OK,

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these are like particles
colliding with particles.

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Their energies shouldn't make
much difference, you know.

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You either collide or you don't
collide, and you bounce back

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or you don't bounce back.

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So they thought that
this would be flat.

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But nevertheless, it actually
went down enormously,

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and then it went up again.

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So they found that for
electrons with about 1 Ev,

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that's very low
energy electrons,

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but they were going pretty fast.

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And E1, Ev electron is
going like at 600 kilometers

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per second.

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So the reflection
was going like this.

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And they had no explanation
why it was so sensitive

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with energy, and why there would
be a funny effect going on,

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that the reflection
would suddenly go down,

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and just basically the
particles would get transmitted.

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But if you think of reflection
here as a continuous line

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and transmission
as a dotted line,

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the transmission that must
alter the reflection to be 1

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would be going up here and
would have reached near 1

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at this value of the energy.

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So the explanation
eventually was this effect,

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that you should
do well and there

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is a resonant effect in the
well, and for some energies

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the resonance is such that it
allows the particles to just go

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through and not scatter.

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So this had to wait some time,
because the experiment was done

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in 1921, and Schrodinger
and everybody started

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doing good work in 1925, and
of wave mechanics took a while.

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But eventually it was
recognized that basically it's

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resonant transmission,
what is happening there.

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Well, if you want to
get the numbers right,

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if you want to get
that Ev better, you

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have to do a spherical model of
the square, finite square well,

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you have this spherical
well, and do it a little more

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precisely.

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But then the agreement
is pretty reasonable.