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PROFESSOR: Today's lecture
continues the thing we're

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doing with scattering states.

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We send in a scattering state.

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That is an energy eigenstate
that cannot be normalized

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into a step barrier.

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And we looked at
what could happen.

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And we saw all kinds of
interesting things happening.

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There was reflection
and transmission

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when the energy was
higher than the barrier.

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And there was just reflection
and a little exponential decay

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in the forbidden region
if the energy was lower

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than the energy of the barrier.

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We also observed when we
did the packet analysis

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that a wave packet
sent in would have

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a delay in coming back out.

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It doesn't come out immediately.

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And that's the property
of those complex numbers

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that entered into the
reflection coefficient.

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Those complex
numbers were a phase

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that had an energy dependence.

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And by the time you're done with
analyzing how the wave packet

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is moving, there was a delay.

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So today we're going to
see another effect that

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is sometimes called
resonant transmission.

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And it's a rather famous.

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Led to the so-called
Ramsauer-Townsend effect.

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So we'll discuss that.

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And then turn for the last
half an hour into the setup

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where we can analyze more
general scattering problems.

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So let's begin with this
Ramsauer-Townsend effect.

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And Ramsauer Townsend effect.

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Before discussing
phenomenologically

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what was involved
in this effect,

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let's do the mathematical
and physics analysis

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of a [INAUDIBLE] problem
relevant to this effect.

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And for that [INAUDIBLE] problem
we have the finite square well.

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We've normalized
this square well.

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Having width to a.

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So it extends from minus a to a.

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That's x equals 0.

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And there's a number minus v0.

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v0 we always define
it to be positive.

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Therefore, v0 is the
depth of this potential.

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And the question, is what
happens if you send in a wave?

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So you're sending in a particle.

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And you want to know
what will happen to it.

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Will it get reflected?

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Will it get transmitted?

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What probabilities
for reflection, what

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probabilities for transmission?

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So of course, we would have to
send a wave packet to represent

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the physical particle.

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But we've learned that dealing
with energy eigenstates

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teaches us the most
important part of the story.

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If a particle has some
energy, well roughly,

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it will tend to behave
the way an energy

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eigenstate of that energy does,
as far as reflection and as far

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as transmission is concerned.

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So we'll set up a wave.

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And there's a coefficient A.
So there's an Ae to the ikx.

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That is our wave.

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And it's moving to the right.

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Because you remember
the time factor,

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e to the minus iet over h bar.

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And when you put
them together, you

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see that it's
moving to the right.

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But presumably, there
will be a reflection here.

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The wave comes in, and
some gets reflected

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and some gets transmitted.

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And the part that gets
transmitted probably

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will get partially reflected
back here and partially

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transmitted forward.

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But the part that is
partially reflected

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will get again reflected here
and partially transmitted back.

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It seems like a
never ending process

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of which, if you think
physically what's happening,

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there's a reflection
at the first barrier.

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Now you say, wait a moment.

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Why would there be a reflection?

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Classically, there would
never be a reflection.

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If the potential goes
down, the particle

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would just be able to continue.

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We had reflection
when we had a barrier.

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Well in quantum mechanics,
any change in the potential

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is bound to produce
a reflection.

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So yes, if you have a
potential like this,

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like a jumping board,
and you come in here,

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there's a tiny
probability that you

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will be reflected as you come
into this potential drop.

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So OK, so this
will be reflection.

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So at the end of the day,
there might be many bouncings.

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If you imagine a
particle doing this.

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Some probability of reflection,
some of transmission.

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But the end of the day,
there will be some wave

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moving to the left here.

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So we'll represent it
by Be to the minus ikx.

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So in this region,
we have A and a wave

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with amplitude B going this way.

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In the middle region,
the same will be true.

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There will be some
wave going here,

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and some wave that bounces
due to this reflection.

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So there will be a--

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I don't know what
letters I used.

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I'd better keep the same.

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C in this direction and
D in this direction.

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And now I would
have Ce to the ikx.

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Well, that e to the
ikx is not quite right.

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Because k here, k squared
represents the energy.

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k is the momentum and
k squared is energy.

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So if you have an
energy eigenstate-- oh,

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my picture is very crowded.

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So I'll do it anyway.

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A is here.

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Maybe I'll put C and D here.

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And now with A and B here I
can write this as the energy.

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You have a particle with
some energy coming in.

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And there is this wave
here and k squared.

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It's 2mE over h squared.

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E is positive scattering states.

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And indeed, E from that equation
is h squared k squared over 2m.

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What do you know?

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But at this point,
the total energy,

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kinetic energy of the
particle is bigger.

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If e is replaced
by e plus v0, which

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is the magnitude of this drop.

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So here there will be a k2x
plus De to the minus ik2x.

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And k2 refers because it's
region two, presumably.

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People use that
name. k2 squared will

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be 2me plus v0 over h squared.

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And finally, to the right
of the potential square

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well there will
be just one wave.

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Because intuitively,
we should be

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able to interpret
this as some wave that

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goes through, but has nothing
to make it bounce or reflect.

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So we try to get
the solution, which

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will have just some wave
going in this direction.

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And it's called Fe to the ikx.

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And I can go back to
the label k because you

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have the same energy
available as kinetic energy

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you had to the left
of the barrier.

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OK, so we've set up the problem.

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The wave function I
would have to write it

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as three expressions.

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One for x less than a, one for
x in between a and minus a,

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and one for x greater than a.

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And those are this one,
two, and three formulas.

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Any questions about
this setup so far?

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OK.

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Well, at this moment
you will eventually

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have some practice on that.

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The thing that you want to do is
relate the various coefficients

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and define some reflection
and transmission coefficients.

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We saw we had to think in
terms of probability current.

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That's the better
way to get an idea

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of what you should call
reflection or transmission

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coefficient.

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So we have to be careful with
the case of the step potential

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when we compared the
meaning of the wave that

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was moving to the right.

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The amplitude divided
by the incoming wave

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was not quite the
transmission coefficient.

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But in this case,
the nice thing is

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that the wave to the left
and the wave to the right

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are experiencing
the same potential.

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So they can be
compared directly.

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So I will be able to conjecture,
it's reasonable to conjecture

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that we can define
reflection and transmission

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coefficients as follows.

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We'll have a
reflection coefficient

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should be B over A squared.

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B represents a reflected
wave, A the incoming wave.

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The transmission
coefficient you may guess

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that it's F over A squared.

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And this all will
make sense if we

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have the reflection plus the
transmission is equal to 1.

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So we have current
conservation, conservation.

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And the current, which
is the net probability

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flow to the left of the barrier
or the depression over here,

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is J on the left is proportional
to A squared minus B squared.

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You remember, we
computed it last time.

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If you compute the probability
current to this Ae to the ikx

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plus Be to the minus ikx,
you get two contributions.

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Essentially A squared
minus B squared.

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There's a factor of h
bar k over m in front.

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At any rate, this should be
equal to the current that is

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flowing out in this direction.

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You see, whatever current
is coming in to the left

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must be the current
going out to the right.

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So this is f squared.

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So current conservation
really tells you

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that A squared minus B
squared is equal to F squared.

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And if you pass the
B to the other side,

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you get A squared equal
B squared plus F squared.

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And dividing by
A squared you get

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1 is equal to B over A
squared plus F over A squared.

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And that's the reflection
plus the transmission

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So the way we've defined
things makes sense.

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Reflection and transmission
are properly defined.

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And this is because current
conservation works well.

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So reflection coefficient
is essentially

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the flux in the reflected,
the probability current

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or in the flux in the
reflected wave compared

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to the flux in
the incoming wave.

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The transmission is the
probability current or flux

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of probability in the
transmitted wave compared

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with the one incoming wave.

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So you've done all
of this set up.

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Now you can't avoid,
however, doing

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a little bit of calculation,
which is boundary conditions.

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You have one, two, three,
four, five variables.

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And somehow, you want to
calculate these ratios.

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So you have to say that the
wave function and the derivative

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00:15:00,850 --> 00:15:02,500
is continuous at this point.

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And the wave function
and the derivative

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00:15:04,670 --> 00:15:06,410
is continuous and this point.

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That would give you
four conditions.

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And that's reasonable.

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We have five variables.

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But you know, the
overall normalization

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could never be determined.

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So things can be
determined in terms of A.

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So you can expect that
with four equations

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you can solve for B, C,
D, and F in terms of A.

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But the overall scale of
this total wave function

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is undetermined.

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Its boundary conditions
will give you constraints,

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but it will never
determine the magnitude,

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the overall magnitude
of an energy eigenstate.