PROFESSOR: Today's lecture continues the thing we're
doing with scattering states.
We send in a scattering state.
That is an energy eigenstate that cannot be normalized
into a step barrier.
And we looked at what could happen.
And we saw all kinds of interesting things happening.
There was reflection and transmission
when the energy was higher than the barrier.
And there was just reflection and a little exponential decay
in the forbidden region if the energy was lower
than the energy of the barrier.
We also observed when we did the packet analysis
that a wave packet sent in would have
a delay in coming back out.
It doesn't come out immediately.
And that's the property of those complex numbers
that entered into the reflection coefficient.
Those complex numbers were a phase
that had an energy dependence.
And by the time you're done with analyzing how the wave packet
is moving, there was a delay.
So today we're going to see another effect that
is sometimes called resonant transmission.
And it's a rather famous.
Led to the so-called Ramsauer-Townsend effect.
So we'll discuss that.
And then turn for the last half an hour into the setup
where we can analyze more general scattering problems.
So let's begin with this Ramsauer-Townsend effect.
And Ramsauer Townsend effect.
Before discussing phenomenologically
what was involved in this effect,
let's do the mathematical and physics analysis
of a toy problem relevant to this effect.
And for that toy problem we have the finite square well.
We've normalized this square well.
Having width to a.
So it extends from minus a to a.
That's x equals 0.
And there's a number minus v0.
v0 we always define it to be positive.
Therefore, v0 is the depth of this potential.
And the question, is what happens if you send in a wave?
So you're sending in a particle.
And you want to know what will happen to it.
Will it get reflected?
Will it get transmitted?
What probabilities for reflection, what
probabilities for transmission?
So of course, we would have to send a wave packet to represent
the physical particle.
But we've learned that dealing with energy eigenstates
teaches us the most important part of the story.
If a particle has some energy, well roughly,
it will tend to behave the way an energy
eigenstate of that energy does, as far as reflection and as far
as transmission is concerned.
So we'll set up a wave.
And there's a coefficient A. So there's an Ae to the ikx.
That is our wave.
And it's moving to the right.
Because you remember the time factor,
e to the minus iet over h bar.
And when you put them together, you
see that it's moving to the right.
But presumably, there will be a reflection here.
The wave comes in, and some gets reflected
and some gets transmitted.
And the part that gets transmitted probably
will get partially reflected back here and partially
transmitted forward.
But the part that is partially reflected
will get again reflected here and partially transmitted back.
It seems like a never ending process
of which, if you think physically what's happening,
there's a reflection at the first barrier.
Now you say, wait a moment.
Why would there be a reflection?
Classically, there would never be a reflection.
If the potential goes down, the particle
would just be able to continue.
We had reflection when we had a barrier.
Well in quantum mechanics, any change in the potential
is bound to produce a reflection.
So yes, if you have a potential like this,
like a jumping board, and you come in here,
there's a tiny probability that you
will be reflected as you come into this potential drop.
So OK, so this will be reflection.
So at the end of the day, there might be many bouncings.
If you imagine a particle doing this.
Some probability of reflection, some of transmission.
But the end of the day, there will be some wave
moving to the left here.
So we'll represent it by Be to the minus ikx.
So in this region, we have A and a wave
with amplitude B going this way.
In the middle region, the same will be true.
There will be some wave going here,
and some wave that bounces due to this reflection.
So there will be a--
I don't know what letters I used.
I'd better keep the same.
C in this direction and D in this direction.
And now I would have Ce to the ikx.
Well, that e to the ikx is not quite right.
Because k here, k squared represents the energy.
k is the momentum and k squared is energy.
So if you have an energy eigenstate-- oh,
my picture is very crowded.
So I'll do it anyway.
A is here.
Maybe I'll put C and D here.
And now with A and B here I can write this as the energy.
You have a particle with some energy coming in.
And there is this wave here and k squared.
It's 2mE over h squared.
E is positive scattering states.
And indeed, E from that equation is h squared k squared over 2m.
What do you know?
But at this point, the total energy,
kinetic energy of the particle is bigger.
If e is replaced by e plus v0, which
is the magnitude of this drop.
So here there will be a k2x plus De to the minus ik2x.
And k2 refers because it's region two, presumably.
People use that name. k2 squared will
be 2me plus v0 over h squared.
And finally, to the right of the potential square
well there will be just one wave.
Because intuitively, we should be
able to interpret this as some wave that
goes through, but has nothing to make it bounce or reflect.
So we try to get the solution, which
will have just some wave going in this direction.
And it's called Fe to the ikx.
And I can go back to the label k because you
have the same energy available as kinetic energy
you had to the left of the barrier.
OK, so we've set up the problem.
The wave function I would have to write it
as three expressions.
One for x less than a, one for x in between a and minus a,
and one for x greater than a.
And those are this one, two, and three formulas.
Any questions about this setup so far?
OK.
Well, at this moment you will eventually
have some practice on that.
The thing that you want to do is relate the various coefficients
and define some reflection and transmission coefficients.
We saw we had to think in terms of probability current.
That's the better way to get an idea
of what you should call reflection or transmission
coefficient.
So we have to be careful with the case of the step potential
when we compared the meaning of the wave that
was moving to the right.
The amplitude divided by the incoming wave
was not quite the transmission coefficient.
But in this case, the nice thing is
that the wave to the left and the wave to the right
are experiencing the same potential.
So they can be compared directly.
So I will be able to conjecture, it's reasonable to conjecture
that we can define reflection and transmission
coefficients as follows.
We'll have a reflection coefficient
should be B over A squared.
B represents a reflected wave, A the incoming wave.
The transmission coefficient you may guess
that it's F over A squared.
And this all will make sense if we
have the reflection plus the transmission is equal to 1.
So we have current conservation, conservation.
And the current, which is the net probability
flow to the left of the barrier or the depression over here,
is J on the left is proportional to A squared minus B squared.
You remember, we computed it last time.
If you compute the probability current to this Ae to the ikx
plus Be to the minus ikx, you get two contributions.
Essentially A squared minus B squared.
There's a factor of h bar k over m in front.
At any rate, this should be equal to the current that is
flowing out in this direction.
You see, whatever current is coming in to the left
must be the current going out to the right.
So this is f squared.
So current conservation really tells you
that A squared minus B squared is equal to F squared.
And if you pass the B to the other side,
you get A squared equal B squared plus F squared.
And dividing by A squared you get
1 is equal to B over A squared plus F over A squared.
And that's the reflection plus the transmission
So the way we've defined things makes sense.
Reflection and transmission are properly defined.
And this is because current conservation works well.
So reflection coefficient is essentially
the flux in the reflected, the probability current
or in the flux in the reflected wave compared
to the flux in the incoming wave.
The transmission is the probability current or flux
of probability in the transmitted wave compared
with the one incoming wave.
So you've done all of this set up.
Now you can't avoid, however, doing
a little bit of calculation, which is boundary conditions.
You have one, two, three, four, five variables.
And somehow, you want to calculate these ratios.
So you have to say that the wave function and the derivative
is continuous at this point.
And the wave function and the derivative
is continuous and this point.
That would give you four conditions.
And that's reasonable.
We have five variables.
But you know, the overall normalization
could never be determined.
So things can be determined in terms of A.
So you can expect that with four equations
you can solve for B, C, D, and F in terms of A.
But the overall scale of this total wave function
is undetermined.
Its boundary conditions will give you constraints,
but it will never determine the magnitude,
the overall magnitude of an energy eigenstate.