PROFESSOR: We can write, however, J left as J of A,
minus J of B. Where J of A would be h bar k over m A squared.
And J of B would be h bar k over m B squared.
You see the current that exists to the left of the barrier
has two components and--
it's very intuitive.
It's the current that would have been brought alone
by the incoming A-wave.
Minus the current that would have existed alone
from the reflected B-wave.
B is the reflected wave.
So that's very nice.
There's no interference between these two terms.
We can really think of a current that
is associated to the incoming wave,
and a current that is associated with the reflected wave.
So this suggests how you should define a reflection
coefficient.
Reflection coefficient R would give me
the amount of current I get reflected,
compared to the amount of current that there is incident.
You see, the incident current is going
to be partially reflected and partially transmitted.
So an idea of a reflection is the value
of the reflected current divided by the incident current.
It's a definition, but it's a reasonable definition.
And then, if it is this ratio--
because of these expressions-- it
happens to be B over A squared.
And that's an interesting number.
Now, there's some physics in it.
It tells me how much of the probability
gets reflected as a function of the probability that
is incident.
So that's a good measure.
If you get a reflection coefficient of 1/10,
then you would expect 1/10 of the particles to be reflected.
Now we don't have particles yet.
This is non-normalizable solution.
But, still, this will be the intuition very soon.
Now we could have a transmission coefficient, as well.
And here is something where we sometimes make a mistake.
T is going to be the transmission coefficient.
Transmission coefficient.
And how should we define it?
There is a temptation to define it-- well, coefficient B over A
gives me this.
Then maybe it should be C over A. But actually--
while c over a gives you some idea
of how big the wave to the right is compared
to the wave of the left--
that's not what we should call a reflection coefficient.
And the reason is that--
I will call this current J C. And that's
the amount of probability--
because it's a current associated to the wave C.
And that's the amount of probability
that is being carried by the transmitted wave.
That is the probability.
Not necessarily C over A.
So the transmission coefficient will
be defined to be J C divided by J A.
And then J C divided by J A--
J C has an h-bar, k-bar.
And J A has a k.
So this is not equal to this ratio,
but is actually k-bar over k, C over A.
So it's not just this number.
The reflection coefficient-- and transmission coefficients--
really originate from probabilities.
And the probabilities for this current.
And, therefore, there is no--
it would have been very hand-wavy, and actually
wrong to think it's C over A.
These definitions-- because after all,
this is a definition--
makes some nice sense.
Because you have J L--
we said is equal to J right, but J L is J A minus J
B, is equal to J C. And therefore R plus T.
The reflection coefficient plus the transmission coefficient--
which is J B over J A plus J C over J A
is equal to J B plus J C over J A.
And you see here that J A plus J C is indeed equal to J B.
I'm sorry.
I got this wrong.
Yes.
Where is the eraser.
I should have passed the B to the other side.
This force implies J A equals J B plus J C.
And this ratio is equal to 1, which is something you usually
want when you define reflection and transmission coefficients.
They should add up to 1.
So now we've got an idea.
Yes, with this solution I can understand the reflection
and transmission coefficients.
But do these apply to particles?
Well, the good news is that it roughly
applies to particles-- as we will see with the wave packets
soon.
If you send the wave packet, it's
going to have some uncertainty and momentum.
It's going to have some uncertainty and energy.
But for some energy--
suppose the wave packet doesn't have that much uncertainty--
basically, the probability that the wave packet bounces
is the reflection at that energy, that is the main energy
that the wave packet sends.
If your wave packet is very broad over energies,
then it's a more complicated thing.
But as long as the wave packet is such that it basically
has one narrow band of energy, the reflection coefficient
associated to this calculation is the reflection coefficient
or reflection probability for the wave packet
that you're sending in.