PROFESSOR: There's one more property of this thing that
is important, and it's something called the correspondence
principle, which is another classical intuition.
And it says that the wave function,
and it addresses the question of what
happens to the amplitude of the wave function.
It says that the wave function should be larger in the regions
where the particle spends more time.
So in this problem, you have the particle going here.
It's bouncing and it's going slowly here,
it's going very fast here.
So it spends more time here, spends a lot of time
here, spends a lot of time here.
So it should be better in these regions
and smaller in the regions that spends little time.
So this was called the correspondence principle,
which is a big name for a somewhat vague idea.
But nevertheless, it's an interesting thing
and it's true as well.
So let me explain this a little more
and get the key point about this.
So we say, if you have a potential, you have x and x
plus dx, so this is dx, the probability
to be found in the x is equal to psi squared dx,
and it's proportional to the time spent there.
So we'll say that it's--
we'll write it in the following way.
It's proportional to the fraction of time spent in dx.
And that, we'll call little t over the period of the motion
in this oscillation.
The classical particle is doing, the period there.
That's the fraction of time it spends there.
Up two factors of 2, maybe, because it
spends going there and there for the whole period,
it doesn't matter, it's anyway approximate.
It's a classical intuition expressed as the correspondence
principle.
So this is equal to dx over v, over the velocity that
positioned the [INAUDIBLE] velocity T.
And this is there for dx.
And the velocity is p over m, so the mass
over period and the momentum.
So here we go.
Here's the interesting thing.
We found that the magnitude of the wave function
should be proportional to 1 over p of x,
or lambda over h bar of x.
So then the key result is that the magnitude of the wave
function goes like the square root
of the position the [INAUDIBLE] de Broglie wavelength.
So if here the de Broglie wavelength is becoming bigger
because the momentum is becoming smaller,
the logic here says that yes indeed, in here,
the particle is spending more time here,
so actually, I should be drawing it a little bigger.
So when I try to sketch a wave function in a potential,
this is my best guess of how it would be.
And you will be doing a lot of numerical experimentation
with Mathematica and get that kind of insight.
They position the [INAUDIBLE] de Broglie wavelength as you have,
it is a function of the local kinetic energy.
And that's what it gives for you.
OK so that is one key insight into the plot of the wave
function.
Without solving anything, you can
estimate how the wave length goes,
and probably to what degree the amplitude goes.
What else do you know?
There's the node theorem that we mentioned, again,
in the case of the square well.
The ground state, the bounce state, the ground state bounce
state is a state without the node.
The first excited state has one node,
the next excited state has two nodes, the next, three nodes,
and the number of nodes increase.
With that information, it already
becomes kind of plausible that you can sketch a general wave
function.