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.
