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PROFESSOR: There's one more
property of this thing that

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is important, and it's something
called the correspondence

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principle, which is another
classical intuition.

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And it says that
the wave function,

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and it addresses
the question of what

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happens to the amplitude
of the wave function.

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It says that the wave function
should be larger in the regions

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where the particle
spends more time.

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So in this problem, you have
the particle going here.

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It's bouncing and it's
going slowly here,

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it's going very fast here.

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So it spends more time
here, spends a lot of time

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here, spends a lot of time here.

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So it should be better
in these regions

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and smaller in the regions
that spends little time.

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So this was called the
correspondence principle,

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which is a big name for
a somewhat vague idea.

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But nevertheless, it's
an interesting thing

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and it's true as well.

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So let me explain
this a little more

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and get the key
point about this.

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So we say, if you have a
potential, you have x and x

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plus dx, so this is
dx, the probability

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to be found in the x is
equal to psi squared dx,

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and it's proportional
to the time spent there.

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So we'll say that it's--

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we'll write it in
the following way.

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It's proportional to the
fraction of time spent in dx.

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And that, we'll call little t
over the period of the motion

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in this oscillation.

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The classical particle is
doing, the period there.

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That's the fraction of
time it spends there.

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Up two factors of
2, maybe, because it

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spends going there and
there for the whole period,

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it doesn't matter, it's
anyway approximate.

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It's a classical intuition
expressed as the correspondence

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principle.

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So this is equal to dx over
v, over the velocity that

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positioned the
[INAUDIBLE] velocity T.

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And this is there for dx.

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And the velocity is
p over m, so the mass

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over period and the momentum.

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So here we go.

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Here's the interesting thing.

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We found that the magnitude
of the wave function

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should be proportional
to 1 over p of x,

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or lambda over h bar of x.

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So then the key result is
that the magnitude of the wave

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function goes like
the square root

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of the position the [INAUDIBLE]
de Broglie wavelength.

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So if here the de Broglie
wavelength is becoming bigger

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because the momentum
is becoming smaller,

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the logic here says that
yes indeed, in here,

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the particle is
spending more time here,

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so actually, I should be
drawing it a little bigger.

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So when I try to sketch a
wave function in a potential,

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this is my best guess
of how it would be.

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And you will be doing a lot
of numerical experimentation

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with Mathematica and get
that kind of insight.

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They position the [INAUDIBLE] de
Broglie wavelength as you have,

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it is a function of the
local kinetic energy.

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And that's what
it gives for you.

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OK so that is one key insight
into the plot of the wave

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function.

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Without solving
anything, you can

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estimate how the
wave length goes,

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and probably to what
degree the amplitude goes.

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What else do you know?

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There's the node theorem
that we mentioned, again,

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in the case of the square well.

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The ground state, the bounce
state, the ground state bounce

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state is a state
without the node.

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The first excited
state has one node,

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the next excited state has two
nodes, the next, three nodes,

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and the number of
nodes increase.

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With that information,
it already

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becomes kind of plausible that
you can sketch a general wave

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function.