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PROFESSOR: We have
found in this solution

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with some energy like this,
that there's a decaying

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exponential over this side.

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And the question is
often asked, well what

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happens if you tried
to measure the particle

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in the forbidden region?

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Must be a problem.

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If you find the particle
in the forbidden region,

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it has energy E that
is less than v0,

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so you you have
found the particle

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with negative kinetic energy.

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How does it look?

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How can it happen?

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What's going on?

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Can you really find the particle
in the forbidden region?

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And then how does this negative
kinetic energy look like?

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The answer is that it's kind
of funny what happens here.

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You can make two statements.

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It would be contradictory,
contradictory

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if you could make--

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could say the following things.

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One, that the particle is
in the forbidden region,

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forbidden region.

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And two, that the particle
has energy less than v0.

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Because then it would mean
negative kinetic energy.

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So if you can say these two
things, it seems contradictory.

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So quantum mechanics
evades this problem.

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Now, this is not
discussed as far

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as I can see, except in
some lecture notes of Gordon

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[? Boehme. ?] And because the
argument is not 100% precise,

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but they think the spirit
of the argument is clear.

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So I want to share it with you.

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So here is the catch.

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This particle,
remember it's governed

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by e to the minus kappa x
is the forbidden region.

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So the length scale here where
you can find it, the particle.

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The length scale is,
this forbidden region

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stretches to about x of
the order 1 over kappa.

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If you are going to find it, it
is in the region of a distance

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1 over kappa.

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At 10 1 over kappa you're
not going to find it.

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The exponential is too small.

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But remember, what was kappa?

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Kappa squared was 2m v0
minus E over h squared.

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That's what it was.

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Now if you want
to see and declare

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that you have this
particle, you would

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have to be able to measure
position with some precision,

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with a precision a
little smaller than this.

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Otherwise if you
measure with precision

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10 times that, well
maybe it's to the left,

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maybe it's somewhere else.

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So you need to measure position
with delta x a little smaller

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than 1 over kappa, otherwise you
cannot really tell it's inside

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the forbidden region.

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But now the problem is that if
you do a position measurement,

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and you localize
the wave function,

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there is some
momentum uncertainty.

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The particle that
you're looking at,

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as opposed to the particle
to the left, has no momentum.

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It's a different kind
of wave function.

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There's no momentum
really associated

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or well-defined momentum to it.

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So because you make
a position, you're

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localizing x, whatever
wave function you have.

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You're going to have some
uncertainty, and some momentum

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that is going to be kind of
bigger than h bar over delta x.

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So a momentum that is bigger
than, or a little bigger,

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than h bar kappa.

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If delta x is less
than that inequality,

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it goes in the same direction.

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So there's going to
be an uncertainty P.

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And therefore, this particle
has now some kinetic energy

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due to this uncertain momentum.

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So uncertainty in the
kinetic energy is how much?

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It's P squared over 2m, where
P is this uncertain momentum.

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So this is equal to h bar
kappa squared over 2m,

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which is equal to v0 minus E.

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So actually, if you think
about it, here is v0.

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This difference is v0 minus
E. And you were going to say,

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oh, I found the particle, it
has negative kinetic energy.

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But no.

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The uncertainty principle
says, you found it localized?

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OK.

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Your kinetic energy,
I'm sorry, no.

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There's an uncertainty.

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How much?

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v0 minus E.

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So whatever you wanted to
prove, it has been disproved.

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You can't do it.

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The total energy,
total energy is now

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E plus the uncertainty
in the energy, which

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is E plus v0 minus
E. And it's therefore

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greater than or equal to v0.

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And no real contradiction.

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So the uncertainty
principle sort of

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conspires to prevent you
from finding a particle

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with negative kinetic energy.

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And if you do detect a particle
in the forbidden region,

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it will have total energy 0,
or total kinetic energy 0.

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It will be a normal particle.

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Nothing strange about it.