WEBVTT
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PROFESSOR: This will
be qualitative insights
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on the wave function.
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It's qualitative,
and it's partially
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quantitative of course,
insights into, let's say,
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real energy eigenstates.
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So whenever you have a
problem and a potential,
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we have what is called the total
energy, the kinetic energy,
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and the potential energy.
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So you have the energy, which
is total, equal kinetic energy
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plus the potential energy.
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Now, the potential
energy, as you've seen,
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sometimes depends on position.
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We did piecewise
continuous end potentials,
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but they could be
more complicated
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and do funny things.
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So this is a function of x.
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And classically speaking,
we speak of the energy.
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You see in quantum mechanics,
the energy is an observable
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and is the result of a
measurement with a permission
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operator.
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Sometimes there could be
uncertainty, sometimes not.
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But in classical physics, which
this intuition will come from,
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you have a total energy.
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It is conserved.
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It's equal to potential
energy and kinetic energy.
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That will also depend
on where the particle is
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in the potential.
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Let's do a very simple case.
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Coordinate x, a potential.
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v of x, this based the
potential. v of x, it's
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a constant, nothing
that complicated.
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And suppose you
have a total energy.
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Now, the total energy
in classical mechanics
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is conserved.
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So when I draw a
line, I'm not implying
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that is a function of x,
that sometimes the energy is
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like that.
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No, it's just a number
there that I fixed.
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Here is the energy.
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And then wherever you
move, if the particle,
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the classical particle,
is here, then it
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has some potential energy, v
of x, and some kinetic energy,
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k of x, building up
the total energy.
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Classically, the kinetic
energy determines the momentum.
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The kinetic energy
is p squared over 2m.
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Now, the kinetic energy
is p squared over 2m.
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In this case, the kinetic
energy is a constant.
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The momentum will be a constant.
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And then what we really want
to just say something about
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is the wave function.
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Well, but if we note
the momentum classically
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it's a momentum p, we can
infer the Broglie wavelength
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of the particle.
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And that the Broglie
wavelength would be h over p.
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And that's for
the wave function.
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So we should expect
a wave function
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that has a wave length
equal to lambda.
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After all, that is
what the Broglie did.
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And from the Broglie, you
got the Schrodinger equation.
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The Schrodinger
equation, in fact,
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says this, if you
look at it again.
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So if you look at
the wave function.
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Well, it must have
wavelength lambda.
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And therefore, I'm talking
about real wave function.
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So it could be a
cosine or a sine that
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has that wavelength lambda.
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Of course in quantum
mechanics, a cosine or a side
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doesn't have exactly--
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it's not an eigenstate
of momentum.
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But it's an eigenstate
state of energy.
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And we want to plug
eigenstates of energy.
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So you will have something
like that, with that lambda.
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And that's intuition.
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You go from the diagram
to a kinetic energy,
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from a kinetic
energy to a momentum,
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from a momentum to a wavelength,
and that's the wavelength
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of your energy eigenstate.
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And maybe it's a good idea that
you try to convince yourself
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this is true by looking again
at the Schrodinger equation.
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For this simple case
of a constant potential
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and an energy that is big,
you will find this result
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very quickly.
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But let's do now a more
interesting case, in which here
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is x.
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And the potential is a
growing function of x.
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And there's a total
energy here still.
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So if you are at
some point here,
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here is the potential of x.
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And now, this is k of x.
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And now comes the
interesting thing.
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You see, as your particle,
or whatever particle,
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is moving here,
the kinetic energy
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is decreasing as you
move to the right.
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So the kinetic
energy [INAUDIBLE]
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velocity and slows down,
slows down, slows down.
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The kinetic energy is
becoming smaller and smaller.
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Therefore the momentum is
becoming smaller and smaller.
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And therefore the wavelength,
the Broglie wavelength,
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must be becoming
bigger and bigger.
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Now that is not exact
because you really
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have to solve the Schrodinger
equation to do this.
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But intuitively, you know that
if a potential is constant,
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this is absolutely true.
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The kinetic energy,
and the momentum,
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and the Broglie waveform
have related in this way.
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It will be sort of true,
or approximately true,
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if the potential is
not changing that fast.
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Because then it's
approximately constant.
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So there's a notion the
slowly changing potential,
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in which we can talk about the
k of x that is decreasing as we
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move to the right, a p of
x that is also decreasing,
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and a lambda of x that
would be increasing,
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a wave with the Broglie
wavelength that is increasing.
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Now I should have
written in here, maybe,
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k of x, p of x, lambda of x.
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This is decreasing,
decreasing, increasing.
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So I can plot it here.
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And I would say, well, I don't
know exactly how this goes.
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But maybe it's the
wavelength is small.
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And then the
wavelength is becoming
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bigger, something like that.
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Well, the wavelength's becoming
bigger in the energy eigenstate
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that you will find is true.
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But there's also the question
whether the amplitude
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of the wave will change or not.
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So we'll answer that
in a couple of minutes.
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But the Broglie
wavelength now is becoming
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a function of position.
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Now, you know that solving
the Schrodinger equation now
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with an arbitrary potential
is a difficult thing.
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With a linear potentially
it's a difficult problem,
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in which the exact solution
exists in terms of Airy
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functions and things like that.
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So this can only be an
approximate statement
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that the Broglie wavelength
is becoming bigger and bigger,
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because the momentum is
becoming smaller and smaller.
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But it's a very
useful statement.
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And whenever you look at
wave functions of potentials,
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you see that thing happening.
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Questions?
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Let me draw another diagram
that illustrates these issues.
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[SIDE CONVERSATION]
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So let's draw a general
picture of a potential
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now, so we can make
a few features here.
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So here it is.
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We'll have a potential that
is like this, v of x, maybe
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some energy, e, and that's it.
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Now what happens
classically, well,
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if your particle
has some energy,
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you know already
this part is v of x.
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This is k of x.
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There is a potential
energy and kinetic energy.
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The kinetic energy cannot
become negative classically.
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So the particle cannot go to the
left of this point called xl,
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x to the left.
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So this region, x left, is
the classically forbidden.
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Similarly on the right,
you cannot go beyond here.
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Because then you would have
negative kinetic energy.
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So this is an x right.
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And everything to the
right [INAUDIBLE],
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x right, is also
classical forbidden.
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These points, x
left and x right,
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are called turning points.
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Because those are
the points where
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a particle, a
classical particle,
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if it lives in this potential,
has to bounce back and turn.
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As we mentioned, at
any general point,
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you have v of x and k of x.
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And this point, for
example, is the point
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with maximum k of x
or maximum velocity.
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This is the point where the
particle is moving the fastest.
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And it always slows down as
it reaches the turning point.
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Because the kinetic energy is
becoming smaller and smaller.
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So as we said, if you
had a constant potential,
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this would be the solution.
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It's constant p,
constant lambda,
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nice, simple wave function.
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If it's not constant, well,
nothing is guaranteed.
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But if it's sufficiently
constant or slowly varied,
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then you're in good shape.
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Now what is the meaning
of slowly varying?
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The meaning of
slowly varying has
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to be said in a precise way.
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And this is what
leads eventually
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to the so-called WKB
approximation of quantum
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mechanics.
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Because we're giving you
the first results of this
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approximation that you
can understand classically
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how they go.
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To mean that you have a
slowly varying potential,
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is a potential whose
percentage change
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is small in the
relevant distances.
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So it's the change
in the potential
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over the relevant
distance must be small
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compared to the potential.
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But what is a relevant distance?
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If we use intuition
from quantum mechanics,
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it's at the Broglie
wavelength at any point.
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That is what the
quantum particle sees.
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So what we need
is that the change
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in the potential over at
the Broglie wavelength--
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take the derivative
multiply it by that.
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The Broglie wavelength
must be much smaller
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than the potential itself.
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And notice, of course, the
potential is a function of x.
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And even lambda is a
function of x there
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at the Broglie wavelength.
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Now, an exact solution will
not be a sine or a cosine.
00:14:21.590 --> 00:14:26.490
So to say has a precise defined
wavelength is an approximation.
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It is the approximation
of slowly varying.
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But it's a nice approximation.
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And this lambda
is the lambda that
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would come as h over p of x.
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And h over p of x is
the square root of 2m
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times the kinetic
energy over h squared--
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no, it's just that,
the square root of 2mk.
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Square root of 2mk of x.
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So the idea is that
you can roughly
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say that the Broglie wavelength
here is of some value here.
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The momentum is small if the
Broglie wavelength is large.
00:15:31.310 --> 00:15:34.380
And so when you draw
things, you adjust that.
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You say, OK, here,
the momentum is large.
00:15:38.710 --> 00:15:40.770
Therefore the Broglie
wavelength is small.
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So you write a short
wavelength thing.
00:15:43.230 --> 00:15:46.620
And then it becomes longer
wavelength and then shorter.
00:15:46.620 --> 00:15:49.620
And you just tried
to get some insight
00:15:49.620 --> 00:15:52.280
into how this thing looks.