WEBVTT

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PROFESSOR: So this is
our adiabatic change.

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So now we can say
several things.

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OK, if omega is changing slowly,
the energy is changing slowly,

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but do we have something that
changes even more slowly,

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something that really
almost doesn't change?

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What you need here
is basically--

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this was a very
important discovery

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in classical mechanics.

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You need like two
things that change.

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Everything is going
to change slowly,

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but then there's going to be
one thing that changes slowly

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and another thing
that changes slowly,

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and they change kind of in
the same way in such a way

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that the ratio or some
combination of them

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doesn't change almost at all.

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That's what we're trying to get.

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Anybody knows in classical
mechanics what quantity here

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doesn't change much?

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Nobody.

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No clue?

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It's not obvious what
doesn't change much,

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but here is the claim.

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Claim is that the quantity
that doesn't change much

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is, in fact, the energy
divided by omega.

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The energy will change slowly.

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Omega will change slowly.

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But the ratio is almost
not going to change at all.

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So here is the claim.

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There is an I of t called
adiabatic invariant, which

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is basically H of t
divided by omega of t,

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and it's almost constant.

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And this quantity has the
units of energy times time.

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I don't want to give
away the whole story.

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But I think it's good
if you, at this moment,

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think a second, well, what
could it mean, or do I even have

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a clue why this could happen?

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And you think oh,
quantum mechanics.

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The harmonic oscillator,
what happened?

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The energy was equal to
h omega times the level.

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So kind of energy
divided by omega

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is kind of a nice quantity.

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It's a quantum number.

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Quantum numbers are
quantized, and they

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don't like to
change, because how

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could an integer change slowly?

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As soon as it changes,
it changes big.

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So a little bit of
what we're getting at

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is the resistance of a system
to change quantum level.

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When something is quantized,
it cannot change slowly,

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and the adiabatic invariant is
exploiting in classical physics

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that quantum
property, if you wish.

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So let's look at that.

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So the claim is that the name
i is for adiabatic invariant,

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and we can verify it, and get
some intuition as to why those

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very slowly.

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Now, I cannot prove that
thing doesn't change.

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That would be too much, but it's
going to change very slowly.

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You will appreciate that.

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Let's see.

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Let's compute the
derivative, di dt.

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So it's a ratio.

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So I have omega squared.

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Omega.

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I'm going to use
dots, and I'm going

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to stop writing the
factor, the key dependents.

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Omega H dot minus H omega dot.

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So what do we have here?

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Omega, H dot was calculated
up there, m omega,

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omega dot x squared minus
H p squared over 2m minus--

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no, minus.

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Plus 1/2 m omega squared
x squared times omega dot

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over omega squared.

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And well, I still remember
when I first saw that.

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I probably wanted the numerator
to cancel and to do something

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very nice and simplify a lot.

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But it doesn't happen.

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So let's see what
really happens.

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Well, you have this
term, omega squared,

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omega dot, x squared
m, omega squared,

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omega dot, x squared m.

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But the factors of 2
don't make it cancel.

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So it's there.

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So let me write what we
get when we simplify this.

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di dt is equal to omega dot
over omega squared times

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1/2 m omega squared x squared
minus p squared over 2m.

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That term is clear.

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The p squared is that, and
here, we cancel the 1, partially

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with a 1/2.

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So we've got this.

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OK, so it doesn't look
like it wants to be 0.

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But it's still very good.

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Let's see why that
result is nice.

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Well, one thing you realize
here is that it actually

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gave you kind of
back the Hamiltonian

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with a different sine there.

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This is negative, and
this will remain positive.

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So let's write this, this
omega dot omega squared.

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And this is the kinetic energy
minus the potential energy,

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well, the potential
energy v of t

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minus the kinetic energy k of
t in the harmonic oscillator.

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Moreover, this quantity
is already very small.

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So this thing is very
small, but the fact

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that the adiabatic
invariant is adiabatic

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that it's really good,
should go beyond this.

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There should be something
suppressing about this factor,

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because you know, this
came from just the fact

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that things vary with omega dot.

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So what is happening?

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This is small and
slowly varying.

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This is neither small, nor
slowly varying, in fact.

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Why?

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Potential minus kinetic energy.

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The potential energy
in an oscillator

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goes up when the
kinetic energy is 0.

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I see the oscillator goes
to the end, stretches

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[INAUDIBLE] potential energy is
large, the kinetic energy is 0.

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As it goes through the
center, the equilibrium point,

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the kinetic energy is larger.

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So this is oscillating.

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And it's very large,
but now, you probably

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remember this fact about
the harmonic oscillators.

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While the potential and
kinetic energies oscillate,

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their averages are the same.

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So that's how this term
is going to help you.

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The average of this quantity
is roughly 0 over any period.

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And a period over a period,
this quantity changes little.

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So this is going to help us.

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Let me remind you here, suppose
you have an oscillation,

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an x equals sine omega t, then
the momentum would be m x dot,

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so m omega cosine omega
t, and the kinetic energy

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minus the potential energy, if
you do this little calculation,

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will go like omega
squared cosine of 2

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omega t, the v minus k.

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I leave for you that
little calculation.

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But it will go like cosine of
2 omega t, twice the period.

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And that thing tends
to have a 0 average.

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So let's see what happens now.

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The idt to see
what happens to it.

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Let's calculate I at t plus
the period minus I at some t.

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So let's see how much
I changes in a period.

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So from here, we
have the derivative.

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So we must do the integral
from t to t plus t of the dI dt

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prime dt prime.

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So this will be the
integral from t to t

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plus capital T of this whole
thing, omega dot over omega

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squared of t times v of t--

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it's all t prime, actually-- vt
prime minus kt prime dt prime.

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Let's see that.

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You have the
derivative of I, so you

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can calculate the change in I by
integrating with the derivative

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of I over mep.

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We've done that, and we've
asked how much does this thing

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change over a period.

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Then we have that I of
t plus t minus I of t,

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we have an integral
over a period.

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We set this quantity very slowly
and very little over a period,

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so roughly speaking, this is
equal to omega dot over omega

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squared at t.

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It didn't change much
over the integral.

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And then we have the
integral over a period

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of the potential energy
minus the kinetic energy.

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And for a normal oscillator
that is time independent,

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this quantity is strictly 0.

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If omega was not changing,
this would be identically 0.

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So if omega is changing
slowly, this quantity

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must be very close to 0.

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It's identically 0
when it doesn't change.

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Therefore, you see you got
an extra suppression factor.

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The change in the adiabatic,
so-called adiabatic invariant

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over time was already small,
because everything goes slow.

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But there is an
extra suppression

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due to the fact that
these two energies have

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the same average over a period.

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So you gain something.

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If the energy changes slowly,
this energy over omega

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changes even much
more slowly than that.

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So this is really exactly 0
for time independent omega,

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approximately 0 for slow
omega, slowly changing omega.

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So that's the extra
suppression factor,

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and that's what makes this an
adiabatic invariant, something

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that really changes
dramatically slower

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in a system in which everything
is already changing slowly.