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PROFESSOR: We're
going to be talking

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about adiabatic approximation.

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It's an interesting
approximation

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that we can do when
some physical parameters

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of your system change slowly.

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And sometimes, things
don't change slowly.

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For example, if you
have a uniform--

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a non-constant magnetic field.

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OK, that's spatial
variation, no time variation.

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But suppose you send
the particle in.

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The particle sees a
changing magnetic field,

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because it's going from one
region of low magnetic fields,

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say, to a region of
high magnetic field.

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So if the particle is
going at slow velocity,

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it's seeing a slowly-changing
magnetic field,

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even though the magnetic
field itself is not changing.

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So there are several
circumstances

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in which slow variations
can be important.

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There can be cases where
the magnetic field is slowly

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changing in time.

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You have a spin, and
the magnetic field

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in which the spin is
located is slowly changing,

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and then you want to
know what happens.

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So when a physical
quantity in your system,

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in your Hamiltonian,
changes slowly,

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you would expect that
your dynamical variables,

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to some degree, will
change slowly as well.

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And they will adjust
themselves in a slow manner

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to these changes.

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And that's all true.

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And we kind of expect it.

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But there are some
things that actually

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change very slowly, not just
slowly, but very slowly.

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And those are things
that are adiabatic.

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Adiabatic changes correspond
to things that change,

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due to some reason, even more
slowly, perhaps, than you

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would have anticipated.

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And that's the kind
of surprising thing

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that we need to understand.

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So we will begin with an
example in classical mechanics

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to illustrate this point.

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And so our subject is
adiabatic approximation.

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And we go, then, to
classical mechanics

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to get a hint of
what can happen.

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And a typical example
could be you're here,

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and you have a pendulum.

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And the pendulum
is going this way.

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But you raise up
and down your hand--

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lift up and down your hand, so
that the length of the pendulum

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varies.

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And therefore, the frequency
of oscillation varies.

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So this is a very
simple system, in which

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you have a Hamiltonian
that depends on x and p,

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and a frequency that
depends on time.

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You've prescribed it.

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It's some frequency
that depends on time.

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So here it is. p squared over
2m plus 1/2 m omega squared of t

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x squared.

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OK, so omega is
changing in time.

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And x and p--

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this is classical motion, so
it's all classical physics.

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This is going to be
classical motion.

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Therefore, x and p are going
to be functions of time.

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Well, let's see
how things change.

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So let's calculate what would be
the change of the Hamiltonian,

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or the total energy
as a function of time.

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Now this is a
time-dependent Hamiltonian.

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There's no such obvious thing
as conservation of energy.

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You're doing some work here
that is changing omega.

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So the amount of energy that
this system has will change.

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So time-dependent
Hamiltonian like

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that, the energy of the
system changes here.

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So let's calculate
the change in time.

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So x and p change in time.

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And therefore, this changes
in time, x changes in time,

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omega changes in time, and
the energy changes in time.

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So let's calculate how this
energy changes in time.

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So what should we do?

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We should do dH dx
times the rate of change

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of x because H depends on x.

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dH dp times the
change in time of p.

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We use dots for
time derivatives.

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And then finally, we also
have to differentiate

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H, with respect to time
to take into account

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the variation of omega,
which is a parameter here.

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So all these things
must be done.

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And let's assume these
are functions of time

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because the system is
doing physical motion.

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So we're trying
to investigate how

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the energy changes if the
system is doing physical motion.

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And here is the 0-th order
result. If somebody would say,

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OK, the frequency is
going to change slow,

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then you could say,
OK, adiabatic result

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is that the energy is
going to change slowly.

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True, but not too interesting.

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We're going to do better than
that, much better than that.

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So let's think a little more.

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If the motion is satisfying
the equations of motion--

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this is physical motion
we're trying to understand,

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how the energy changes
as this particle,

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its doing its motion--

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we can use equations of
motion of the system.

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These are the Hamiltonian
equations of motion that

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say dH dp is equal to x dot.

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And dH dx is equal
to minus p dot.

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These are the
Hamilton's equations

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of motion of
classical mechanics.

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It is a property of
your education at MIT

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that you probably are
less familiar with those

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than the quantum equivalent.

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So let me remind you of
the quantum equivalent.

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If you have iH
bar d dt of x, you

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remember that by Ehrenfest, this
is xH commutator expectation

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value.

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That's Ehrenfest theorem.

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And here, you also
remember that x

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can be thought as iH bar d
dp, such in the same way as p

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can be thought as
H bar over i d dx.

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So this commutator,
the way we compute it,

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is as iH bar d dp of
H expectation value.

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And that's-- cancel the iH's.

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And here, you've got x
dot is dH dp, the quantum

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version of that.

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You have also seen
that iH bar d dt of p

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is equal to expectation
value of p with H.

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And that would be H
bar over i dH dx using

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that p as that derivative.

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And that, canceling
the H bars and noticing

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you have an i and
a 1 over i here,

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gives you this equation
with a minus sign.

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So these are
Hamilton's equations

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from classical mechanics that,
in case you have not seen them,

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you know they're
quantum analogs.

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And you probably believe
that that is the case.

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Now, that has an important
consequence on all of this,

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that the first two terms
in this expression cancel,

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because dH dx gives
you a minus p dot.

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So you get minus p dot x dot.

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And from here, you get p dot x
dot, so the two terms cancel.

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And we get that the
dH dt is just dH dt.

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It's kind of nice because if
there was no explicit time

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dependence in the Hamiltonian,
the energy should be conserved.

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And therefore, it's nice
that all that is left

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is just dH dt, which we can
evaluate from the formula

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up there.

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We just have to
differentiate omega.

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So this gives you m omega
omega dot x squared.

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It's a result that
is of interest.

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So, so far so good.

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The energy will change.

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And if omega changes slowly,
the energy will change slowly.

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Nothing too dramatic.

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We need to do better.

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So let's think more
precisely, what

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do we mean by adiabatic change?

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So basically, it means that
the time scale for change

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is much bigger than the time
scale for an oscillation

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of your system.

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So this omega is going
to change in time.

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But now you can ask-- given an
omega there is a period, maybe

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a second.

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So probably,
adiabatic change will

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hold if the change in omega
is small over a second--

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if the change in omega
maybe happens over a year,

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with omega being
about one second.

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So this is a little like we
had in the WKB approximation.

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I'll make the point here.

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So here is, for example, a
typical graph that one draws.

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Here is omega of t.

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And that's a constant,
then it changes a bit.

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And it stabilizes after a
while in some time, tau,

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for a finite change
in omega of t

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from some initial value
to some later value.

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In this case, the
timescale from change, tau,

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should be much
bigger than the time

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period of isolation,
which is 2 pi over omega

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of t, which is the period.

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But let's be a
little more precise,

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like the way we did
for the WKB case.

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What did we have
in the WKB case?

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We said that the change
in the Broglie wavelength

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over the Broglie
wavelength was much smaller

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than to the Broglie wavelength.

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Or the change in energy
over the Broglie wavelength

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was much smaller than
the energy of the system.

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So here, we can say the
change in omega of t

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over a period is much
smaller than omega of t.

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So how do we say that?

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Here is the rate
of change of omega.

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This is, roughly, the change
in omega over a period.

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So rate of change in
omega-- this is the period,

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so that's, roughly, the change
in omega over a period--

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must be much smaller
than omega itself.

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So this is 2 pi
over omega squared.

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d omega dt is much
smaller than 1.

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And we can write it
in two different ways.

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This says, actually, that
omega dot over omega squared

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is much less than 1.

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That's one way of writing it.

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Forget the 2 pi.

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And the other way is
by saying that this

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is d dt of 2 pi over
omega much less than 1.

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That's correct, because when
you differentiate 1 over omega,

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you get d omega dt times
1 over omega squared.

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And therefore, this is dT
period dt is much less than 1,

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which might remind you
of the WKB's d lambda dx

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was much less than 1.

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This was WKB, very,
very analogous.