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PROFESSOR: Let's do
it biogeometrically.

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There's a nice geometric
interpretation to this thing.

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So it comes from thinking
of this in phase space.

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So this is motion
in the x, p plain.

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So let's think of the
motion of this oscillator

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in the x, p plane.

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Here, we have that the energy is
equal to p squared over 2m plus

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1/2 m omega squared x squared.

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So this is an ellipse in the x--

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a closed orbit.

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A constant omega solution
is an ellipse in this plane.

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That's because it's
some something squared

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plus something squared
with different coefficient.

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So here it is.

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It's some sort of
ellipse like that.

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Semi-major axis,
semi-minor axis.

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I actually don't know which is
the major and which is a minor.

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But two semi axes.

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Well when p is equal to 0, what
is the value of x defines this.

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So a is the value of x when p0--

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so it's 2 square root of
2E over m omega squared.

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And b is the value of
p when x is equal to 0.

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So it's just square root of 2mE.

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And here is the particle doing
this motion in this orbit.

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As time goes by,
the position x goes

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from a maximum to a minimum and
return with the momentum going

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like that.

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It's a nice representation
of the physical motion

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as moving on the ellipse.

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That's what this
system is doing.

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So you could ask, when you
have something like that,

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you would ask, OK,
you have an ellipse,

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what's the area of the ellipse?

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Area is pi ab.

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That's the formula for
the area from an ellipse.

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Pi times the product of the
semi-major and semi-minor axes.

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That clearly generalizes
correctly to a circle.

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And it's the right formula.

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And then when we multiply
it, look what happens.

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There is-- do I have--

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yeah, I think I have
everything here--

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pi ab-- I get 2 pi.

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The m's cancel.

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The E's don't cancel.

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E over omega, hey, that's
our adiabatic invariant.

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2 pi E over omega,
the area of this thing

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is our adiabatic invariant.

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That's a very nice
classical picture.

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You have motion in phase space.

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And as omega changes, maybe
the ellipse will change.

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But the area tends
to keep constant.

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That's what's happening.

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That's a statement of this
result. So that's nice.

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It also can be written as a
formula, which is kind of neat.

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So the area of the
ellipse is this.

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But the area of the
ellipse, it can also

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be written in a
slightly different way.

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Let's assume the orbit is
going like this, for example.

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The motion is going like that.

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And then, what is the
area of the ellipse?

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The area of this top part
is the integral of pdx,

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is the integral of the top part.

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But we'll write it,
that's just the top.

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But if I think of
this as an integral

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over the whole
counter, I would be

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having the whole
integral of pdx like that

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into over the whole counter.

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I integrate here and I
get the area of the top.

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And when they
integrate down here,

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I'm having the x's
that are negative

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and p's that are negative.

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So I'm getting the area
of the bottom part.

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So actually this
full counter integral

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over the whole boundary gives
you the full area of the thing.

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The top area in one part, the
bottom area in the bottom part.

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So that's the area
of the ellipse.

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So we get the idea that
the integral of pdx

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is roughly equal to
2 pi E over omega.

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That's exactly equal when the
system is time independent.

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But then if it's not
time independent,

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this is an equation that can
help us think of this system

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and identify an
adiabatic invariant,

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because we identify
this quantity

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as an adiabatic invariant.

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The more general statement
in classical mechanics

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is that this kind of integral
is an adiabatic invariant.

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So in classical
mechanics, people

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search from adiabatic invariants
by integrals over phase space.

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It's a nice way
to think of them.

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But let's go quantum mechanical.

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It's the analogies that
we mentioned before.

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Let's use them.

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So here we go.

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We've said a little about this.

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And we'll say a bit more.

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So for quantum mechanics,
what do we have?

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Well, we had the oscillator.

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We mentioned it.

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And we said that E over omega
was h bar omega occupation

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number plus 1/2 over omega
and is therefore h n plus 1/2.

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So in quantum mechanics,
the adiabatic invariant

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becomes a quantum number.

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And the adiabatic theorem
in quantum mechanics

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is essentially going to say that
if you have quantum numbers,

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you are almost guaranteed, if
the system is slowly varying,

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to remain in that quantum state.

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We're going to try to make that
clearer, but that's the spirit.

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Quantum numbers don't change
under adiabatic approximation.

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Quantum numbers don't change.

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So in some sense, this whole
story was developed today.

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The classical
intuition, maybe it's

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a little less obvious, is
integrals over phase space,

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a trajectory in phase space of
a particle conserves the area.

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And here, in quantum
mechanics is the idea

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that if you have
a quantum number,

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you're going to find
it difficult to have

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a change in quantum numbers.

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But that's all we've
been doing with time

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dependent perturbation theory,
change of quantum number.

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So we'll think a little bit
about it why that happened.

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Now, there's more here
that is interesting.

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You remember your
WKB approximation.

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You did the quantization
when you had a system,

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say, with two turning
points, a and b.

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Bohr-Summerfeld
quantization, remember

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you assume there's a decaying
thing here, therefore

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a cosine in the middle
with a pi over 4,

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a decaying, another
close n with a pi over 4.

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And the compatibility gave you
the quantization condition.

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Now, what was that
quantization condition?

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It was 1 over h bar integral
of the local momentum px a/b

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equals n plus 1/2 pi.

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So if you multiply
by 2 this integral,

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that is the full integral
over the back and forth

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of p of x dx is equal to
2 pi h bar n plus 1/2.

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That formula, if you remember,
gave the oscillator exactly.

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But look how nice
you see the intuition

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that this quantity of classical
mechanics that we said

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is the idea adiabatic
invariant also

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shows in semi-classical
quantization telling you

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that that quantity,
yes, doesn't want

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to change because, in fact, it
represents a quantum number.

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So the WKB approximation is
also reinforcing the idea

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that first this quantity,
this integral over phase space

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is an adiabatic
invariant and second

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that it represents a
quantum number that

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doesn't want to change.

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Let me make one last
comment before we

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start a real calculation in
quantum mechanics about this.

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Transitions, we
studied transitions.

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And transitions are
the kind of things

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that don't happen
easily when you have

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an adiabatic approximation.

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So what did we have
for transitions?

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For transitions, we
had the probability

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to go from some initial
to some final state

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was the integral from
0 to t e to the i omega

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f i t prime delta h of t prime
f i over i h bar v t prime.

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Assume you have a
constant perturbation.

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So you can take this quantity
out of the integral, delta h

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f i of t prime squared
over h squared.

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And you get this integral of
e to the i omega f i t prime.

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And that integral
can roughly be done.

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And you've done it a few times--

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f i t minus 1 over
omega f i squared.

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So this is what we got for
a constant perturbation.

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And a constant
perturbation finds

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it hard to induce energy
jumping transition.

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00:14:11,170 --> 00:14:16,530
So if you have a
discrete system,

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making that transition is
hard, because however time you

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let go, this quantity is--

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maybe there's a square here.

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Yes, there's a square--

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this quantity is
bounded in time.

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This doesn't grow beyond
the particular quantity.

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But here you have a suppression
because the energies

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are different.

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00:14:39,300 --> 00:14:42,030
And if the energies
are fairly different,

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this is very suppressed.

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The way our calculation
escape that,

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we'd said, oh, if
you have a continuum

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behind these
discrete states, then

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00:14:56,070 --> 00:14:58,770
you can make a
transition because you

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don't have to have a
large change of energy.

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00:15:04,320 --> 00:15:11,220
So our transitions
are things that

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illustrate a little
bit what we're

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getting to that it's difficult
to change energy levels

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for slowly varying processes.

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In fact, if this Hamiltonian
was not exactly a time constant,

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00:15:27,810 --> 00:15:31,770
it's still difficult to bank a
transition, because, you know,

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if it varies slowly, this
is still roughly true.

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00:15:36,420 --> 00:15:38,730
Over some period of time,
you could say, well,

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it's the average value.

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00:15:40,440 --> 00:15:44,160
In order to get an efficient
transition between two energy

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levels, you had to
put the Hamiltonian--

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a cosine of omega t at
the right frequency.

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And then you induce
the transition.

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00:15:55,050 --> 00:15:59,070
But slowly varying
Hamiltonian finds

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it difficult to
induce transitions.

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All right, so this is the end of
our introduction to the subject

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of adiabatic evolution.

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00:16:16,240 --> 00:16:19,180
And now we're going
to try to calculate

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how a quantum state changes
under adiabatic evolution.