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PROFESSOR: I want to just
elucidate a little more what

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are the eigenstates here.

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So with angular momentum,
we measure L squared

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and we measure Lz.

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So with spin, we'll measure
spin squared and Sz.

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And Sz is interesting.

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It would be spin or angular
momentum in the z direction.

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So let's look at
that, Sz, this is

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the operator, the measurable.

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It's this time nothing
else than a simple matrix.

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It's not the momentum operator.

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It's not angular momentum
operator with derivative.

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It's an angular
momentum operator,

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but it seems to have
come out of thin air.

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But it hasn't.

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So here it is.

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Oh, and it's diagonal already.

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So the eigenstates
are easily found.

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I have one state--

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I don't know how I
want to call it--

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I'll call it 1, 0.

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It's one state.

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And that's an eigenstate of it.

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We'll call it, for
simplicity, up.

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We'll see why.

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Sz, acting on up, is equal to
h bar over 2, 1, 0; 0, minus 1

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acting on 1, 0.

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That's h bar over 2.

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And the matrix is at 1, 0.

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So it is an eigenstate
because it's h bar over 2 up.

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The 1, 0 state again.

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So this thing, we call it up,
because it has up component

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of the z angular momentum.

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So it's a spin up state.

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What is the spin down state?

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It Would be 0, 1.

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It's a spin down.

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And Sz on the spin
down, it's also

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an eigenstate, this time with
minus h bar over 2, spin down.

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And we call it spin 1/2
because of this 1/2.

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And you'd say, no, you
just put that constant

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because you want it there.

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Not true, if I would have
put a different constant here

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in defining this, I would
not have gotten this

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without any constant, that it's
how angular momentum works.

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So if I use two-by-two matrices,
I'm forced to get spin 1/2.

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You cannot get anything else.

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The 1/2 of the spin
is already there.

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The component of angular
momentum is h bar over 2.

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If you have a photon,
it has spin 1.

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The components of angular
momentum is plus h or minus h,

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if you have the two
circularly polarized waves.

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So this is actually interesting.

54
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But it begs for another
question because we

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have a good intuition.

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And this is spin up
along the z direction

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because it has a Sz component,
eigenvalue h over 2.

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So the last question
I want to ask

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is, how do I get a spin state
to point in the x direction

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or in the y direction.

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You see, the interpretation
of this spin state

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is that it's a spin state
that has the spin up

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in the z direction,
because that's

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what you can
measure, or spin down

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in the down direction of Sz.

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Can I get spin states that
point along the x direction or y

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direction?

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And here's where the problem
seems to hit you and you say,

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I'm in trouble.

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I have this state spin
up and spin down along z.

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And it's a two-dimensional
vector space,

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because two-by-two matrices, and
Sx, Sy, Sz is three dimensions.

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How am I going to get
three dimensions out

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of two dimensions?

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You just have spin states
along z, up and down.

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Now the spin up and
spin dow, moreover,

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are orthogonal states.

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These two are orthogonal states.

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You see, you do the inner
product, transpose this,

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you get this, and times that.

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So they are
orthogonal, unless you

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imagine this vector
plus this vector

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is a full basis for
the vector space,

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because the vector
space is a, b.

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And now you see that this is
a times up plus b times down.

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So anything is a
superposition of up and down.

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So how do I ever
get something that

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points along x, or something
that points along y?

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Well, let's try to see that.

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Well, consider Sx, you
have an Sx operator, which

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is h bar over 2, 0, 1, 1, 0.

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And then you can
try to analyze this,

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but it's more entertaining to
imagine other things, to say,

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look, if I've gotten this
vector 1, 0, which is up,

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and 0, 1, which is down,
I can try maybe a vector

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that has the up and the down.

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Maybe the up and the down is
a vector that points nowhere.

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Who knows, whatever.

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If I want to normalize
it, I have to put a 1

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over square root of 2.

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And now I know, it's 1
over square root of 2,

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up, plus down.

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That's what this vector is.

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But let's see what
Sx does on it.

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Sx on 1 over square root
of 2, 1, 1 is h bar over 2,

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1 over square root of
2, 0, 1, 1, 0, on 1, 1.

107
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So h bar over 2, 1
over square root of 2.

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And let's see, that gives
1, that gives me another 1.

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Oops, I got the same
vector I started with.

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It an eigenstate.

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So this thing, this plus
and down, superimposed,

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is an eigenstate of Sx.

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So this is actually a
spin that points up,

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but in the x direction.

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Whenever we don't put anything,
we're talking about z.

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But this is the spin
up in the x direction.

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And these appeared
as the sum of a spin

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up and spin down
in the z direction.

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It may not be too
surprising for you

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to imagine that if you put
1 over square root of 2,

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1 minus 1, that vector is
orthogonal to this one.

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Yes, you do the transpose.

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And this one is orthogonal.

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So this is 1 over square
root of 2, up, minus, down.

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That is the down spin along x.

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So the up and down spins
along x come out like that.

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We form the linear combinations.

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So finally, you would say,
well, I'm going to push my luck

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and try to get spins
along the y direction.

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But I now form those
linear combinations.

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What else could I do?

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These linear
combinations are there.

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And I've got already two things.

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And you say, well, that's fair,
you're a two-dimensional vector

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space, so you're getting two
things, spin states along x

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and spin states along z.

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But actually, we didn't
run out of things to try.

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We could try a state of the
form 1 over square root of 2,

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something like this.

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We could try the state up.

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And then, we've put
a plus, but now we

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could put a plus i, state down.

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So this would be a
state of the form 1, i.

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And what does it do?

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Well, let's see what
it does with Sy.

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1, i.

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And the Sy matrix is h bar
over 2 minus i, i, 0, 0, 1, i.

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And there's 1 over
square root of 2.

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So it's 1 over square root
of 2, or h bar over 2,

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1 over square root of 2.

151
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And let's see what we get.

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Minus i times i is one 1.

153
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And the second one is i.

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We get the same state.

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Yes, it is an eigenstate.

156
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So with a plus i here,
this is this spin up

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along the y direction.

158
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And the spin down
along the y direction

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would be up, minus i, down.

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This is orthogonal
to that vector.

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It's 1 minus i.

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And it's the spin down
in the y direction.

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You can calculate
the eigenvalue,

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it's minus h bar over 2,
and it's pointing down.

165
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So your complex numbers
play the crucial role.

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If you didn't have
complex numbers,

167
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there was no way you could ever
get a state that this pointing

168
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in all possible directions.

169
00:12:31,200 --> 00:12:35,510
And you also see,
finally, that this thing

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has nothing to do with your
usual wave functions, functions

171
00:12:40,190 --> 00:12:42,620
of x, theta, phi.

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No, spin is an additional
world with two degrees

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of freedom, an extra thing.

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It doesn't have a
simple wave function.

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The spin wave functions are
these two column vectors.

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00:12:57,590 --> 00:13:02,330
But there is angular momentum in
there, as you discovered here.

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00:13:02,330 --> 00:13:05,790
There is a commutation
relations of angular momentum,

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00:13:05,790 --> 00:13:08,760
the units of angular
momentum, the eigenvalues

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00:13:08,760 --> 00:13:10,610
of angular momentum.

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And this great thing is
such a nice simple piece

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of mathematics.

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It has an enormous utility.

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It describes the
spins of particles.

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So it's an introduction,
in some sense,

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00:13:29,360 --> 00:13:32,900
to what 805 is all about.

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Spin systems are
extremely important,

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practical applications.

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00:13:38,690 --> 00:13:41,570
These things, because they
have basically two states,

189
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are essentially qubits
for a quantum computer.

190
00:13:46,780 --> 00:13:49,420
Within these systems,
we understand,

191
00:13:49,420 --> 00:13:52,330
in the simplest
way, entanglement,

192
00:13:52,330 --> 00:13:55,780
Bell inequalities,
superposition,

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00:13:55,780 --> 00:13:58,840
all kinds of very, very
interesting phenomena.

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So it's a good place to stop.