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00:00:01,180 --> 00:00:04,990
We spoke about
superposition, and we

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showed how, when you have two
states that are superimposed,

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the resulting state
that is built up

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00:00:15,730 --> 00:00:20,410
doesn't have properties that
are intermediate between the two

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00:00:20,410 --> 00:00:22,210
states that you're
superimposing.

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But rather, when you
do a measurement,

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you obtain the result
that you would sometimes--

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you sometimes obtain
the result that you

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would have with
one of the states,

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and some other times with
different probabilities,

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the result as if you
had the other state.

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So it's a strange kind of
way in which things are

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combined in quantum mechanics.

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So the next thing we have to
say is a physical assumption

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that is made here.

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And it is that if
you have a state

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and you superimpose
it to itself,

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you haven't done anything.

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So the superposition
of a state with itself

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has no physical import.

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So we can say this.

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A physical assumption
superimposing a state to itself

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does not change the physics.

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So if I have a state, this
is physically equivalent--

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I'll write physically
equivalent with this symbol--

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to the state a plus a,
which would be 2 times a.

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It's physically equivalent
to the state minus a.

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It's physically equivalent
to the state ia on anything.

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It's not equivalent
to 0a, because that

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would be the zero state.

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So it's physically
equivalent as long

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as you have a
non-zero coefficient.

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All these states are supposed
to be physically equivalent.

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And that will eventually
mean that we sometimes

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choose a particular one in
those collection of states

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that is one that is
convenient to work with.

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And that will be called a
normalized state, a state that

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satisfies other
properties having

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to do with the norm
squared of the state.

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That will come later.

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But it's important
that the number that

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is multiplying the physical
state of your system

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has no relevance.

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And you could say, well, why all
of the sudden you tell us this.

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Could this be shown
to be necessary?

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Or it's a physical
assumption, so can we test it?

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Does it make some sense?

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And we can make some sense of
this assumption at this level.

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And we do it with
states of light.

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So remember, we spoke about
photons hitting a polarizer.

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And we could speak of two
independent kind of photons--

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photons polarized along the
x-axis and a photon polarized

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along the y-axis.

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And those are two quantum
mechanical states.

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Now suppose I decide
to superimpose

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those states to create the
most general photon state.

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I would have an alpha,
which is a number here,

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a complex number,
and a beta there.

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And I would say, OK, here is
my most general photon state.

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And how many parameters
does this state have?

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It has two complex parameters,
alpha and beta, and therefore,

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four real parameters.

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And then you think about
polarization states,

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how many parameters they have.

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And as we'll review in a second,
it's well known that photons--

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their polarization
state can be expressed

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with just two real parameters.

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So some counting is not
going very well here.

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But here comes the help.

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If the overall coefficient
here doesn't matter--

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if I can change it, I
can multiply everything

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by 1 over alpha, and therefore
get that the state is just

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the same, physically
equivalent to this state,

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beta over alpha photon y.

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So all the physics is contained
in this state as well.

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And if all the physics is
contained in that state,

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I must look how many
parameters it has.

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It still looks like
there's two numbers here,

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but only the ratio appears.

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So if you call beta over
alpha, the number gamma

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is just one complex parameter.

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And therefore, thanks
to this assumption,

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you now get that the most
general photon polarization

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state has just one
complex parameter, or just

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equivalently, two
real parameters.

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And that is the correct number.

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Indeed, if you have a
polarization, a wave that

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has some polarization, the
most general polarization state

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of a wave is an
elliptical polarization.

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You probably did study a lot
about circular polarizations,

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or maybe you also heard
about the elliptical one

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in which the electric field--
in a circular polarization,

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the electric field at any
point traces a circle.

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But if you have an
elliptical polarization,

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the electric field
traces an ellipse.

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And that ellipse has an
angle that is one parameter.

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And for an ellipse, the other--

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the size doesn't matter.

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The size depends
just on the magnitude

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of the electric field.

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It's not a parameter of the
polarization of the wave.

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Since the size
doesn't matter, it's

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the shape of the
ellipse that matters.

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And that's characterized by the
eccentricity or by the ratio

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a over b of the semi-major axis,
so parameters, two parameters,

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and they are a over b and theta.

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So an elliptically
polarized wave,

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which is the most general state
of polarization of a wave,

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has two real parameters.

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And now, thanks to this physical
assumption, we get this right.

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And this is important
because that's

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something we're going
to use all the time,

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that the overall factor in a
wave function does not matter.

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So if we have
superpositions, I want

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to emphasize one more
thing about superpositions.

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And for that, I'm
going to use spins.

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So what is spin?

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Spin is a property of
elementary particles

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that says that actually,
even if they're not

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rotating around
some other particle,

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they have angular momentum.

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They have intrinsic
angular momentum,

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as if they would be made
of a tiny little ball that

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is spinning.

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I say as if because
nobody has ever

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constructed a model of
an elementary particle

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where you can really make
it spin and calculate

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how it works.

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Somehow, this
elementary particle

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has angular momentum is born.

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Even if it is a point particle,
it has angular momentum,

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and it's spin.

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And spin is very
quantum mechanical.

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And we can't quite
understand it without it.

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So what happens is that you can
measure the spin of a particle.

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And then if you
measure it, you have

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to decide, however, since
angular momentum is a vector,

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what direction you should use.

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And suppose you
use the z direction

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to measure the
spin of a particle.

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You may find that the
particle has either spin up

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or the particle has spin down.

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Spin.

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And the spin is the direction
of the angular momentum.

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And that's a funny thing
that happens with most matter

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particles.

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These are spin 1/2 particles.

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The spin can be up or it can
be down along the z direction

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that you measure.

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You measure it, and you never
find it's 0 or a little bit.

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It's just either up
with the full magnitude

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or down with the full magnitude.

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That is a spin 1/2 particle.

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And the state where it is
up, we sometimes denote it

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with an arrow up and call it
z because it's up along z.

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And this would be down,
an arrow down along z.

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If those are possible
quantum states,

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you could build a
new quantum state

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by superposition which would be
up along z plus down along z.

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Now, if I wish to
normalize it, I

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would put the factor
in front of this.

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I will not talk about
normalizations at this moment.

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They're not so important.

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If you are faced with
this quantum state--

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so suppose you have an electron
that is not in this state

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nor in this state, but is
in this state, in a quantum

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superposition.

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So you go and you decide
to try to measure it.

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Now, since you cannot predict
what that electron is going

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00:12:00,110 --> 00:12:01,280
to be doing--

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we cannot predict things
in quantum mechanics with

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certainty--

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we, since we're going
to do this experiment,

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avail ourselves of 1,000 copies
of this electron, all of them

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in this peculiar quantum state.

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So you have the 1,000 copies,
and you start measuring.

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And you decide to measure
the spin in the z direction.

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And now what do you get?

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Well, we mentioned last time
that you don't get an average,

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or since this is up and
this is down, you get 0.

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You measure the first
particle and you find it up.

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Measure the second, up, the
third, up, the fourth, down,

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five, down.

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And then you get a
series of measurements.

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At the end of the 1,000
particles, you find about 495

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up and 505 down,
about half and half.

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And if you did it
with 10,000 particles,

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maybe it would be closer.

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Eventually, you'll
find 50% in this state

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and find 50% in this state.

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And if you think this is
strange, which you probably do,

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well, you could be justified.

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But here would
come Einstein along

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and would say all this
stuff of this superposition

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is not quite right.

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00:13:55,610 --> 00:13:59,280
You had this 1,000 particles.

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00:13:59,280 --> 00:14:04,940
But actually, those 1,000
particles, half of them

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00:14:04,940 --> 00:14:09,300
were with a spin up and half
of them were with a spin down.

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00:14:09,300 --> 00:14:11,600
So here you have
your 1,000 particles,

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00:14:11,600 --> 00:14:14,030
your quantum state, this.

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00:14:14,030 --> 00:14:20,150
But Einsten says, no, let's make
an ensemble of 1,000 particles,

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00:14:20,150 --> 00:14:25,730
500 up, 500 down, and
do the same experiment.

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00:14:25,730 --> 00:14:27,870
And the result is
going to be the same.

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00:14:27,870 --> 00:14:31,980
So how do you know you
really have this as opposed

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00:14:31,980 --> 00:14:39,080
to somebody has given you 1,000
particles, 500 up, 500 down?

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00:14:39,080 --> 00:14:41,520
How can you tell?

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00:14:41,520 --> 00:14:44,120
And in fact, he
would say even more--

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00:14:44,120 --> 00:14:49,310
whenever Einstein
used the word realism

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00:14:49,310 --> 00:14:55,130
to say if I measure a
spin and I find it up,

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00:14:55,130 --> 00:15:01,600
it's because before I
measured it, the spin was up.

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00:15:01,600 --> 00:15:04,160
It's almost like learning
something about an object.

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00:15:04,160 --> 00:15:08,410
If I look at this page
and I find the color red,

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00:15:08,410 --> 00:15:11,740
it's because before I
looked at it, it was red.

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00:15:14,510 --> 00:15:17,870
But then in quantum
mechanics, that

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00:15:17,870 --> 00:15:19,590
doesn't seem to be the case.

216
00:15:19,590 --> 00:15:22,500
The state is this mix.

217
00:15:22,500 --> 00:15:25,460
And it was this mix
before you measured.

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00:15:25,460 --> 00:15:27,800
And after you
measure, it's this.

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00:15:27,800 --> 00:15:31,130
So there is no such
thing as you learn

220
00:15:31,130 --> 00:15:36,810
by doing one measurement what
the state of the particle was.

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00:15:36,810 --> 00:15:41,450
So we will not resolve
Einstein's paradox

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00:15:41,450 --> 00:15:45,540
completely here because we would
have to learn more about spins,

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00:15:45,540 --> 00:15:48,500
which you will do soon enough.

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00:15:48,500 --> 00:15:53,090
But here's the catch
that actually happens.

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00:15:53,090 --> 00:15:58,640
If, instead of having an
ensemble of quantum states,

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00:15:58,640 --> 00:16:03,080
you would have an ensemble of
those states that half of them

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00:16:03,080 --> 00:16:05,570
are up and half
of them are down,

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00:16:05,570 --> 00:16:10,550
you could now decide to measure
the spin of the particle

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00:16:10,550 --> 00:16:13,670
along the x direction.

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00:16:13,670 --> 00:16:18,630
You take these particles,
and you measure along x.

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00:16:18,630 --> 00:16:21,510
And what you will calculate
with quantum mechanics

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00:16:21,510 --> 00:16:22,590
later in this course--

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00:16:22,590 --> 00:16:26,610
if you measure along
x, in this state,

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00:16:26,610 --> 00:16:32,880
you will find all of them to
be pointing along plus, up

235
00:16:32,880 --> 00:16:36,370
along x, all of them.

236
00:16:36,370 --> 00:16:43,990
While on this Einstein ensemble
of 50% up and 50% down,

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00:16:43,990 --> 00:16:50,230
you would find 50% up along
x and 50% down along x.

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00:16:50,230 --> 00:16:55,210
So there is an experiment
that can tell the difference,

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00:16:55,210 --> 00:16:58,740
but you have to look
in another direction.

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00:16:58,740 --> 00:17:01,170
And that experiment,
of course, can be done.

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00:17:01,170 --> 00:17:03,120
And it's a calculation
that can be done,

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00:17:03,120 --> 00:17:06,930
and you can decide whether
these quantum states exist.

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00:17:06,930 --> 00:17:10,369
And they really seem to exist.