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PROFESSOR: Determinism.

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And it all begins with photons.

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Einstein reluctantly
came up with the idea

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that light was made of quanta--

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quanta of light called photons.

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Now when you think of photons,
we think of a particle.

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So everybody knew
that light was a wave.

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Maxwell's equations
had been so successful.

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Nevertheless,
photoelectric effect--

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Planck's work-- all were leading
to the idea that, in some ways,

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photons were also particles.

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So when you think of a
particle, however, there

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is an important difference
between a particle

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in the sense of Newton, which
is an object with zero size that

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carries energy and
has a precise position

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and velocity at any time, and
the quantum mechanical idea

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of particle, which is just some
indivisible amount of energy

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or momentum that propagates.

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So light was made of photons--

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packets of energy.

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And a photon is a particle--
a quantum mechanical particle.

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Not in the sense that maybe
it has position and velocity

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determined or it's
a point particle,

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but more like a packet
that is indivisible.

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You cannot decompose
it in further packets.

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So Einstein realized
that for a photon,

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the energy was given by h nu.

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Where nu is the
frequency of the light

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that this photon is
helping build up.

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So if you have a
beam of light, you

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should think it's
billions of photons.

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And according to the
frequency of that light

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that is related to
the wavelength--

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by the equation frequency
times wavelength

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is velocity of light--

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you typically know, for
light, the wavelength,

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and you know the
frequency, and then

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you know the energy of
each of the photons.

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The photons have very,
very little energy.

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We have very, very
little energy,

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but your eyes are very
good detectors of photons.

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If you're in a totally
dark room, your eye,

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probably, can take as
little as five photons

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if they hit your retina.

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So it's a pretty good
detector of photons.

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Anyway, the thing that
I want to explain here

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is what happens if a beam
of light hits a polarizer.

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So what is a polarizer?

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It's a sheet of plastic
or some material.

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It has a preferential direction.

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Let me align that preferential
direction with the x-axis,

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and that's a polarizer.

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And if I send light
that is linearly

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polarized along the x-axis,
it all goes through.

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If I send light linearly
polarized along the y-axis,

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nothing goes through.

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It all gets absorbed.

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That's what a polarizer
does for a living.

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In fact, if you send
light in this direction,

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the light that comes out
is identical to the light

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that came in.

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The frequency doesn't change.

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The wavelength doesn't change.

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It's the same light,
the same energy.

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So far, so good.

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Now let's imagine that we send
in light linearly polarized

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at some angle alpha.

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So we send an electric
field E alpha,

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which is E0 cosine alpha x
hat plus E0 sine alpha y hat.

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Well, you've studied
electromagnetism,

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and you know that this thing,
basically, will come around

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and say, OK, you can go
through because you're

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aligning the right
direction, but you

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are orthogonal to my
preferential direction,

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or orthogonal I absorbed,
so this disappears.

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So after the polarizer, E is
just E0 cosine alpha x hat.

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That's all that is left
after the polarizer.

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Well here is something
interesting--

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you know that the energy
on electromagnetic field

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is proportional to the
magnitude of the electric field

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square, that's what it is.

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So the magnitude of
this electric field--

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if you can notice,
it's the square root

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of the sum of the
squares will give you

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E0 as the magnitude of
this full electric field.

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But this electric field has
magnitude E0 cosine alpha.

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So the fraction of power--

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fraction of energy through
is cosine alpha squared.

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The energy is always
proportional to the square.

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So the square of this is E0
squared cosine squared alpha.

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And for this one, the
magnitude of it is E0,

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so you divide by E0 and cosine
alpha is the right thing.

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This is the fraction
of the energy.

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If alpha is equal to 0,
you get cosine of 01.

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You get all the energy 1.

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If alpha is equal to
pi over 2, the light

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is polarized along
the y direction,

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nothing goes through--

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indeed, cosine of pi over 2 is
0, and nothing goes through.

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So the fraction of
energy that goes through

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is cosine squared alpha.

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But now, think what
this means for photons.

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What it means for
photons is something

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extraordinarily strange.

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And so strange that it's
almost unbelievable that we

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get so easily in trouble.

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Here is this light
beam over here,

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and it's made up of photons.

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All identical photons,
maybe billions of photons,

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but all identical.

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And now, think of sending
this light beam over there--

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a billion identical photons--
you send them one by one

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into the state, and
see what happens.

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You know what has to happen,
because classical behavior is

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about right.

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This fraction of the
photons must go through,

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and 1 minus that
must not go through.

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You see, it cannot be there
comes a photon and half of it

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goes through, because there's
no such thing as half of it.

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If there would be
half of it, it would

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be half the energy and,
therefore, different color.

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And we know that
after a polarizer,

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the color doesn't change.

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So here is the situation.

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You're sending a billion
photons and, say, one-third

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has to get through.

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But now, the photos
are identical.

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How can that happen
in classical physics?

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If you send identical photos,
whatever happens to a photon

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will happen to all, but the
photon either gets absorbed

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or goes through.

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And if it gets absorbed,
then all should get absorbed.

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And if it goes through,
all should go through

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because they are all identical.

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And now you have
found a situation

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which identical
set of experiments

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with identically
prepared objects

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sometimes gives you
different results.

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It's a debacle.

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It's a total disaster.

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What seems to have
happened here--

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you suddenly have
identical photons,

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and sometimes they go
through, and sometimes they

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don't go through.

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And therefore, you've
lost predictability.

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It's so simple to show
that if photons exist,

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you lose predictability.

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And that's what
drove Einstein crazy.

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He knew when he
entered these photons

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that he was getting in trouble.

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He was going to get in trouble
with classical physics.

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So possible ways out--

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people speculate about it--

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people said, well, yes,
the photos are identical,

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but the polarizer
has substructure.

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If it hits in this interatomic
part, it goes through,

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and in that interatomic
part, it doesn't go through.

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People did experiments
many times.

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It's not true.

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The polarizer is like that.

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And then came a more
outrageous proposition

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by Einstein and others--

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that there are hidden variables.

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You think the photons
are identical,

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but a photon has a hidden
variable-- a property

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you don't know about.

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If you knew that property
about the photon,

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you would be able to
tell if it goes through

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or it doesn't go through.

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But you don't know it,
so that's why you're

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stuck with probabilities.

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It's because the quantum
theory is not complete.

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There are hidden variables.

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And once you put the
hidden variables,

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you'll discover the photon
has more something inside it,

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and they are not the same,
even though they look the same.

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And that's a hidden
variable theory.

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And it sounds so philosophical
that you would think, well,

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if you don't know about
them, but they are there,

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these properties, how could
you ever know they are there?

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And the great progress of John
Bell with the Bell inequalities

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is that he demonstrated that
that would not fix the problem.

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Quantum mechanics cannot be
made deterministic with hidden

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variables.

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It was an unbelievable result--

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the result of John Bell.

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So that's something we will
advance towards in this course

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but not quite get there.

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805 discusses this
subject in detail.

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So at the end of the day,
we've lost determinism.

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We can only predict
probabilities.

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So photons either
gets through or not,

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and can only predict
probabilities.

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Now we write, in classical
physics, a beam like that.

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But how do we write the
wave function of a photon?

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Well, this is quite interesting.

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We think of states of a
particle as wave functions.

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And I will call them,
sometimes, states;

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I will call them,
sometimes, wave functions;

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and I sometimes will
call them vectors.

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Why vector?

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Because the main thing
you do with vectors

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is adding them or multiplying
them by numbers to scale them.

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And that's exactly what you
can do with a linear equation.

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So that's why people think
of states, or wave functions,

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as vectors.

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And Dirac invented
a notation in which

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to describe a photon
polarized in the x direction,

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you would simply write
something like this.

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Photon colon x and this object--

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you think of it as some
vector or wave function,

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and it represents a
photon in the x direction.

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And we're not saying yet
what kind of vector this is,

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but it's some sort of vector.

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It's not just a symbol,
it represents a vector.

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And that's a possible state.

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This is a photon
polarized along x.

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And you can also
have, if you wish,

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a photon polarized along y.

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And linearity means that
if those photos can exist,

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the superposition can exist.

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So there can exist a state
called cos alpha photon x

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plus sine alpha photon y,
in which I've superposed

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one state with another--

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created a sum-- and this I
call the photon state polarized

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in the alpha direction.

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So this is how, in quantum
mechanics, you think of this--

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photons-- we will elaborate that
and compare with this equation.

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It's kind of interesting.

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What you lose here is this ease.

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There's no ease there
because it's one photon.

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When you have a
big electric field,

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I don't know how many
photons there are.

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I would have to calculate
the energy of this beam

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and find the frequency
that I didn't specify,

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and see how many photons.

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But each photon in this
beam quantum mechanically

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can be represented as
this superposition.

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And we'll talk more about
this superposition now

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because our next subject
is superpositions

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and how unusual they are.

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Well the hidden
variable explanation

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failed because Bell
was very clever,

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and he noted that
you could design

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an experiment in which
the hidden variables would

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imply that some measurements
would satisfy an inequality.

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If the existed hidden
variables and the world

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was after all classical,
the results of experiments

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would satisfy a Bell inequality.

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And then a few years
later, the technology

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was good enough that
people could test the Bell

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inequality with an experiment,
and they figured out

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it didn't hold.

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00:16:30,750 --> 00:16:35,670
So the hidden variables
lead to Bell inequalities

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00:16:35,670 --> 00:16:39,870
that are experimentally
shown not to hold.

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00:16:39,870 --> 00:16:41,850
And we will touch
a little bit on it

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00:16:41,850 --> 00:16:43,710
when we get to untangle them.

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00:16:43,710 --> 00:16:56,640
After the polarizer, the photon
is in the state photon x.

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00:16:56,640 --> 00:16:59,850
It's always polarized
along the x direction,

260
00:16:59,850 --> 00:17:05,069
so it's kind of similar that
this doesn't go through.

261
00:17:05,069 --> 00:17:08,040
This goes through, but
at the end of the day,

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00:17:08,040 --> 00:17:11,060
as we will explain very
soon, the cosine alpha

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00:17:11,060 --> 00:17:12,630
is not relevant here.

264
00:17:12,630 --> 00:17:15,480
When it goes through, the
whole photon goes through.

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00:17:15,480 --> 00:17:17,520
So there's no need
for a cosine alpha.

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00:17:17,520 --> 00:17:21,900
So that's what goes
out of the polarizer.