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Okay so quantum mechanics is a very
strange theory. Nothing you have studied

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about in classical physics. Nothing in the
experience you have with the physical

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world can prepare you for it. It's just
the case that, that at the level of

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elementary particles. Nature behave in a
very strange way and this strange way is

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described by this theory called quantum
mechanics. In fact this, this theory is so

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strange that Fineman once, once said,
nobody understands quantum mechanics. By

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which he meant, not that nobody
understands it but nobody intuitively

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understands why it behaves, nature behaves
the way it does at quantum level. Okay so,

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let me describe some of these, some of the
funny aspects of quantum mechanics. So

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the, the first aspect of quantum mechanics
that's, that's strange is that it's

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inherently probabilistic. So, when you
make a measurement, the result is you

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know, when you, when you measure something
about, about an element or particle, the

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result is always a sample from a
probability distribution. It's not a

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determined quantity it's, it's going to
be, you know. It's, you, you only get

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probabilistic information about the
system. A second aspect of quantum

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mechanics which is bizarre is that you
cannot make a measurement of a system

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without disturbing it. So and then there
are many other funny aspects of quantum

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mechanics so, elementary particles like
electrons and photons behave like nothing

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at all that you're used to in the
classical world so they behave neither

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like particles. Nor like waves even though
they have, they share some of the features

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of each of them but they, but they don't
behave like either of these two entities.

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What's interesting though is that you
know, all these particles so photons,

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electrons, etcetera they all behave in the
same way. Not like particles not like

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waves but like something else and it's
always the same. So in this lecture and

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the rest of the lecture, I'll describe a
particular experiment called the

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double-slit experiment. Which highlights
both the , the commonality as well as the

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differences between the behavior of
quantum particle you know, quantum

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entities like electrons, photons, on the
one hand. And, particles and waves on the

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other hand so we'll, we'll, we'll be able
to see in what sense quantum mechanics

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gives us particle like behavior in what
sense it gave us wave like behavior and in

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what, what sense it deviates from both of
these. Now this, this experiment the

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double-slit experiment is a very basic
experiment and it illustrates many of the

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fundamental features of quantum mechanics.
So it's, you know even though we won't

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directly rely on this experiment for
anything that follows. It'll actually help

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you a lot if you try to understand this
experiment for what follows. On the other

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hand, you know for, for some of you, if it
doesn't make any sense, that's okay.

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Because as I said, we won't rely on it.
And starting on the next lecture I'll stop

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with by describing cubits or quantum bits.
And, and we'll, we'll do an entirely self

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contained exploration of basic quantum
mechanics in terms of quantum bits. One

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other thing for those of you who have
trouble, you know intuitively grasping

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this double-slit experiment, it would
still be useful if a week or two from now

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once we have you know, once you. Actually
understand the basics about quantum bits.

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It would be worthwhile if you come back
and review this lecture and see if the

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two, double-slit experiment makes more
sense at that point. Okay, so let's, let's

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get started. Okay. In the, in the double
slit experiment, we have a source of, you

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know of either light from photons or
electrons and, and then there's, there's a

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screen with a single small slit in it.
Through which, through which these, these

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electrons or photons have to pass. And
then at some distance from it which, which

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I guess I have shrunk to make it all fit.
There's another screen with two slits

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carved into it. We'll label this slits
slit one and slit two. And then. A long

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way from there, there's another screen
which is you know, which is where we are

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going to detect where these, where these
electron or photons ended up. So now

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let's, what we're going to do is later
I'll describe to you how, you know how

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this experiment, what the out come really
does look like if your source actually was

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shooting out electrons. But first let's,
let's imagine you know let's, let's try to

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understand what the behavior of this
experiment would be if the source was

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shooting classical particles which we'll
model as you know, bullets. So let's think

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of the source as a machine gun which is
firing bullets, you know bullets because

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they are you know, they are discrete
objects. They are indestructible. We think

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of them as discreet indestructible
objects. And so this split then becomes an

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armor plate with, with a hole in it and
the detector is just the sand box which,

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which we place at the distance x. From,
from some, from some fixed reference

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point. We'll also assume that this, you
know this machine gun is sort of jiggly.

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It's not held very, very firm so that the
bullets sort of spray at, you know along

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different trajectories. And moreover, I'll
assume that this, the, this, this slit

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here is, is narrow enough that the bullets
actually tend to strike the edge and spray

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off the edge so, so that, so that so that
the trajectory of the bullet of the, if

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the bullet happens to go through this
whole they actually spray in some,

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according to some, some angle out here.
And then some of these, some of these

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bullets then make it, you know get to one
of these slits one and slip two and once

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again, they end up. Going along, you know
they, they actually end up ricocheting off

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the edges and going along some random
path. So, now of course you can what, what

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you can do is, is you an you can keep the
machine gun firing and, at some rate. And

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you can see how many bullets end up at
this detector at location x. And of

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course, this is going to be a random
whatever because as we said, the bullets

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ricochet off at random angles so you can
only talk about how many bullets end up in

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this detector over a certain period of
time. Let's say, over a period of a minute

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and let's say that on average at x you
have some number of bullets, let's say

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three, three per minute so we'll, we'll
think of that as the average rate at which

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bullets and going to end up. Now of course
this, this three does not have to be

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natural number, a whole number. It can
also be some number like 3.5 or 3.1 and

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what that means is on average on the
course of an, of a minute you get 3.1

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bullets. Meaning, if you actually get the
measurement over a period of ten minutes

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and you got 31 bullets during that time,
you would say that greater of which

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bullets are ending up in this detector is
3.1 per minute. And so now you can plot,

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how many bullets end up in this detector
as a function of x? And you might get some

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like this. So, it might have two
different, two humps in it, two peaks

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corresponding let's say to straight line
parts going through slit one or slit two.

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And it falls off on either end. Now you
can also close one of these slits and ask,

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how many, how many bullets end up if you
only have slit one open or how many end up

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if you only have slit two open? And you
get these two different curves, curves

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like this. And so if we label this curve
as n1(x). And this curve as n2(x),

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alright? Then when both slit one and slit
two are open, we get this curve n12. And

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the equation that it satisfies is that n12
= n1 + n2. So the number of bullets that

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end up at x if both slits are open is the
umber that ends up if only slit, slit one

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was open plus the number if only slit two
was open. It makes a lot of sense, that's,

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that's very simple. So now let's, let's
repeat this experiment and now let's,

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let's imagine that a source instead is you
know that, that, that we are, we are, we

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are now, imagine that we are, we are doing
this experiment with water waves, with

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waves. And so, so what we have is a little
pond. The source is, is some vibrating

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object which sets up this you know this
little waves and let's say the vibrating

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object is vibrating at, at a fix frequency
so that the waves are nice and steady and

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they go to you know and, and you have some
sort of a barrier here with the little

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slit through it so that, so that so that
as the, as the wave gets through this,

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this little slit, it start spreading and
then it spreads all the way until it gets

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to the second barrier over the two slits.
And again, and the wave starts spreading

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through this two slits. And now our
detector is just going to detect what's

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the intensity of the, of the disturbance
at the point x. What's the energy of the

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wave at the point x. So one way to measure
it is by putting some object like a cork

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in the water and you, you see how
vigorously it gets disturbed. What's the

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energy of the cork and that's what we're
going to measure and so when we measure it

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as a function of x, that intensity which
let's say they call that I12, say denoting

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the fact that you know the function of x.
Denoting the fact that both slit one and

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slit two are open. It, it forms this very
funny curve which, which you would

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probably recognize from maybe high school
physics as the interference pattern, okay.

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So, so what happens in this interference
pattern? So remember, we said that we have

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a wave coming through each of the two
slits and now if you look at the point

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right in the middle here where you have,
you have this, this crest, this big crest.

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Right? So, what happens at this point in
the detector? Now what happens is, since

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it's equidistant from both the slits. And
since, since the two waves from, from the

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two slits are, are completely in sync.
Crest from both the waves appears

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simultaneously here because they have an
equal distance to travel from here to

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here. And perhaps appear also
simultaneously so that you have

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constructive interference between the
crest and constructive interference

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between the, between the tops and so, the
crest become higher and the tops become

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lower and so the energy of this you know,
the disturbance here is, is really, is

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really much h igher you know the, the
waves reinforce each other. On the other

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hand, if you move a little further away
from the center, then about the time the,

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the [inaudible] crest arise from slit two
is the time that the, that the G - first

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top arise from slit one because they're
off by exactly half of wave length. And so

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the more or less cancel each other at, you
know, perhaps, perhaps the wave from slit

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one is, is, is a little bit stronger than
the one from slit two here and so they

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don't quite cancel out but they very
nearly cancel out. And so you get nearly,

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completely destructive interference and,
and the water is, is essentially close to

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being still. And so the cup core is
imparted almost no energy at all. And

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similarly if you, if you move a little
further and again you'll get constructive

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interference here because, because now
when the, when the when the fourth class

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arrives from slit two. It's at the same
time that the third crest arise from slit

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one and you get constructive interference
again. Okay, so that, that accounts for

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this sort of you know this sort of
pattern. On the other hand, if you were

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to, if you were to open only slit one and
ask, what's the intensity as a function of

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x? You get this, you know this curve that
we saw with bullets i1(x) and with slit

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two you get, you get this curve i2(x) and
i12(x) is not equal to i1(x) + i2(x).

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Okay, so how do you get i12 from i1 and
i2? Well, it's actually very simple. So

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the intensity at, at x, the intensity of
the energy of the wave is just the square,

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is proportional to the square of the
height of the wave. And so, what you have

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is that the height when both slits are
open is exactly equal to the height when

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only the first slit is open plus the
height when the second slit is open, okay?

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But, but of course, what this means is
that, that i12(x) which is each twelve(x)

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all thing squared is not equal to h1(x)^2
+ h2(x)^2. In particular, if h1 is very

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nearly equal to -h2, then h12 is zero,
very close to zero so I want this very

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close to zero even though i1 and i2 are
each not close to zero. Okay, so that's

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the case with, with waves. Now let's look
at the situation when, when we have a

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source of electrons or photons. So let's
say it's a source of electrons, it's an

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electron gun and. Again we have, we have
you know we have the same sort of set up

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with a detector in the back here which we
can think of as you know the screen in the

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back is fluorescent so that whenever an
electron here to get, you get little burst

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of light and now we can ask what's the
intensity? What's, what's the intensity

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of, of, of, of the electrons arriving at,
at point x? How much light do we see? How

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many we need to detect? And the, the thing
that happens here is that well, one thing

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that you notice is as you turn down the
intensity of the source of electrons. You

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start noticing that, that as you turn it
down further and further. The electrons

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start arriving at point x at discrete
points in times. So you see a flash and

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then nothing for a while, and then another
flash, nothing for a while and so on. And

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as you turned down the intensity, the
flash has don't get any, any less intense.

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What changes is the frequency of which
what you see the flashes. So, again what,

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what we, what we have to say is that you
know, that would, that would seem to tell

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us that this electrons are real particles.
They, you know, they are, they are

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particles with you know, they are these
charge particles and, and you know, as you

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turn down the electric, intensity of the
electron source, there are fewer and fewer

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electrons going out per unit time. And as
they go out and they are not, not through

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you know, they are deflected through the
edges of these slits. They are randomly

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sharp at some point x. And, and, and the,
the probability that you see in electron

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here depends upon you know as you turn
down the intensity of the source you see

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them less and less frequently but they
arrive as discrete objects, as discrete

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lumps. And, and so you can talk about the
probability of detecting the, the electron

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at point x or you can talk about you know,
you can call it its intensity i(x). And

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so, now we would you know, given that,
given that they are like, like you know

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like discrete particles or bullets. What
kind of behavior would we expect? And so

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again, if we, if we open only slit one,
the intensity as a function of x. Looks

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like this. If you open only slit two, the
intensity as a function of x, the

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probability that we see, see in electron
looks like this and. Since, since

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electrons. Are behaving like particles
like bullets, you would imagine that both

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slits are open, you should see this curve
which is the i12 which is sum of i1 and

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i2. But in fact. What you end up seeing is
this interference pattern like we did in

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the case of waves. I12 is not equal to i1
+ i2. And that's the strange thing about

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quantum mechanics. So how could it be if
electrons are traveling, if they are, if

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hey are like particles. If they are, if
they are, if, if, if they are discrete

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objects, discrete you know, indestructible
objects. How could it be that. That when

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both slits are open, you do not get to see
the sum of these two curves as the

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probability of, of the electron ending up
at x. And you could say well, let's,

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let's, let's reason about this a little
more carefully and say well clearly, the

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electron was fired through this source. It
went through this solid called deflected,

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and then it either went through slit one
or through slit two. And if it went

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through slit one, it ended up at x with
probability equal to i1(x). If it went

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through slit two, it ended up at x with
probability i2(x). Surely, if both slits

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were open. It should end up at x with
probability i1 + i2(x) because after all

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if it went through slit one, why should it
matter to it were the slit two was open or

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not? And the answer is, we don't know but,
but when you do the experiment, you get to

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see the interference pattern. Now, in
quantum mechanics we have a way of

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explaining this. What we can do is we can
say, well actually there's. There's an

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amplitude with which the e lectron goes
like one and ends up at x. And that

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amplitude is a1(x) and actually the, the
probability that we detect the photon at

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point x i1(x) is actually the square if
a1(x). And similarly, there's an amplitude

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with which it goes through to slit two and
ends up at x. And if only slit two is

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open, i2(x) is just the square of this
amplitude. And similarly, if both slits

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are open, then ai2 is a1 + a2. So that the
amplitude with which the photon ends up at

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x is just a1(x) + a2(x). And of course,
the probability that you detect the photon

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is a12(x)^2. So this is just like the
water waves case where we have the height

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of the water wave and the intensity is the
square of the height of the water wave.

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Except that there is no height here so
what is this amplitude? Well, we don't

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know but this is how nature behaves. The
electrons behave as though there was some

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comp, some amplitude with which it ends up
at x. And this amplitude can be positive

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or negative leading to this kind of
interference pattern. Okay, that's the

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funny thing about quantum mechanics.
That's how electrons and photons behave.

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So, let's summarize what we've learned.
So, we did this double slit experiment

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three times, in three different settings.
First, we considered it. Where we, where

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it was a source of particles of bullets
which we think of as bullets. Then we

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repeated this experiment with waves, with
water waves. And finally, we repeated it

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with, with quantum objects like with
elementary particles like photons,

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electrons. So of course, in the case of
bullets we have discrete objects that come

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you know, bullets come as discrete chances
as, as, as units. In the case of waves,

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the energy arrives not as discrete objects
but it's, it's continuous. And as we saw

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in the case of photons and electrons, they
behave this quickly. They, they, they

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arrive in discrete chunks which we think
of as electrons or photons which are

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particles of light. So, discrete. In the
case of bullets, we talked about the

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probability of arrival at x. In the case
of waves, we measured the intensity

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ornergy. In the case of electrons or
photons, well again we measured the

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probability of arrival. Which we said is
proportional to the intensity. In the case

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of bullets, when we have both slits open,
we saw no interference. In the case of

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waves, we saw interference. In the case of
photons and electrons, we again have

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interference and this is the funny thing.
So even though photons and electrons

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arrive as discrete entities, and we think
they should have gone through either slit

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one or slit two, they actually, we do get
the interference pattern and this is part

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of the mystery. This is where we have this
strange behavior in quantum mechanically.

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In the case of bullets, when both slits
are open and one, two is n1 + n2. In the

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case of waves, i12 was not equal to i1 +
i2. But, what we have was that h12, the

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height of the wave, did add and the
intensity was the square of the height. In

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the case of photons or electrons, again we
had i12 is not equal to i1 + i1. The

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probability did not add but then we, we
came up with this notion of an amplitude

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which is just some invented notion and so
a12 = a1 + a2. And that the intensity or

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the probability is just the square of a
and I actually put the square in, inside

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the absolute values because, because in
fact the amplitude can also be a complex

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number not just positive or negative. And
we can, we can go back and try to

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understand this, this strange behavior of
photons and electrons and a little bit

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more closely by, by trying to understand
this proposition. It says, the electron.

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Either went through slit one or it went
through slit two. So to try to test this

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proposition, what we can do. Is we can run
this experiment again with electrons but

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now we'll try to detect which slit it went
through. So what we can do is. Put a

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little source of light here which you
know, across slit one or across each of

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the slits so that when the electron is
going through the slit, going through the

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slit, you know the light actually bounces
off of it and, and we get to see that the

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electron went by. So now what we'd imagine
is not only are we going to get the

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interference pattern but we'll also going
to actually detect what actually happened

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better than you know, this proposition
that the electron went through either slit

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one or through slit two. So when you
actually do this experiment, what you

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realize is that in fact, and that it
didn't go through board slits and in fact

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what you realize is, you know, the result
of this experiment they'll show you that

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every time you see the electron go past
through slit one, in fact it doesn't go

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through slit two. Every time you measure
it is going through slit two, it doesn't

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go through slit one. But on the other
hand, when you do the experiment this way.

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The count at this detector suddenly
changes. You no longer see this

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interference pattern but instead you see,
you see this, this, this pattern we saw

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with bullets. So this is a very strange
thing. As long as you don't try to see

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which slit the electron went through then
you get the interference pattern. But if

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you try to see, if you try to see which
slit it went through. To confirm this you

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know, to confirm this hypothesis that the
bullet, the electron ether went through

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00:35:15,009 --> 00:35:22,000
slit one or through slit two. Well then
you, you, you do actually detect that it

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went through either one of the set, of the
other but now the interference pattern

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disappears. Of course you can turn down
the intensity of light here by these two

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slits to make sure that you, you don't
really disturb, you try not to disturb the

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electron as its going through. And as you
turn down the intensity, what you end up

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getting is some combination of this
interference pattern and this trade

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edition. And in fact, the extent of, of
this combination you get is exactly

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proportional to how, what fraction of the,
of the electrons you, you, you, you are

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able to detect at the slit one or slit
two. So nature can perfectly hide its

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tracks. So if you, if you try to, try to
measure which slit the electron went, went

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through, then the interference pattern
disappears. If you. If you try to detect

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only a little bit. Then the interference
pattern disappears a little bit. Exactly

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when you actually do the detection. Okay,
so, so let's summarize what we've learned

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and by quantum mechanics. First thing
we've learned is, in quantum mechanics

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measurements are probabilistic. So,
whether electron or photon ends up is,

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you, you have to do to detect it, you have
to do a measurement, and the measurement

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is inherently probabilistic process.
Second thing, you cannot measure without

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disturbing the system. Whenever you make a
measurement you can, you can make the

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measurement as subtle as you want but you
still disturb the system. The third thing

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we learned is that, elementary particles
behave in a very strange way. They behave

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like no classical entities with that we
have seen they behave near like particles

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00:37:44,025 --> 00:37:49,058
nor like waves but it behave in a
completely strange way. Some of the

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characteristics seem as though they are
like characteristics of particles. They

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behave like bullets. So other
characteristics are those of waves but

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really they, they have, they behave like
some funny combination of the two which is

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like neither of the two at all. The fourth
thing is that when we do this experiment,

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we can say that there's a moment when,
when the electron keaves the source, the

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electron gun. And then from then, then on,
we cannot really say what part the

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00:38:35,046 --> 00:38:43,050
electron took or whether it took multiple
parts at the same time to arrive at, at,

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at the point x. And so, we really cannot
say what the trajectory of this electron

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is. So what quantum mechanics allows us to
do is its you know, we start the

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00:38:56,000 --> 00:39:01,082
experiment, it happens quantumly and as
soon as we look, as soon as we measure

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that disturbs the system and that gives us
the outcome of the, of the experiment. So

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there's this black box nature to, to a
quantum experiment where we, where we

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start the experiment to the source of
electrons and then something happens and

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then we do this, do a measurement when we
see the outcome that the electron ended up

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at point x. But in the middle here, we
cannot really see what happened and what,

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00:39:34,052 --> 00:39:42,032
what the formulation tells us is, is that
if we had double-slit here then, then the

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photon has, the electron has some
amplitude. A1 with which it goes through

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00:39:50,067 --> 00:39:58,085
slit one and ends up at x. Some amplitude
a2 with which it goes through slit two and

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00:39:58,085 --> 00:40:05,044
ends up at x. And the amplitude with which
it ends up at x is a1(x) + a2(x) and the

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00:40:05,044 --> 00:40:13,008
probability of detecting it is the square
of this amplitude. Okay, so that's, that's

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00:40:13,008 --> 00:40:22,075
the you know that's, that's the strange
behavior of, of quantum particles and okay

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00:40:22,075 --> 00:40:32,005
so starting next time we'll, we'll, we'll
start from scratch and talk about quantum

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00:40:32,005 --> 00:40:37,006
bits and the basic axioms of quantum
mechanics.