Okay so quantum mechanics is a very
strange theory. Nothing you have studied
about in classical physics. Nothing in the
experience you have with the physical
world can prepare you for it. It's just
the case that, that at the level of
elementary particles. Nature behave in a
very strange way and this strange way is
described by this theory called quantum
mechanics. In fact this, this theory is so
strange that Fineman once, once said,
nobody understands quantum mechanics. By
which he meant, not that nobody
understands it but nobody intuitively
understands why it behaves, nature behaves
the way it does at quantum level. Okay so,
let me describe some of these, some of the
funny aspects of quantum mechanics. So
the, the first aspect of quantum mechanics
that's, that's strange is that it's
inherently probabilistic. So, when you
make a measurement, the result is you
know, when you, when you measure something
about, about an element or particle, the
result is always a sample from a
probability distribution. It's not a
determined quantity it's, it's going to
be, you know. It's, you, you only get
probabilistic information about the
system. A second aspect of quantum
mechanics which is bizarre is that you
cannot make a measurement of a system
without disturbing it. So and then there
are many other funny aspects of quantum
mechanics so, elementary particles like
electrons and photons behave like nothing
at all that you're used to in the
classical world so they behave neither
like particles. Nor like waves even though
they have, they share some of the features
of each of them but they, but they don't
behave like either of these two entities.
What's interesting though is that you
know, all these particles so photons,
electrons, etcetera they all behave in the
same way. Not like particles not like
waves but like something else and it's
always the same. So in this lecture and
the rest of the lecture, I'll describe a
particular experiment called the
double-slit experiment. Which highlights
both the , the commonality as well as the
differences between the behavior of
quantum particle you know, quantum
entities like electrons, photons, on the
one hand. And, particles and waves on the
other hand so we'll, we'll, we'll be able
to see in what sense quantum mechanics
gives us particle like behavior in what
sense it gave us wave like behavior and in
what, what sense it deviates from both of
these. Now this, this experiment the
double-slit experiment is a very basic
experiment and it illustrates many of the
fundamental features of quantum mechanics.
So it's, you know even though we won't
directly rely on this experiment for
anything that follows. It'll actually help
you a lot if you try to understand this
experiment for what follows. On the other
hand, you know for, for some of you, if it
doesn't make any sense, that's okay.
Because as I said, we won't rely on it.
And starting on the next lecture I'll stop
with by describing cubits or quantum bits.
And, and we'll, we'll do an entirely self
contained exploration of basic quantum
mechanics in terms of quantum bits. One
other thing for those of you who have
trouble, you know intuitively grasping
this double-slit experiment, it would
still be useful if a week or two from now
once we have you know, once you. Actually
understand the basics about quantum bits.
It would be worthwhile if you come back
and review this lecture and see if the
two, double-slit experiment makes more
sense at that point. Okay, so let's, let's
get started. Okay. In the, in the double
slit experiment, we have a source of, you
know of either light from photons or
electrons and, and then there's, there's a
screen with a single small slit in it.
Through which, through which these, these
electrons or photons have to pass. And
then at some distance from it which, which
I guess I have shrunk to make it all fit.
There's another screen with two slits
carved into it. We'll label this slits
slit one and slit two. And then. A long
way from there, there's another screen
which is you know, which is where we are
going to detect where these, where these
electron or photons ended up. So now
let's, what we're going to do is later
I'll describe to you how, you know how
this experiment, what the out come really
does look like if your source actually was
shooting out electrons. But first let's,
let's imagine you know let's, let's try to
understand what the behavior of this
experiment would be if the source was
shooting classical particles which we'll
model as you know, bullets. So let's think
of the source as a machine gun which is
firing bullets, you know bullets because
they are you know, they are discrete
objects. They are indestructible. We think
of them as discreet indestructible
objects. And so this split then becomes an
armor plate with, with a hole in it and
the detector is just the sand box which,
which we place at the distance x. From,
from some, from some fixed reference
point. We'll also assume that this, you
know this machine gun is sort of jiggly.
It's not held very, very firm so that the
bullets sort of spray at, you know along
different trajectories. And moreover, I'll
assume that this, the, this, this slit
here is, is narrow enough that the bullets
actually tend to strike the edge and spray
off the edge so, so that, so that so that
the trajectory of the bullet of the, if
the bullet happens to go through this
whole they actually spray in some,
according to some, some angle out here.
And then some of these, some of these
bullets then make it, you know get to one
of these slits one and slip two and once
again, they end up. Going along, you know
they, they actually end up ricocheting off
the edges and going along some random
path. So, now of course you can what, what
you can do is, is you an you can keep the
machine gun firing and, at some rate. And
you can see how many bullets end up at
this detector at location x. And of
course, this is going to be a random
whatever because as we said, the bullets
ricochet off at random angles so you can
only talk about how many bullets end up in
this detector over a certain period of
time. Let's say, over a period of a minute
and let's say that on average at x you
have some number of bullets, let's say
three, three per minute so we'll, we'll
think of that as the average rate at which
bullets and going to end up. Now of course
this, this three does not have to be
natural number, a whole number. It can
also be some number like 3.5 or 3.1 and
what that means is on average on the
course of an, of a minute you get 3.1
bullets. Meaning, if you actually get the
measurement over a period of ten minutes
and you got 31 bullets during that time,
you would say that greater of which
bullets are ending up in this detector is
3.1 per minute. And so now you can plot,
how many bullets end up in this detector
as a function of x? And you might get some
like this. So, it might have two
different, two humps in it, two peaks
corresponding let's say to straight line
parts going through slit one or slit two.
And it falls off on either end. Now you
can also close one of these slits and ask,
how many, how many bullets end up if you
only have slit one open or how many end up
if you only have slit two open? And you
get these two different curves, curves
like this. And so if we label this curve
as n1(x). And this curve as n2(x),
alright? Then when both slit one and slit
two are open, we get this curve n12. And
the equation that it satisfies is that n12
= n1 + n2. So the number of bullets that
end up at x if both slits are open is the
umber that ends up if only slit, slit one
was open plus the number if only slit two
was open. It makes a lot of sense, that's,
that's very simple. So now let's, let's
repeat this experiment and now let's,
let's imagine that a source instead is you
know that, that, that we are, we are, we
are now, imagine that we are, we are doing
this experiment with water waves, with
waves. And so, so what we have is a little
pond. The source is, is some vibrating
object which sets up this you know this
little waves and let's say the vibrating
object is vibrating at, at a fix frequency
so that the waves are nice and steady and
they go to you know and, and you have some
sort of a barrier here with the little
slit through it so that, so that so that
as the, as the wave gets through this,
this little slit, it start spreading and
then it spreads all the way until it gets
to the second barrier over the two slits.
And again, and the wave starts spreading
through this two slits. And now our
detector is just going to detect what's
the intensity of the, of the disturbance
at the point x. What's the energy of the
wave at the point x. So one way to measure
it is by putting some object like a cork
in the water and you, you see how
vigorously it gets disturbed. What's the
energy of the cork and that's what we're
going to measure and so when we measure it
as a function of x, that intensity which
let's say they call that I12, say denoting
the fact that you know the function of x.
Denoting the fact that both slit one and
slit two are open. It, it forms this very
funny curve which, which you would
probably recognize from maybe high school
physics as the interference pattern, okay.
So, so what happens in this interference
pattern? So remember, we said that we have
a wave coming through each of the two
slits and now if you look at the point
right in the middle here where you have,
you have this, this crest, this big crest.
Right? So, what happens at this point in
the detector? Now what happens is, since
it's equidistant from both the slits. And
since, since the two waves from, from the
two slits are, are completely in sync.
Crest from both the waves appears
simultaneously here because they have an
equal distance to travel from here to
here. And perhaps appear also
simultaneously so that you have
constructive interference between the
crest and constructive interference
between the, between the tops and so, the
crest become higher and the tops become
lower and so the energy of this you know,
the disturbance here is, is really, is
really much h igher you know the, the
waves reinforce each other. On the other
hand, if you move a little further away
from the center, then about the time the,
the [inaudible] crest arise from slit two
is the time that the, that the G - first
top arise from slit one because they're
off by exactly half of wave length. And so
the more or less cancel each other at, you
know, perhaps, perhaps the wave from slit
one is, is, is a little bit stronger than
the one from slit two here and so they
don't quite cancel out but they very
nearly cancel out. And so you get nearly,
completely destructive interference and,
and the water is, is essentially close to
being still. And so the cup core is
imparted almost no energy at all. And
similarly if you, if you move a little
further and again you'll get constructive
interference here because, because now
when the, when the when the fourth class
arrives from slit two. It's at the same
time that the third crest arise from slit
one and you get constructive interference
again. Okay, so that, that accounts for
this sort of you know this sort of
pattern. On the other hand, if you were
to, if you were to open only slit one and
ask, what's the intensity as a function of
x? You get this, you know this curve that
we saw with bullets i1(x) and with slit
two you get, you get this curve i2(x) and
i12(x) is not equal to i1(x) + i2(x).
Okay, so how do you get i12 from i1 and
i2? Well, it's actually very simple. So
the intensity at, at x, the intensity of
the energy of the wave is just the square,
is proportional to the square of the
height of the wave. And so, what you have
is that the height when both slits are
open is exactly equal to the height when
only the first slit is open plus the
height when the second slit is open, okay?
But, but of course, what this means is
that, that i12(x) which is each twelve(x)
all thing squared is not equal to h1(x)^2
+ h2(x)^2. In particular, if h1 is very
nearly equal to -h2, then h12 is zero,
very close to zero so I want this very
close to zero even though i1 and i2 are
each not close to zero. Okay, so that's
the case with, with waves. Now let's look
at the situation when, when we have a
source of electrons or photons. So let's
say it's a source of electrons, it's an
electron gun and. Again we have, we have
you know we have the same sort of set up
with a detector in the back here which we
can think of as you know the screen in the
back is fluorescent so that whenever an
electron here to get, you get little burst
of light and now we can ask what's the
intensity? What's, what's the intensity
of, of, of, of the electrons arriving at,
at point x? How much light do we see? How
many we need to detect? And the, the thing
that happens here is that well, one thing
that you notice is as you turn down the
intensity of the source of electrons. You
start noticing that, that as you turn it
down further and further. The electrons
start arriving at point x at discrete
points in times. So you see a flash and
then nothing for a while, and then another
flash, nothing for a while and so on. And
as you turned down the intensity, the
flash has don't get any, any less intense.
What changes is the frequency of which
what you see the flashes. So, again what,
what we, what we have to say is that you
know, that would, that would seem to tell
us that this electrons are real particles.
They, you know, they are, they are
particles with you know, they are these
charge particles and, and you know, as you
turn down the electric, intensity of the
electron source, there are fewer and fewer
electrons going out per unit time. And as
they go out and they are not, not through
you know, they are deflected through the
edges of these slits. They are randomly
sharp at some point x. And, and, and the,
the probability that you see in electron
here depends upon you know as you turn
down the intensity of the source you see
them less and less frequently but they
arrive as discrete objects, as discrete
lumps. And, and so you can talk about the
probability of detecting the, the electron
at point x or you can talk about you know,
you can call it its intensity i(x). And
so, now we would you know, given that,
given that they are like, like you know
like discrete particles or bullets. What
kind of behavior would we expect? And so
again, if we, if we open only slit one,
the intensity as a function of x. Looks
like this. If you open only slit two, the
intensity as a function of x, the
probability that we see, see in electron
looks like this and. Since, since
electrons. Are behaving like particles
like bullets, you would imagine that both
slits are open, you should see this curve
which is the i12 which is sum of i1 and
i2. But in fact. What you end up seeing is
this interference pattern like we did in
the case of waves. I12 is not equal to i1
+ i2. And that's the strange thing about
quantum mechanics. So how could it be if
electrons are traveling, if they are, if
hey are like particles. If they are, if
they are, if, if, if they are discrete
objects, discrete you know, indestructible
objects. How could it be that. That when
both slits are open, you do not get to see
the sum of these two curves as the
probability of, of the electron ending up
at x. And you could say well, let's,
let's, let's reason about this a little
more carefully and say well clearly, the
electron was fired through this source. It
went through this solid called deflected,
and then it either went through slit one
or through slit two. And if it went
through slit one, it ended up at x with
probability equal to i1(x). If it went
through slit two, it ended up at x with
probability i2(x). Surely, if both slits
were open. It should end up at x with
probability i1 + i2(x) because after all
if it went through slit one, why should it
matter to it were the slit two was open or
not? And the answer is, we don't know but,
but when you do the experiment, you get to
see the interference pattern. Now, in
quantum mechanics we have a way of
explaining this. What we can do is we can
say, well actually there's. There's an
amplitude with which the e lectron goes
like one and ends up at x. And that
amplitude is a1(x) and actually the, the
probability that we detect the photon at
point x i1(x) is actually the square if
a1(x). And similarly, there's an amplitude
with which it goes through to slit two and
ends up at x. And if only slit two is
open, i2(x) is just the square of this
amplitude. And similarly, if both slits
are open, then ai2 is a1 + a2. So that the
amplitude with which the photon ends up at
x is just a1(x) + a2(x). And of course,
the probability that you detect the photon
is a12(x)^2. So this is just like the
water waves case where we have the height
of the water wave and the intensity is the
square of the height of the water wave.
Except that there is no height here so
what is this amplitude? Well, we don't
know but this is how nature behaves. The
electrons behave as though there was some
comp, some amplitude with which it ends up
at x. And this amplitude can be positive
or negative leading to this kind of
interference pattern. Okay, that's the
funny thing about quantum mechanics.
That's how electrons and photons behave.
So, let's summarize what we've learned.
So, we did this double slit experiment
three times, in three different settings.
First, we considered it. Where we, where
it was a source of particles of bullets
which we think of as bullets. Then we
repeated this experiment with waves, with
water waves. And finally, we repeated it
with, with quantum objects like with
elementary particles like photons,
electrons. So of course, in the case of
bullets we have discrete objects that come
you know, bullets come as discrete chances
as, as, as units. In the case of waves,
the energy arrives not as discrete objects
but it's, it's continuous. And as we saw
in the case of photons and electrons, they
behave this quickly. They, they, they
arrive in discrete chunks which we think
of as electrons or photons which are
particles of light. So, discrete. In the
case of bullets, we talked about the
probability of arrival at x. In the case
of waves, we measured the intensity
ornergy. In the case of electrons or
photons, well again we measured the
probability of arrival. Which we said is
proportional to the intensity. In the case
of bullets, when we have both slits open,
we saw no interference. In the case of
waves, we saw interference. In the case of
photons and electrons, we again have
interference and this is the funny thing.
So even though photons and electrons
arrive as discrete entities, and we think
they should have gone through either slit
one or slit two, they actually, we do get
the interference pattern and this is part
of the mystery. This is where we have this
strange behavior in quantum mechanically.
In the case of bullets, when both slits
are open and one, two is n1 + n2. In the
case of waves, i12 was not equal to i1 +
i2. But, what we have was that h12, the
height of the wave, did add and the
intensity was the square of the height. In
the case of photons or electrons, again we
had i12 is not equal to i1 + i1. The
probability did not add but then we, we
came up with this notion of an amplitude
which is just some invented notion and so
a12 = a1 + a2. And that the intensity or
the probability is just the square of a
and I actually put the square in, inside
the absolute values because, because in
fact the amplitude can also be a complex
number not just positive or negative. And
we can, we can go back and try to
understand this, this strange behavior of
photons and electrons and a little bit
more closely by, by trying to understand
this proposition. It says, the electron.
Either went through slit one or it went
through slit two. So to try to test this
proposition, what we can do. Is we can run
this experiment again with electrons but
now we'll try to detect which slit it went
through. So what we can do is. Put a
little source of light here which you
know, across slit one or across each of
the slits so that when the electron is
going through the slit, going through the
slit, you know the light actually bounces
off of it and, and we get to see that the
electron went by. So now what we'd imagine
is not only are we going to get the
interference pattern but we'll also going
to actually detect what actually happened
better than you know, this proposition
that the electron went through either slit
one or through slit two. So when you
actually do this experiment, what you
realize is that in fact, and that it
didn't go through board slits and in fact
what you realize is, you know, the result
of this experiment they'll show you that
every time you see the electron go past
through slit one, in fact it doesn't go
through slit two. Every time you measure
it is going through slit two, it doesn't
go through slit one. But on the other
hand, when you do the experiment this way.
The count at this detector suddenly
changes. You no longer see this
interference pattern but instead you see,
you see this, this, this pattern we saw
with bullets. So this is a very strange
thing. As long as you don't try to see
which slit the electron went through then
you get the interference pattern. But if
you try to see, if you try to see which
slit it went through. To confirm this you
know, to confirm this hypothesis that the
bullet, the electron ether went through
slit one or through slit two. Well then
you, you, you do actually detect that it
went through either one of the set, of the
other but now the interference pattern
disappears. Of course you can turn down
the intensity of light here by these two
slits to make sure that you, you don't
really disturb, you try not to disturb the
electron as its going through. And as you
turn down the intensity, what you end up
getting is some combination of this
interference pattern and this trade
edition. And in fact, the extent of, of
this combination you get is exactly
proportional to how, what fraction of the,
of the electrons you, you, you, you are
able to detect at the slit one or slit
two. So nature can perfectly hide its
tracks. So if you, if you try to, try to
measure which slit the electron went, went
through, then the interference pattern
disappears. If you. If you try to detect
only a little bit. Then the interference
pattern disappears a little bit. Exactly
when you actually do the detection. Okay,
so, so let's summarize what we've learned
and by quantum mechanics. First thing
we've learned is, in quantum mechanics
measurements are probabilistic. So,
whether electron or photon ends up is,
you, you have to do to detect it, you have
to do a measurement, and the measurement
is inherently probabilistic process.
Second thing, you cannot measure without
disturbing the system. Whenever you make a
measurement you can, you can make the
measurement as subtle as you want but you
still disturb the system. The third thing
we learned is that, elementary particles
behave in a very strange way. They behave
like no classical entities with that we
have seen they behave near like particles
nor like waves but it behave in a
completely strange way. Some of the
characteristics seem as though they are
like characteristics of particles. They
behave like bullets. So other
characteristics are those of waves but
really they, they have, they behave like
some funny combination of the two which is
like neither of the two at all. The fourth
thing is that when we do this experiment,
we can say that there's a moment when,
when the electron keaves the source, the
electron gun. And then from then, then on,
we cannot really say what part the
electron took or whether it took multiple
parts at the same time to arrive at, at,
at the point x. And so, we really cannot
say what the trajectory of this electron
is. So what quantum mechanics allows us to
do is its you know, we start the
experiment, it happens quantumly and as
soon as we look, as soon as we measure
that disturbs the system and that gives us
the outcome of the, of the experiment. So
there's this black box nature to, to a
quantum experiment where we, where we
start the experiment to the source of
electrons and then something happens and
then we do this, do a measurement when we
see the outcome that the electron ended up
at point x. But in the middle here, we
cannot really see what happened and what,
what the formulation tells us is, is that
if we had double-slit here then, then the
photon has, the electron has some
amplitude. A1 with which it goes through
slit one and ends up at x. Some amplitude
a2 with which it goes through slit two and
ends up at x. And the amplitude with which
it ends up at x is a1(x) + a2(x) and the
probability of detecting it is the square
of this amplitude. Okay, so that's, that's
the you know that's, that's the strange
behavior of, of quantum particles and okay
so starting next time we'll, we'll, we'll
start from scratch and talk about quantum
bits and the basic axioms of quantum
mechanics.