**Professor Ramamurti
Shankar:** Well, this is just informal discussion till everybody's in here. So any
questions on the subject? What?

**Student:** Every question.

**Professor Ramamurti
Shankar:**
Everything. Okay, well, you know what, you guys should stop and ask more
things as you go along, because there is just no way you could get all of
this. And it's a little strange and only by talking about it, you will at
least know what's going on. There's no way to make it reasonable. It's not a
reasonable world out there. I can only tell you what it is. I take that view;
when I teach quantum mechanics, just tell the rules and say, "This is what
happens. This is how we calculate things." And whether you like the
formulas or not, it's not my concern. And the fact that it doesn't look like
daily life, also not my concern, because this is not daily life. Strange
things happen. But you have to keep me informed on how much you're following
and what you are understanding, at any stage. Don't wait for this to end,
because it's not something where you can go on the last day and figure
everything out. And I will try to repeat at every stage what has gone up to
that point, because the whole thing is only a few lectures, maybe six or so. I
can afford to go back every time to the beginning.

But I know that it makes sense to me, because I've seen it, and I don't know how it sounds to you. I have no clue. You know that and so you have to speak up. You can ask any question you want, and I will try to answer you, if it's within the realm of possibility. Okay, so what have I said so far? So let me summarize. Even if you never came to last lecture, here is what you should know about the last lecture, okay? Here's what I said. First thing I said is, everything is really particles, all things, electrons, photons, protons, neutrons. They are all particles, so let there be no doubt about that. By that, I mean if one of them hits your face, like an electron, you will feel it in only one tiny region, one spot. Electron dumps all its charge, all its momentum, all its energy to one little part of your face. So there's nothing wavelike about that. It's not like getting hit by a boxing glove, which can hit your whole face. An electron hits one dot, or if it's an electron-detecting screen, only 1 pixel is hit by the electron. And into that pixel is given all the charge, all the momentum, all the energy of that electron.

That's exactly what particles do. So when you encounter an electron, it is simply a particle. So where does the problem come in? Where does the quantum mechanics come in? It comes in when you do the famous double slit experiment. That's the key. The entire quantum mystery is in the double slit. Part of the resolution is in the double slit, but the rest are a little more difficult, and I'll try to tell you. First I want to tell you what goes wrong with Newtonian mechanics. After all, if everything is a particle, what's the big deal, what's the problem? The double slit experiment is a problem. That's what puts the nail on the coffin for Newtonian physics, and here it is in the basic version. You've got two slits.

By the way, I'm going to call the particle the electron. They're all doing the same thing, so what applies to one, applies to all of them. There is a source, like an electron gun, that emits electrons. In the old days, televisions had the electron gun. And the gun emits the electrons, they go and hit the screen, they make a little dot, and then the dot moves around, and you see your favorite show. Okay, this is the electron gun, and the electron gun has been engineered to send electrons off a definite momentum. That you can get by accelerating the electrons over a definite potential, and the gain of so many electron volts will turn into kinetic energy. As for direction, if this gun is really far away to the left, in principle 1 mile, then the only way electrons are going to go 1 mile and hit the screen is they're all basically moving in the horizontal direction. Then you put a row of detectors in the back, which will detect electrons. Then this is slit 1 and this is slit 2. You block slit 1.

In fact, let me say the following thing: what do we really know when we do the experiment? Once in a while this gun will emit an electron, and we know it's emitted the electron, because it will recoil one way, just like a gun, rifle. It will recoil. That's when we know the electron left. Then we don't know anything, and suddenly, one of these guys says click. That means electron's arrived here. This is what we really know. Everything else you say about the electron is conjecture at this point. You know it was here, you know it was there. The question is, what was it doing in between? Now if you say, "Look, things cannot go from here to there, except by following some path, I don't know what path it is." Maybe if it's an ordinary particle, like a Newtonian particle, it will take some straight line, hit that slit, or go through that slit and arrive here. So you might say, "I don't know the trajectory, but it's got to be some trajectory, maybe like that, or maybe like that." So the electron takes some path and you can label the path as either through slit 1 or through slit 2.

Okay, now here is the
problem. Suppose I do the experiment with slit 2 blocked, so you cannot even
get through this one, and I sit at a certain location for a certain amount of
time, maybe 1 hour and I see how many electrons come, and I get 5 electrons,
with only 1 slit open. And if I move that observation point, I get some
pattern, pretty dull, looking like that, and I'm going to call it *I*_{1}.
That's the count, as a function of position up and down that wall of
detectors. Then I repeat the experiment with this guy closed and that guy
open, and I get another count, looks like that. Now I'm going to pick a
location. These are not drawn to scale or anything, so I'm not responsible for
any of that. Maybe I'll at least show you one thing, which is pretty
important. This graph will be big in front of the second slit, which is
somewhere here. It will look like that. So I get *I*_{1} when 1
is open and I get *I*_{2} when 2 is open. This is *I*_{1},
this is *I*_{2}. Now I'm going to open the two slits and I'm
going to pick a particular location. It doesn't happen everywhere. I'm going
to pick one location called *x*. Where I used to get 5 electrons per hour
with one thing open, and 5 electrons per hour with the second thing open.

Now I want to open both
and ask, what will I get? In Newtonian mechanics, there's only one possible
answer to that question, and that is 10, because we've got 5 this way and
you've got 5 that way. And you open both, whoever is going this way will keep
going that way; whoever is going this way will keep going this way. They will
add up to give you 10. Now I told you, some people may say, "Well, maybe
it's not 10, because with both slits open, maybe someone from here can collide
with someone from there. How do you know that will not happen?" So I'm
saying, do the experiment with such a feeble beam of electrons, there's only
one electron at a time in the whole lab. It's not going to collide with
itself. Then you wait long enough, and you have to get 5 + 5 = 10. And what I'm
telling you is that if you go to the location marked *x*, where you've got
5 with each one open, when you open both, you will get 0. You don't get
anything. That is a great mystery.

That is the end of
Newtonian physics. And I told you that something like that never happens in
your daily life. I gave an example with machine guns. This is a machine gun.
This is a concrete wall with 2 holes in it, and there's some target here, you.
And then you see how many are coming through this and how many are coming
through that. Then you go there and you wait. And both are open. Somehow,
nothing comes. With the second hole in the wall, you are safe. With one hole
in the wall, you're not safe. That can never happen with bullets. So these
electrons are not following any path, because the minute you commit yourself to
saying it follows one path, either through this one or that, you cannot avoid
the fact that with both of them open, the intensity with 1 + 2 has to = *I*_{1}
+ *I*_{2}. That's the Newtonian prediction. And *I*_{1}
+ *I*_{2} gives you 5 + 5 = 10 here and you get 0. In other
places, instead of 10, you will get 20. Some places you get more, but more
dramatic thing is where you get less, where you get nothing.

Therefore you abandon
the notion that electrons have any trajectories. You don't want to abandon it,
but you have to, because that assumption, which is very reasonable, just
doesn't agree with experiment. Then you say, "Okay, what should I do?
Newtonian mechanics is wrong. What's going to take its place?" To find
that, you have to move away from this *x* and move up and down this row
here and see what you get, and I think I told you what you get. You get a
pattern that looks like this. So the real I1 + 2 looks like this: it's got ups
and downs and ups and downs. And let's say these downs really correspond to
0. That means nobody comes here, a lot of them come here, no one comes here, a
few come here and so on. That's what you find. I'm just telling you what
happens when you do the experiment. So you put yourself in the place of a
person who did the experiment. You thought of moving away from that point *x*
and you plot it, it makes no sense in the language of particles.

But this is such a
familiar pattern. If you're a trained physicist, which you guys are, you will
say, "Hey, this reminds me of this wave interference, with water waves or
sound waves or any waves. " Obviously there's some wavelength. The minute
you give me wavelength and a slit separation, I can calculate this pattern. *dsinθ
= **l* and whatnot, and
from that sin*θ*, where you get a minimum or a maximum, and if there's a
certain separation to the screen, you can find the precise location of these
maxima or minima. Or given the maxima and minima, you can work back and find
the wavelength. And the wavelength happens to be some number called *ℏ,*
which is 10^{-34} joule seconds, divided by the momentum. *p* is
the momentum. In other words, you find that if you send more energetic
electrons, accelerate them through bigger voltage, increase the p, *l* goes down, the pattern gets
squeezed. You slow down the electrons, *p* reduces, *l* increases, the pattern spreads out,
and the dependence on momentum is inversely proportional to momentum. And you
fool around and find out the coefficient of proportionality, which people used
to call *h* in the old days is now written as *2**p** xℏ*, but it doesn't
matter, it's some constant. And the number is 10

So you can successfully reproduce this pattern, but what does it tell you about what's going on? What good is that pattern? The pattern tells you that if you repeat the experiment with this electron gun a million times or a billion times and you plotted the histogram patiently, the histogram will eventually fill out and take this shape. So this wave is not the wave associated with a huge stream of electrons. A single electron in the lab is controlled by this wave. You need a whole wave for 1 electron, so it's obviously not a wave of electrons. It's not a wave of charge, like the wave of water. It's a mathematical function and you are drawn to it, because the only way you know how to get this wiggly graph is to take something with definite wavelength and let it interfere. So you're forced to think about this wave. And the intensity of the wave, the brightness if you like, the square of its height, gives you what? Gives you the graph you will get if you repeat the experiment many times. And what does it mean for the individual trial? What does it mean for the millionth + one electron? For the millionth + one electron, it gives you the odds of where it will land on that screen, okay?

You can never tell
exactly where it will be. You can tell what the odds are, and the only way to
test the statistical theory is to do the experiment many times. And if you do
it, it works, and it seems to work for everything, for electrons, for protons,
for photons, whatever it is, the wavelength and momentum are connected in this
fashion. So this wave is forced upon us, and it gives you the odds of finding
the electron somewhere. And we say that the probability-- I'll be a little
more precise in a minute on what I mean by the probability to find it at a
location *x*, but let's just say, if you draw the graph of *Y** ^{2}*, wherever it is big,
the probability is larger; wherever it is small, probability is small; wherever
it is 0, probability is 0. So there seems to be a function whose amplitude or
whose square gives you the probability. That function is called the wave
function, and we know it exists, because it's the only way to calculate the
result of this experiment.

Once you tell me that the fate of a particle is controlled by a wave, you're immediately led to some other conclusions, so I'm going to tell you what they are.

First conclusion is
this: if I make a single slit, let's call this the *x* direction, let's
call this the *y* direction, and I'm sending a bunch of particles in the *x*
direction with some momentum *p*_{0}. In Newtonian mechanics, I
can manufacture for you an electron of known momentum and known position, or
known to arbitrary accuracy. If *D**x* is the uncertainty in my position, and *D**p* is the uncertainty
in momentum, I can make each of them as small as I like. So here's an actual
practical way to prepare such a state. If I say, "Give me an electron of
known position, known momentum," here's what I will do. I will take a
slit with a very tiny hole in it. The width of that hole is *d*, and
whoever comes out on the other side, what can I say about that particle? Its
position has an uncertainty of order *d*, because if it was not--I'm
sorry, *D**y* now, because this is
the *y* direction. Right? Anything who came out of hole right after it
had to have a y position, whose known to within *d*. It's got to come
anywhere within the slit, but that's it. So that is how I have prepared for
you, that's how we filter electrons of definite position. And you can make *D**y* as small as you like
by making the slit as thin as you like. What about its momentum? If you had a
momentum *p _{0}* in the

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Right, but how will I calculate what will happen in this experiment? What will
decide? Yes?

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Yes, this is one hole, and a light beam is coming from the left--I mean, a wave
is coming from the left, and if the particle has momentum* p _{0}*, it's got some wavelength, which
is

Now a particle cannot
go from this slit to there unless it had a momentum, which had a component in
the *y* direction. You cannot get there from here, unless you have y
momentum. It's the *y* momentum is uncertain to within that
cone. So what's the uncertainty in the *y* momentum? For a vector of
length *p _{0}* when it gets shifted by an angle D

That's just wave
theory. People knew this about waves hundreds of years back. What is novel is
that this wave is going to tell you where the particle will end up. This wave
is going to control the odds of where the particle will end up, and the odds
are pretty much concentrated in this cone, not of 0 opening angle, but an angle
*θ*, so that *dsinθ = **l*. l in turn is connected to the momentum
of the particle. This is where the uncertainty principle comes in. What
Heisenberg said is, “You had in your mind the classical notion that you can
have a particle of known position and known momentum. Let me see you produce
that particle. So you try to do it by putting a slit and catching guys with a
very narrow range in *y*, but now you find out that the momentum gets
broader and broader, and that's the result of the wave associated with the project.
“ It was not in the Newtonian picture. Yes?

**Student:** Why with the
single slit, like you've drawn there, do you get the little wiggle at the
outside--?

**Professor Ramamurti
Shankar:**
You mean, why would it have those wiggles? Okay, so if I look at the single
slit, I could think of many little point sources. In the forward direction, if
you go very far, so you treat them as roughly parallel, they are all in step.
You've got a big maximum. In another direction where this difference is l, *d*sin*θ* = l, this guy and this guy are differing
by *l*/2.

**Student:** So you're assuming
the slit is large enough for them to do that ________.

**Professor Ramamurti
Shankar:**
These are little mathematical dots inside the slit. You know, when you make a
slit, every point on the slit looks like a source of light, a point source of
light. They are not really light bulbs, but if you make any hole in the wall,
a light comes through that hole, looks like it's a source of light. I'm taking
every point on the hole to be a source of light. And what I'm saying is, there
is a direction in which it will cancel, but if you go a little further out, it
won't cancel completely. This is the direction for perfect cancelation, where
I can pair them, this one with this one, that one to that one, and so on. They
pair to give 0. But if you move further up, you no longer cancel completely,
but you don't add perfectly either, so things will get better, then they'll get
worse again, and better and worse and so on. So that's the origin of that
pattern. Yes?

**Student:** So you're saying
this is the first minimum. After that, it would increase _______.

**Professor Ramamurti
Shankar:**
Yes. For a single slit diffraction, the big thing in the middle is pretty much
all you have. It's not like double slit experiment with two holes, where you
get many times the pattern. That's because--let's understand why that is
true. Here, if these two differ by 1 wavelength, they don't differ at all. I
can find another angle where they differ by 2 wavelengths. They don't differ
at all. There's only 2 sources, so you can engineer them to differ by either 1
wavelength or 2 wavelength or 3 wavelengths. Here in a single slit, each point
is like a source of light. You got them all to agree. You can get them to agree
only in the forward direction. In any other direction, you can get them to
neutralize each other, but never for perfect reinforcement. So you cannot get
it more than--this is the only real maximum here. Everything else is tiny.
Yes?

**Student:** Why can't you do
the same trick with the double slit, where each of the one slits has its
_______?

**Professor Ramamurti
Shankar:**
No, in double slit, what's happening is, we take--in double slit, the number *d*
I used in double slit was not the size of a slit, but the space in between the
slit, okay? So there I took the size of every slit to be vanishingly small.
That means a light coming out of this slit spreads out completely in all
directions, okay? It's as if this packet became that broad. Likewise a light from
this one, we are still in the first maximum of that slit. So you've got to
understand, in a single slit experiment of diffraction, the slit size is what
you are varying. In a double slit experiment, the slit is taken to be
mathematically point like, so it fans out completely. It's the interference
between those two point sources that you're adding. They can add and cancel,
add and cancel, many, many times as you move along this line.

Okay, so what we
learn is it's when you combine waves and particles and go back and forth that
you run into the situation. So you cannot make a state of perfect momentum.
By the way, I said one thing, I thought about it, which is incorrect, which is,
in the microscope, I said if you want to locate the position of an electron in
a microscope, take a microscope with an opening, and electron is somewhere on
this line. I said you're shining light down here. It hits the electron, but it
goes in through the slit by spreading out. So the photon that came in goes
into the eyepiece with a certain uncertainty in its final angle. That means we
know the incoming momentum, but we don't know the outgoing momentum of the
photon. The lens picks up everything inside that cone. That means we don't
know how much momentum it gave to the electron. It gives an indefinite amount
of momentum to the electron. Therefore the *x* momentum of the electron
is uncertain by that little shape, the conical shape of the momentum. And if
you do the uncertainty principle argument, you'll again find *D**x**D**p is h, h*.

Now what I don't like about my experimental setup is that I had the incoming light also coming from inside the microscope, but that means incoming light, when it comes through this hole here, will itself spread. Then it will hit this guy, and that will go to the aperture. That will spread some more. This uncertainty in incoming momentum is unnecessary. We can do better than that, because in other words, when the light is picked up, it is picked up by this tiny hole. There's no reason it should also come from the tiny hole. It can come from a source far away, say on the other side, so that it is a well defined direction, it's not diffracting at all. So I want it to come in through a very broad hole, so it's got well defined direction, so the light here has known momentum. It hits the electron and goes into the microscope. It is the final momentum of the photon I don't know. And I cannot make it better. If I make it better, I've got to open this eyepiece a lot. If I open the eyepiece a lot, I don't know where I caught this guy. So again the problem between taking a very tiny eyepiece, so that if I see a flicker, I know the electron was in front of it, but the light coming from the reflected electron fans out more and more.

Okay, so anyway, this
is the uncertainty principle and the uncertainty principle told us something
very interesting. I asked you, what can be the function here that produces
this interference pattern in the double slit? We know the wavelength.
Wavelength was *2**p*__ℏ__*/p*. And you know from
basic physics that a function like cosine *2px/**l* has got wavelength l.
So let's put in the formula for l
here. You get cosine 2*p**x*. l is 2*p*__ℏ__*/p*. Cancel the 2 p's, you get *A* cosine *px/ℏ*.
That function, when you throw it at a double slit, will form two little
wavelets, and they will interfere, that produce an interference pattern of the
type you want.

Do you understand
that the experiment only showed you there's a wavelength. It did not tell you
what the actual function is? That's very, very important. When Young did the
experiment with the double slit, he found the oscillations and he could read
off the wavelength. It's just geometry. But he didn't know what was
oscillating. He didn't know there's an electric and magnetic field underneath
all of that. But you can always read out the wavelength without knowing what's
going on. Likewise, we have the wavelength. We know it comes from a function
with a well defined wavelength, so I make my first guess to be this function.
But I told you what was wrong with this choice. You guys remember that? I said
this function violates the uncertainty principle. The uncertainty principle
says if you know the position to an accuracy D*x*, and if you know the momentum to accuracy D*p*, the product must be
at least as big as * hx* some number of order 1. We have taken the
particles to have well defined momentum. If they have well defined momentum, D

But then
what came to the rescue is the following function, not a cosine but *A*e^{ipx}^{/ℏ},
rather than cosine *xp/ℏ*. Look at this function. This function, I've
told you many, many times, if you don't know your complex numbers, you're
definitely going to have trouble. It looks like a vector of length *A*
and angle *θ*, which is px/ℏ. As you vary *x*, this *x* changes
and this will rotate round, but as it rotates the amplitude of this complex
number, absolute value of *Y** ^{2}*, which
is

That cancels out, you
just get *A*^{2}. In other words, the complex number describing
the wave function changes in phase but not amplitude. It's the amplitude that
gives the probability. Now there is no problem with this guy having a
wavelength, because this oscillates in *x*. Its real part and imaginary
part both oscillate, but the square of the real + the imaginary square is 1
[A^2]. That's why the amplitude doesn't change.

So we are driven now
to the very interesting result that the wave function for a particle of
definite momentum *p* is this. So this is a very important lesson. Let
me label this function by label *p* to tell you, "Hey, I'm not
talking about any old wave function." This guy has a definite momentum *p*.
Its wave function looks like e^{ ipx/ℏ}
times any number *A* you want in front of it. That's a very important
thing to know. This is called a plane wave, and a plane wave with a *p*
right where it is describes a particle of momentum *p* in particle
mechanics. And I told you particles of momentum *p* are everywhere. Every
machine produces them, every accelerator produces them, and if you want to
describe them in quantum mechanics, you have to know complex numbers. There's
just no way you can get a real answer to our predicament. It's complex.

So that's roughly where I left you, and I want to remind you of a few other things, this further discussion of the result we have, okay? The discussion is, if the world is really this messed up at the microscopic level, why do I think it's the world I see in the macroscopic level? Where are all these oscillations? Why is it that when there's a concrete wall, making another hole is bad for me and not good? Why do all these things happen? Why do I think particles have definite momentum and position? Why do I think that if I make a hole in the wall and I send a beam, the beam will go on the other side of the wall with a shape precisely like the shape of the hole, no spreading out? It all has to do with the size of the object. The laws of physics are always quantum mechanical laws, but when you apply it to an elephant, you get one kind of answer; when you apply it to an electron, you get another kind of answer. You don't have separate laws for big and small things. The real question is, how do these very same laws, when applied to big things--by big things, I mean things you see in daily life--give the impression that the world is Newtonian? So let's look at the double slit experiment.

Here's a double slit
and we are told, "Send something. See what happens on the other side."
And the prediction is that you get these oscillations, with the peculiar
property that with two holes open, you don't get anything somewhere. We don't
seem to see that in daily life, and you can ask, "Why is that so?"
Well, you remember that the condition for the next minimum is like *d*sin*θ*,
is *l*/2. So if *θ*
is very small, it's like *θ* x*d* is l/2, or *θ* = l/*d*.
That's the angle you've got to go through from the central maximum. That's the
central maximum to the first minimum here. That tiny angle is given by *l*/*d*. The reason you don't see
the oscillations is when you put in the values for *l* and *d*. Let's pick a
reasonable value for this angle *θ*. Do you understand what *θ* is?
In that maximum, there are some oscillations. I want to go to the first
minimum near that. The distance between these two is roughly the spacing
between maxima and minima, maxima and minima. That angle is *l*/2. *l* is 2*p** ℏ* over the momentum and that is

You find this number
is 10^{-34} radians. That means that the angular difference between
the maximum and the minimum and the maximum and the minimum is 10^{-34}
radians. What that means is, if you put a screen 1 meter away, the distance
between one maximum and the next maximum or one minimum to the next minimum and
so on, that spacing will be 10^{-34} meters, because that's how a
radian is defined. If that angle is* θ*, that distance is just the
distance to the screen times *θ*. That will be 10^{-34} meters.
That means the wavelength of the oscillation on your screen is 10^{-34}
meters. Can you see it? Well, make the world's smallest detector. It's as
big as 1 proton, okay. Nothing can be smaller. That's your whole detector,
all the parts, everything, 1 proton. Size of a proton is 10^{-15}
meters. That means you will have 10^{19} oscillations inside your tiny
detector. So don't be fooled by the 19. Let's take a minute to savor this.

That's how many
oscillations you have, okay? You've got enough now? 3, 6, 9, 12, well, I
don't have enough time. That's a lot of oscillations. You should check the
numbers though, okay? I'm saying they typical angle will be 10^{-34}
radians and if you put the screen 1 meter away, the spacing will be 10^{-34}meters,
and you look at it with an object, a detector, whose size is 10^{-15}
meters, which looks very small, size of a proton. But look, 10^{19}
fit into that length, so your proton detector looks huge. In fact, I cannot
even show it here. So you don't see the oscillations; you see the average
only. If you see only the average, you can show that with 2 slits open, the
sum is the sum of the two averages, so you don't see the oscillation. That's
the first reason. Now you can say, "You took a kilogram. Let me take a
gram." I said, "Go ahead. Take a gram, take a milligram and take a
slit which is not 1 meter wide but--1 meter apart, but 1 millimeter apart."
It doesn't matter. You're playing around with factors like 10 and 100 and
1,000. I got 10^{19} here. So nothing you do will make any dent on
that. So in the macroscopic world, you will not see this interference.

Another reason you won't see it is that the particle should have a definite momentum. It's got an indefinite momentum, it's coming in with different momenta, then each will have its own interference pattern and they'll get washed out. Finally, I told you, if you ever try to see which slit the particle took by putting a light beam here, the minute you catch the electron going through one slit or the other, this pattern is gone. It will do this "I'm not here and I'm not there" routine only if you never catch it being anywhere. That's very interesting. The electron behaves like it does not go through any one particular slit, as long as you don't catch it going through one slit. You put enough light to catch every electron, then you can add the numbers and you must get the sum of the two numbers. Now for the atomic world, it's possible for the electron to go for a long time without encountering anything, and the interference effects come into play. In a macroscopic world, there is no way a macroscopic object can travel for any length of time without running into something. It will run into other air molecules. It will run into cosmic ray radiation. It can collide with black body radiation from the big bang, anything. The minute you have any contact with it, this funny thing will disappear. So that's one reason you don't see it.

Now we can go on and
on and give other numbers. I've given examples in my notes, which I will post
later on. One of them is the uncertainty principle. Why does it look like the
uncertainty principle is not important? Take again, this is 10^{-34}.
Everything is in MKS units. So take an object of mass 1 kilogram whose
location is known to the accuracy of 1 proton. Okay? So this number is 10^{-15}
meters, do you understand? You take an object made of 10^{23} protons
and you know its location to the width of 1 proton.

That's all you don't
know about its location. That's your *D**x*. What's the *D**p*? Well, *D**p* is now 10 to the, what, 19? 10^{-19}. Now *D**p* is *m* times *D*v. That's 10^{-19}. If this
is 1 kilogram, *D**v* is 10^{-19}meters
per second. You don't know its velocity to 1 part in 10^{19}. Now how
bad is that? Well, suppose I start a particle off exactly known velocity, I
know where it will be forever. But suppose I don't know the velocity to this
accuracy, and I let it run for 1 year. So I don't know precisely where it is,
but how bad is it? How badly do I not know? Well, 1 year is 10^{7}
seconds, so if it runs for 1 year, it will be unknown to 10^{-12 }meters.
10^{-12}meters is what, let's see? It's 1/100 of an atom size, 1/100 the
size of an atom.

So you see, these uncertainties are not important in real life. So everything that you think has a definite position and momentum actually has a slight uncertainty, but the uncertainties don't lead to any measurable consequences over any distances that you can actually have. So what I'm trying to tell you is, there are these waves. They do all kinds of things, they do interference and everything, but the condition for them is really the microscopic world. The minute the masses become comparable to gram or kilogram and distances and slits and so on, or like a meter or a centimeter, these effects get washed out. But in the atomic scale, they are seen.

Now the final thing I want to mention before moving on to a completely new topic is the role of probability in quantum mechanics. We have seen that quantum mechanics makes probabilistic predictions. It says if you do the double slit experiment, I don't know where this guy will land, but I'll give you the odds. Okay, now that looks like something we have done many times in classical mechanics. For example, if you have a coin and you throw the coin, you flip it and you say, "Which way will it land?" well, it's a very difficult calculation to do, but it can be done in principle, because a coin, once released from your hand, can only land in one way. That's the determinism of Newtonian mechanics. If you knew exactly how you released it with what angle or momentum, what's the viscosity of air, whatever you want, if you give me all the numbers, I'll tell you it's head or tails. There's no need to guess. In practice, no one can do the calculation. What you do in practice is, you throw the same coin 5,000 times and you find out the odds for head or tails and you say, "I predict that when you throw it next time, it will be .51 chance that it will be heads." That's how you give statistical predictions. Now you did not have to use statistics. You use it, because you cannot really in practice do the hard calculation. In principle, you can. Secondly, if you toss a coin and I hide it in my hand, I don't show it to you, it's either head or tails, and I say, "What do you think it is?" you'll say, "It's .1 chance that it's heads." And I look at it, it may be head or it may be tail. Suppose I got head. It means that it was head even before I opened my hand, right? The correct answer's already inside my hand.

I just didn't know it. I'm using odds, but when I look at it, I get an answer. That's the answer it had even before I looked.

So I'll give another analogy here. So this is a probability of locating me somewhere. This is my home town, Cheshire, this is Yale, and this is the infamous Route 10. So somebody has studied me for a long time and said, "If you look for this guy, here are the odds." Either he's working at home or he's working at Yale, and sometimes he's driving, okay? This is the probability. First thing to understand is the spread out probability does not mean I am myself spread out, okay? Unless I got into a terrible accident on Route 10, I'm in only one place. So probability's being extended doesn't mean the thing you're looking at is extended. I am in some sense a particle which can be somewhere. These are the odds.

Well, suppose you catch me here on one of your many trials. If you catch me only once, you don't know if the prediction's even good, so you repeat it. You study me over many times and you agree the person had got the right picture, because after observing me many, many days you in fact get the histogram that looks like this. The important thing is, every time you catch me somewhere, I was already there; you just happened to catch me there. I had a definite location. It was not known to you, but I had it. I had a definite location because in the macroscopic world I'm moving in, my location is being constantly measured. You didn't ask or you didn't find out, but I'm running through air molecules. I've slammed into them. They know that. I ran over this ant. That was the last thing the ant knew, okay? So I'm leaving behind a trail of destruction and they all keep track of where I am. My location is well known. You just happened to find out.

But now let's change
this picture and say this is not me. This is an electron and it's got two
nuclei. This is nucleus 1 and this is nucleus 2. It can be either near this
nucleus or that nucleus, and this is the *Y** ^{2}* for the electron. That means it's
the probability you'll catch it here and you'll catch it there. Once again, if
you catch the electron, you will catch all of it in one place. It is wrong to
think the electronic charge is somehow spread out around the atom. It's not
true. The charge is in one place. The odds are spread out. That looks just
like my case. But the difference is, if you catch the electron--let's in fact
simplify life and say there are only 2 possibilities. Either it is near atom
1--nucleus 1 or nucleus 2, only 2 discrete choices. If you catch it near 2, it
is wrong to think that it was there before you got it. So where was it? It
was not in any one place. It had no location till you found its location.
That's very strange.

We think of
measurement as revealing a pre-existing property of the object. But in quantum
theory, it's not that you don't know where the electron is. It does not know.
It is not anywhere. It's the act of measurement that confers a location or
position on the electron. That state of being, where you can be either here or
there, or simultaneously here or there, has no analog in the classical world.
If anybody tries to give you an example, don't believe it, because there are no
examples in the macroscopic world that look like this. No analogies should
satisfy you, because this has no analog in the real world, okay? So this is
the interesting thing in quantum mechanics. If this is a possible wave
function *Y*, electron near
nucleus 1, and that's a possible wave function *Y*, electron near nucleus 2, you can add
the two functions. That's another possible function. But what does that
describe?

It describes an electron which upon measurement could be found here and could be found there. It's not like finding me in Cheshire or finding me in New Haven, because in those cases, on a given day on a given measurement, you can only get one answer, depending on where I am. Right now if they look for me, they can only find me here. They cannot find me anywhere else. But in the case of the electron, the one and the same electron, on a given trial, at a given instant, is fully capable of being here or there. It's like tossing a coin and it's in my hand. We all know that when I reveal it to you, you can only get one answer, now that the toss has been done, it's got one answer. If it's a quantum mechanical coin, you don't know, and it doesn't have a value till you look. When you look, it has a definite value. Before you look, it doesn't have a definite value. That's exactly like saying, when you looked, it goes through a definite slit. When you don't look, it's wrong to assume it went through a definite slit. Yes?

**Student:** Say you had a
double slit experiment but instead of having a screen that went all the way in
both directions, you just sort of had _____ screen. So then you would only be
looking at the final location of a _______ electron. The other half you would
know. How would that work, because location for some of them has to be
____________.

**Professor Ramamurti
Shankar:**
The minute you find the location to the accuracy of knowing which slit it went
through, you've got 2 slits or only 1 in the experiment?

**Student:** You have 2 slits
but only a half screen, and nothing on the other one.

**Professor Ramamurti
Shankar:**
Oh, you've got a screen that only comes to here, you mean?

**Student:** Yes.

**Professor Ramamurti
Shankar:**
Yes, the real problem of location that I'm talking about here is not when it
hits the screen, but here, when you try to see which hole it went through, by
putting a light source here. I was referring to the fact that if you have the
right kind of light to know which hole it went through, if you give it enough
momentum to wash out the pattern. As far as the screen goes, once it comes
out, it's the sum of these 2 waves coming from the 2 holes, and it also doesn't
have a well defined position. The probability for finding it may look like this.
In fact, the probability will look like this. Forget your screen. This is the
probability. The minute you catch it, it is found there, that's the location
after that measurement. Prior to the measurement, it can really be anywhere
where the function is not 0. There are many wave functions. There's the 1 to
the left of the slit and there is 1--then it becomes 2 wave functions coming
from the 2 slits. They form the interference pattern and that gives you the
odds that if you looked for it, you will find it.

Now till you find it, it's not anywhere. It can be anywhere on this line at that instant. It's only the act of measurement, or hitting a detector that tells you that's my location. So you will have to get used to that. You'll have to get used to the fact that things don't have position, momentum, angular momentum, energy or anything, until you measure it. Okay, so I've got to tell you a little more now about just position. So let's take--by the way, any questions so far? Yes?

**Student:** Can you explain
again how you can tell which hole the electron goes through with the light?

**Professor Ramamurti
Shankar:**
Well, you just see it. You see a flicker and if it's near this hole, you know
this guy went through that. And if it's near that one, you know it went
through that. But to have such good resolution, the wavelength should be much
smaller than the space in between the slits. Otherwise you'll get a big blur
and you won't know which one it went through. That's a soft measurement that
doesn't do you any good. It won't destroy the pattern. That's because you
don't know which hole it went through. If you make it fine enough to know
which hole it went through, you will disrupt the electron's momentum enough to
wipe out the pattern. Yes?

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Oh, here? You mean what happened to the 2? Oh yeah, forget the 2s. There are
2p's I forgot, right?
There's a lot of 2p's too you've got to
put in. 10^{-34 }is not exactly the answer, because I got 2p there. I'm just saying, look, if
it's 10^{-34}, suppose you're talking about how much money Bill Gates
has. It's 10^{9}. Now is it 2 x 10^{9}, 3 x 10^{9}?
I don't know, but I'm not worried about his financial wellbeing, because it's
up there in the 10^{9}s. So whenever I do these arguments, you should
get used to this notion, it's very common for physicist when they argue in
quantum mechanics, will use the symbol that's not quite an =. It looks like a
wiggle and =. Basically it means, I'm not quite sure, but the number is in
this ballpark. And if this ballpark is 1,000 miles from that ballpark, you
just have to know it's in the ballpark. You don't have to know where it is.
Some things are definitely macroscopic; some things are definitely
microscopic. But something very interesting is happening at Yale right now.
People are asking the following question: how small an object has to be before
I can see its quantum mechanical fluctuations? We know that if it's like an
electron, it's completely fluctuating. You don't know anything. If it's like a
bowling ball, it seems to have a well defined position. Make the objects
smaller and smaller and smaller. How small can it be before it's first
beginning to show quantum effects, like quantization of energy or quantization of
momentum, depending on the problem, or quantum fluctuations in position? So
those measurements are now being carried out at Yale. It's a very exciting time
and it's so many years after the discovery of quantum mechanics. Because we
knew the really big world and we knew the really small world, but now we're
trying to go, because of nanotechnology, continuously from big to that small,
and how small is small?

That's the question? Can a little macroscopic object simultaneously be in two places? Most of them seem to have a well defined location. Can you create a situation when it's capable of being found here and found there? It's very hard, because you have to isolate the particle from the outside world. That's the first condition. That's what ruins everything.

A quantum computer,
you must know, has got these bits called qbits and unlike the bits in your
laptop, which are either 0 or 1, a qbit can also give you 0 or 1, but it can
also be in a state where it can give either 0 or 1 on a given trial. The bits
in your computer, the particular bit right now is either a 0 or a 1. Maybe you
don't know it, but it can only give you that answer because that's what it is.
Because that bit is in contact with the world and the world is constantly
measuring its value. A qbit is a quantum system which can do one of two
things, but it's isolated and it's neither this nor that. It's like the
electron going through both slits. So a quantum bit can explore many
possibilities. If you build a computer with 10 qbits it can be doing 2^{10}
things at the same time. And if it's got a million bits, it's 2^{million}
operations, things it can be exploring at the same time.

That's why, as you
know, one of the ways to securely send your credit card information is to use
very large numbers, on the assumption that no one can factorize them. You can
always multiply a 100 digit number by a 100 digit number on your computer
instantaneously. But if I gave you the 200 digit number and told you to find
the two factors you won't find it. You won't find it in the age of the
universe. It's amazing, but that simple problem of factorization cannot be
done if the numbers are 100 digits long, and that's the reason why people
openly broadcast the product, they may broadcast one of the numbers, and only
the other person knows the second number. Now there is something somebody
called Shore, Peter Shore, showed that if you have a quantum computer, made up
of these qbits, it can actually factor the number exponentially faster, namely,
instead of taking 10^{10 }seconds, it will take 10 seconds. So if you
build a quantum computer, you have two options. Either you can become famous,
or get tenure at Yale, maybe, or you can go on the biggest shopping spree of
your life, because you can get anybody's credit card number. So that's the
choice. When you come to that fork, you decide which way you want to go.
Maybe you can go through both choices, I don't know. That's something in your
future.

So why is it so hard to build a quantum computer? There are many, many quantum systems which can do one or two things, and can be the state, but they are both this and that. The problem is, they cannot be in contact with the outside world, because single contact with them is like a dream. Think about it, it's gone. Same thing. Any measurement destroys it. Any unintended measurement also destroys it, so you've got to keep your system fully isolated. A system that is not talking to the outside world, unfortunately, is also not talking to you. So you cannot ask it any questions, and if it knows the answer, it cannot tell you. So sometimes you want it to talk. It's like relationships. Sometimes you don't want it to talk. So what do you do? You've got to build a system where sometimes, in a controlled way, you can make contact with your system, namely give it the problem. Then it does its quantum thing, then you've got to make a measurement to find out what the answer is. Then you want to be able to get into it again. So the challenge for quantum computers is how to keep them isolated long enough to do the calculation. That's a challenge, how to keep it from--how to keep it in what's called a quantum coherent state. A coherent state is really when it's doing many things at the same time.

All right, so I want
to tell you now more formally how to do more quantum mechanics. So let's take
a simple example, a particle living on a line. That's the function *Y*(x). Let's ask the following
question: how do we do business in Newtonian mechanics? We say, "Here's a
particle. That's its *x*. Here's the momentum. That's its *p*."
Given that, I know everything I need to know right now. Angular momentum, *r* x *p*, Kinetic energy, *p*^{2}/2*m*.
Everything is given in terms of the coordinates and momentum.

In quantum theory,
you don't even tell me where it is. For every possible *x*, there is a
function whose square, if you now square this guy, everything will be now
positive, definite. I don't know, it's something like this. This is *Y** ^{2}*. We say the height
is proportional to finding it everywhere. What is the condition on the
function side?

The answer is,
whatever you like. Anything I can draw, with no special effort, is a possible
function for an electron. There are no restrictions. It's like saying, what
should the position of the particle be? *x*. Any *x* you want is a
possible *x*. Likewise, any Y
you draw is a possible Y. That's a set of
all possible ampli--it's called wave
functions--whose square is the set of all possible probabilities. So I said *Y** ^{2}*, Y at the point x

So what we really
mean by *Y** ^{2}* is called the
probability density. That means draw the

So when you multiply Y by any number, you don't change the
basic predictive power of the theory. It is just that if your original Y had a square integral = 1, the new
one may not have, but the information is the same. It's really the relative
height of the function. That's another shocking thing in quantum mechanics.
If Y stood for a string
vibrating, 2 × Y (this is Y, this is 2 × Y) is a totally different configuration
of the string. But in quantum mechanics, Y
and any multiple of Y are physically
equivalent, because what we extract from *Y*
is the relative probability of finding it here and there and there and there.
So scaling the whole thing by a factor, 2 or 4 or any number, it doesn't
matter. That's a very new thing. That's why the *Y* is not very physical. If you took a
string and you pulled it by twice as much, it's a totally different situation.
If you took the electric field and made it twice as big, that's a different
situation. Forces on electrons are doubled now. In the quantum mechanical *Y*, when you double it, it stands for
the same physical condition of the electron, because the odds of being here versus
being there are not altered. The only job of *Y* is to give you the odds.

Therefore it's like
saying in 2 dimensions, that's a vector, that's a different vector. But
suppose you only care about the direction of the vector. For some reason, you
don't care how long it is, you just want to know which way it points, then of
course all of these are considered equal. And that's really how it is for
quantum mechanics. Every *Y*
and every multiple of it stands for one situation only. So what one normally
does is from all these vectors in that direction, you may pick one whose length
is 1 and say, "Let me use that member of the family to stand for the
situation." That's like saying of all the *Y*'s obtained by scaling up and down,
I'll pick one whose square integral is 1.

So let me do a
concrete example, so you know what I'm talking about. So let's take a function
that looks like this. It is 0 everywhere, and it has a height A between +a and
-a. That's my *Y*. So *Y*(*x*) = A for absolute value(*x*)
less than a and = 0 outside. That's a possible wave function *Y*. Now what does it tell you in
words? If it's a word problem, what does it tell you about the electron? Can
anybody tell me? What can you say about the electron given by this function?
What do you know about it? Yes?

**Student:** It must be found
within + or -a.

**Professor Ramamurti
Shankar:**
It must be found within + or -a and more than that.

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
That is correct.

**Student:** With the same
probability.

**Professor Ramamurti
Shankar:**
With the same probability, okay? The probability is it restricted to -a to +a,
and it's the same throughout the interval. After all, if you just set it
restricted to -a to +a, it's true for this function too, but that's not the
same everywhere. I've got a guy who's same everywhere. Do you agree that this
function has exactly the same property, restricted to -a to +a, and the
probability's constant? So there are many, many functions you can draw, all
with the same statement that this object has got equal likelihood to be in this
interval and 0 outside. Of this family, we are going to pick one guy whose
square integral is 1. So I'm going to keep this number *A*, the height of
the function, as a free parameter, and I'm going to choose is so that *A ^{2}*--so
that the

That tells me that *A*
must be chosen to be 1/(2*a*)^{1/2}, where this little *a*, 2*a*,
is now the width of this region. Therefore from this whole family, the
normalized *Y* will look like 1/(2*a*)^{1/2
}for mod *x* < or = *a* and 0 outside. And we can all see at
a glance that if you squared this normalized *Y* and integrated from -a to +a you will get
1. This is normalization. So sometimes, people will give you a wave function
and they will say as a first step, "Normalize this wave function."
What you have to do is, you've got to square the wave function and then put a
number in front of it, and choose it so that the number makes the square
integral 1.

Let me give you
another example. There's a very famous function, called the Gaussian
function. It looks like this. The function *e ^{-α(x (squared))}dx*
from - to + infinity happens to have an area which is square root of

So I will choose *A*
so that 1 = *Y*^{2}*xdx*. This being real, I
don't need the absolute value of *Y*.
That gives me *A ^{2}e ^{-x(squared)}*/

Normalization is
just a discipline. You discipline yourself to take all functions and normalize
them, because why do you normalize them? If you normalize them to 1, then *Y** ^{2}* is directly the
absolute probability density. That means when you add it all up, you'll get
1. If you don't normalize it to 1,

Okay, this is just
practice in normalization. So I'm going to give you a little hint on what is
going to happen next, but I won't do it now, so you guys don't have to take
down anything. Just ask the following question and we'll come back to it on
Wednesday. I've told you that in Newtonian mechanics, every particle has an *x*
and it has a *p*. In quantum theory, instead we traded for a function *Y**(x)* and we learned the
meaning of the *Y**(x)* is that absolute
value of *Y** ^{2}* is the probability
density, meaning

Now we can say,
"Okay, that's enough about position. What about momentum?” I can measure
the momentum of a particle. You talked about momentum on and off in the
lecture. If I measure momentum, what answer will I get?" What are the
odds for getting this or that answer? So given *Y**(x)* that looks like
this, you square it, you get *Y**(P(x))*. The question is, *x* is not the only
thing we're interested in. Even in Newtonian mechanics, *x* and *p*
were equally important. What do you think will happen now? How do I find out
what happens if, instead of being interested in where I find it, I ask, with
what momentum will I find it? Can you imagine a guess on what the answer might
be or in what form the answer will be given to you? This is a wild guess.
Nobody expects you to invent quantum mechanics in 30 seconds, so make a wild
guess. Yes? Anybody there want to make a wild guess? No? Go ahead, yes,
you're smiling. Make a guess. I want the odds for different values for
momentum. How do you think that information will be contained in this theory?

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Pardon me?

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Maybe, based on the uncertainty principle, but I want for every value of
momentum a probability, right? I want the odds of getting this *p* or
that *p* or that. So what do you think we need to get the odds for every
momentum? Yes?

**Student:**
<<inaudible>>

**Professor Ramamurti
Shankar:**
Pardon me?

**Student:** If you have a more
defined location _________ calculate the probabilities of momentum, same way we
did it for location?

**Professor Ramamurti
Shankar:**
Right. So what you will need, it seems reasonable to think, that this guy
contained all the information on where you will find it, maybe there's a
different function of momentum, whose square will give you the probability
density that you get--if that function looks like this and you square that,
that's the odds for getting one momentum versus another momentum. After all,
every variable in classical mechanics you can measure in the quantum theory and
you can give the odds. And for every variable, it looks like you need a
function. What I will show you is that you don't need that. *Y**(x)* itself contains
information on what happens when you measure momentum, what happens when you
measure energy, what happens when you measure anything--and how do you extract
it is what we'll talk about.

[end of transcript]