Prof: Well,
this is just informal discussion till everybody's in
here. So any questions on the subject?
What?
Student: Every question.
Prof: 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
sinθ, 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 2p xℏ,
but it doesn't matter, it's some constant.
And the number is 10^(-34).
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 Dx is the
uncertainty in my position, and Dp 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, Dy 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
Dy 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 x direction,
no momentum in the y direction,
therefore py was strictly 0,
no uncertainty. Dpy is 0 and the
fellow I catch here is moving horizontally with that momentum
p_0 whose y position is within
d, and I can make the d as
small as I like. This is Newtonian physics.
But we have now learned that
the fate of the particle is not in its own hands.
It's contained in this wave.
So what should I do in this
context to find out what it will do?
Any idea what I should do to
find out what will happen in this experiment,
given what we learned? Yes?
Student:
> Prof: Right,
but how will I calculate what will happen in this experiment?
What will decide?
Yes?
Student:
> Prof: 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 2p ℏ/p_0.
But if you want to know what
will happen on the other side of this slit, I have to find the
fate of that wave. Yes, you can put a screen,
but what will I see on a screen?
Will the wave just hit this
region? You know it will spread out
from diffraction. I've told you,
the light will spread out. There are tiny wiggles we
ignore, and this point, where you get most of the
action, that angle, θ,
satisfies dsinθ= l.
This is just wave theory.
That's when you can pair up the
points in the slit, in the hole,
to things shifting, differing by half a wavelength,
so for every one I can find a partner that cancels it,
you will get 0 here. Beyond that,
you may get a few more rises, but it's pretty much dead
outside this cone. That means you will observe
this particle anywhere in this angular width.
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
Dθ or by an angle θ,
it is just p_0xθ.
Or, if you like,
precisely, p_0 times sinθ.
But p_0
sinθ, sinθ
is controlled by dsinθ
= l, so this is l/d.
But l is 2p
ℏ/p_0 x d. You cancel that,
you find D py X d = 2pℏ.
That means D py D y is
roughly--forget the 2p's--of our
ℏ. So you should understand this
much completely without any doubt: if the future of the
particle, the fate of the particle,
is controlled by the wave, you try to narrow the location
of the particle by making the hole smaller and smaller,
the wave fans out more and more. 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--? Prof: 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, dsinθ = 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
________. Prof: 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
_______. Prof: 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 _______? Prof: 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
DxDp 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 2p
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
2px. l is 2pℏ
/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 Dx,
and if you know the momentum to accuracy Dp,
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, Dp is 0. Dp is 0,
Dx is infinity. Now I told the square of the
wave function is the probability to find it somewhere,
and you have no idea where it is.
In other words,
a particle of perfectly known momentum has totally unknown
position. So the probability should look
flat, Y^(2) should look flat.
But the Y^(2),
due to cosine, of course does this.
It prefers some locations to
another, but you're not supposed to have any preference for any
x, so we have a problem. How do I put in a wavelength
into a function whose square is flat?
That's the problem we have.
When you think about it,
you realize your trigonometric functions will not do the trick.
If they have a wavelength,
their square is not flat. The square is also oscillating.
But then what came to the
rescue is the following function,
not a cosine but Ae^(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/â„ . 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 Y times Y', which is A--
I'm taking A to be real here,
so A' is this, times e^(ipx/â„ ),
times e^( −ipx/â„ ). 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
dsinθ, is l/2.
So if θ is very
small, it's like θ xd 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
2pℏ over the momentum and that is
d. In a microscopic world,
p is mv, and let's take an object of
mass 1 kilogram moving at 1 meter per second and a slit hole
is 1 meter. The size of the slit is 1 meter.
Put everything equal to 1 for a
typical estimate. 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 Dx.
What's the Dp?
Well, Dp is now
10 to the, what, 19?
10^(-19).
Now Dp is
m times Dv. That's 10^(-19).
If this is 1 kilogram,
Dv 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 ____________.
Prof: 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.
Prof: Oh,
you've got a screen that only comes to here,
you mean? Student: Yes.
Prof: 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? Prof: 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:
> Prof: 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)/2m. 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^(2), is connected to the probability
of finding it at x. That has to be in fact to be
refined. That's not precisely the story.
I'll tell you why.
If a statistical event has got
6 possible answers, like you throw the die,
you want to get any number from 1 to 6,
you can give the probability for 1,
probability for 2,3, 4,5 and 6. These are all the odds for
getting any number from 1 to 6. Since there are only a finite
number of things that can happen,
I call them I = 1 to 6, there's a probability for each
I, and if you add all the
probabilities, you should get 1.
But if the set of things that
can happen is a continuous variable like x,
in other words, the location of the electron is
not a discrete set of numbers. It's any real number is a
possible location. Then you cannot give a finite
probability for any one x.
If that was finite,
since the number of points is infinite, you cannot make the
total probability 1. So what we really mean by
Y^(2) is called the probability density.
That means draw the
Y^(2), let this be the height of
Y^(2). Take a little sliver of width
dx. The area under the graph,
P(x) dx, that is the probability of
finding the electron, or whatever particle,
between x and x dx.
You understand?
It's called a density.
So you don't give a finite
probability to each point. For an infinitesimal region,
you give it infinitesimal probability, which is the
function P(x) dx. And the statement that the
particle has to be somewhere, namely, all the probabilities
add up to 0, is the statement that when you
integrate this probability density from - to infinity,
namely, Y^(2)dx, from - to infinity,
that should be 1. This is called normalization.
It's a mathematical term.
Norm is connected to length in
some way, and these can be viewed as length squared.
Anyway, this is called the
normalization. Now if somebody gave you a Y,
which did not obey this condition, here is a Y.
This already tells you a nice
story, right? It tells you the odds are
pretty big here, pretty small there,
0 here. Now take a function that's
twice as tall. That gives you the same
relative odds, you understand.
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. Prof: It must be found
within or -a and more than that. Student:
> Prof: That is correct.
Student: With the same
probability. Prof: 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
Y^(2)dx from - to infinity,
I want it to be 1. And I'll pick A so that
that is true. Well, we can do this integral
in our head. What is this integral?
This is just the square of
A times the width of this region.
That's got to be 1.
That tells me that A
must be chosen to be 1/(2a)^(1/2),
where this little a, 2a, is now the width of
this region. Therefore from this whole
family, the normalized Y will look like
1/(2a)^(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 p/a.
That is just one of those
tabulated integrals. So here's a bell shaped
function with this property. Now I want to make a quantum
mechanical wave function that looks like the
Y(x) = A e^(- x) squared over 2 D
squared. That's a possible wave
function, right? Nothing funny about it,
but what do you know about the wave function?
It's biggest at x = 0.
It's symmetric between and
-x. And it dies off very quickly,
but how far should you go? You can easily guess that when
x is much bigger than D, this function is gone,
because x/D^(2) is
going on the exponent. So if that number's big,
it's e to the - big, which is very small.
So roughly speaking the width
of this graph is of order D or 2D.
I'm just going to call it
D, just to give you an order of magnitude.
So that's an electron whose
location is roughly known to an amount D.
But this is not normalized,
because if I take the square of this, I won't get 1.
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))/ D^(2)dx.
The 2 went away,
because I squared the function Y, so don't forget that.
Now I look at the table of
integrals and what is a? When I compare these 2,
a is just 1/D^(2). So it's square root of
pD^(2). This is an easy thing,
because I'm already giving you the integral you need to do,
but I want you to get used to it.
So this whole thing should be 1.
That means A is
1/pD^(2) to the fourth root, the power ¼.
Therefore the normalized wave
function, Y normalized,
looks like 1/pD^(2) to the ¼,
e^( −x(squared ))/2D^(2).
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,
Y^(2) is the relative probability density.
It will still tell you the
relative odds of this and that, but you cannot say this
interval from here to here, the chances are 30 percent for
catching it. You must take the region that
you're integrating, divide by the whole thing.
But you don't have to divide by
the whole thing if you've normalized 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 P(x) dx is the
probability of finding it between x and x
dx. 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:
> Prof: Pardon me?
Student:
> Prof: 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:
> Prof: Pardon me?
Student: If you have a
more defined location _________ calculate the probabilities of
momentum, same way we did it for location?
Prof: 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.
