WEBVTT

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Prof: Well,
this is just informal

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discussion till everybody's in
here.

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So any questions on the subject?

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What?

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Student:  Every question.

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Prof: Everything.

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Okay, well, you know what,
you guys should stop and ask

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more things as you go along,
because there is just no way

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you could get all of this.

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And it's a little strange and
only by talking about it,

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you will at least know what's
going on.

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There's no way to make it
reasonable.

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It's not a reasonable world out
there.

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I can only tell you what it is.

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I take that view;
when I teach quantum mechanics,

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just tell the rules and say,
"This is what happens.

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This is how we calculate
things."

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And whether you like the
formulas or not,

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it's not my concern.

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And the fact that it doesn't
look like daily life,

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also not my concern,
because this is not daily life.

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Strange things happen.

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But you have to keep me
informed on how much you're

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following and what you are
understanding,

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at any stage.

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Don't wait for this to end,
because it's not something

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where you can go on the last day
and figure everything out.

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And I will try to repeat at
every stage what has gone up to

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that point, because the whole
thing is only a few lectures,

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maybe six or so.

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I can afford to go back every
time to the beginning.

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But I know that it makes sense
to me, because I've seen it,

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and I don't know how it sounds
to you.

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I have no clue.

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You know that and so you have
to speak up.

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You can ask any question you
want, and I will try to answer

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you, if it's within the realm of
possibility.

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Okay, so what have I said so
far?

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So let me summarize.

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Even if you never came to last
lecture, here is what you should

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know about the last lecture,
okay?

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Here's what I said.

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First thing I said is,
everything is really particles,

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all things, electrons,
photons, protons,

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neutrons.

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They are all particles,
so let there be no doubt about

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that.

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By that, I mean if one of them
hits your face,

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like an electron,
you will feel it in only one

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tiny region, one spot.

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Electron dumps all its charge,
all its momentum,

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all its energy to one little
part of your face.

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So there's nothing wavelike
about that.

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It's not like getting hit by a
boxing glove,

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which can hit your whole face.

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An electron hits one dot,
or if it's an

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electron-detecting screen,
only 1 pixel is hit by the

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electron.

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And into that pixel is given
all the charge,

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all the momentum,
all the energy of that

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electron.

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That's exactly what particles
do.

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So when you encounter an
electron, it is simply a

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particle.

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So where does the problem come
in?

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Where does the quantum
mechanics come in?

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It comes in when you do the
famous double slit experiment.

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That's the key.

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The entire quantum mystery is
in the double slit.

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Part of the resolution is in
the double slit,

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but the rest are a little more
difficult, and I'll try to tell

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you.

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First I want to tell you what
goes wrong with Newtonian

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mechanics.

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After all, if everything is a
particle, what's the big deal,

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what's the problem?

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The double slit experiment is a
problem.

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That's what puts the nail on
the coffin for Newtonian

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physics, and here it is in the
basic version.

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You've got two slits.

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By the way, I'm going to call
the particle the electron.

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They're all doing the same
thing, so what applies to one,

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applies to all of them.

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There is a source,
like an electron gun,

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that emits electrons.

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In the old days,
televisions had the electron

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gun.

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And the gun emits the
electrons, they go and hit the

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screen, they make a little dot,
and then the dot moves around,

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and you see your favorite show.

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Okay, this is the electron gun,
and the electron gun has been

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engineered to send electrons off
a definite momentum.

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That you can get by
accelerating the electrons over

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a definite potential,
and the gain of so many

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electron volts will turn into
kinetic energy.

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As for direction,
if this gun is really far away

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to the left,
in principle 1 mile,

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then the only way electrons are
going to go 1 mile and hit the

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screen is they're all basically
moving in the horizontal

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direction.

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Then you put a row of detectors
in the back, which will detect

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electrons.

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Then this is slit 1 and this is
slit 2.

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You block slit 1.

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In fact, let me say the
following thing:

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what do we really know when we
do the experiment?

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Once in a while this gun will
emit an electron,

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and we know it's emitted the
electron, because it will recoil

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one way, just like a gun,
rifle.

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It will recoil.

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That's when we know the
electron left.

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Then we don't know anything,
and suddenly,

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one of these guys says click.

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That means electron's arrived
here.

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This is what we really know.

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Everything else you say about
the electron is conjecture at

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this point.

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You know it was here,
you know it was there.

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The question is,
what was it doing in between?

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Now if you say,
"Look, things cannot go

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from here to there,
except by following some path,

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I don't know what path it
is."

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Maybe if it's an ordinary
particle,

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like a Newtonian particle,
it will take some straight

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line,
hit that slit,

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or go through that slit and
arrive here.

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So you might say,
"I don't know the

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trajectory, but it's got to be
some trajectory,

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maybe like that,
or maybe like that."

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So the electron takes some path
and you can label the path as

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either through slit 1 or through
slit 2.

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Okay, now here is the problem.

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Suppose I do the experiment
with slit 2 blocked,

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so you cannot even get through
this one,

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and I sit at a certain location
for a certain amount of time,

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maybe 1 hour and I see how many
electrons come,

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and I get 5 electrons,
with only 1 slit open.

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And if I move that observation
point, I get some pattern,

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pretty dull,
looking like that,

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and I'm going to call it
I_1.

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That's the count,
as a function of position up

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and down that wall of detectors.

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Then I repeat the experiment
with this guy closed and that

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guy open, and I get another
count, looks like that.

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Now I'm going to pick a
location.

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These are not drawn to scale or
anything, so I'm not responsible

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for any of that.

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Maybe I'll at least show you
one thing, which is pretty

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important.

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This graph will be big in front
of the second slit,

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which is somewhere here.

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It will look like that.

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So I get I_1
when 1 is open and I get

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I_2 when 2 is
open.

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This is I_1,
this is I_2.

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Now I'm going to open the two
slits and I'm going to pick a

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particular location.

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It doesn't happen everywhere.

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I'm going to pick one location
called x.

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Where I used to get 5 electrons
per hour with one thing open,

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and 5 electrons per hour with
the second thing open.

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Now I want to open both and
ask, what will I get?

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In Newtonian mechanics,
there's only one possible

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answer to that question,
and that is 10,

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because we've got 5 this way
and you've got 5 that way.

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And you open both,
whoever is going this way will

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keep going that way;
whoever is going this way will

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keep going this way.

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They will add up to give you 10.

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Now I told you,
some people may say,

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"Well, maybe it's not 10,
because with both slits open,

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maybe someone from here can
collide with someone from there.

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How do you know that will not
happen?"

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So I'm saying,
do the experiment with such a

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feeble beam of electrons,
there's only one electron at a

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time in the whole lab.

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It's not going to collide with
itself.

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Then you wait long enough,
and you have to get 5 5 = 10.

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And what I'm telling you is
that if you go to the location

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marked x,
where you've got 5 with each

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one open, when you open both,
you will get 0.

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You don't get anything.

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That is a great mystery.

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That is the end of Newtonian
physics.

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And I told you that something
like that never happens in your

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daily life.

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I gave an example with machine
guns.

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This is a machine gun.

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This is a concrete wall with 2
holes in it, and there's some

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target here, you.

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And then you see how many are
coming through this and how many

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are coming through that.

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Then you go there and you wait.

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And both are open.

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Somehow, nothing comes.

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With the second hole in the
wall, you are safe.

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With one hole in the wall,
you're not safe.

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That can never happen with
bullets.

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So these electrons are not
following any path,

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because the minute you commit
yourself to saying it follows

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one path,
either through this one or

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that, you cannot avoid the fact
that with both of them open,

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the intensity with 1 2 has to =
I_1

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I_2.

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That's the Newtonian prediction.

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And I_1
I_2 gives you 5

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5 = 10 here and you get 0.

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In other places,
instead of 10,

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you will get 20.

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Some places you get more,
but more dramatic thing is

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where you get less,
where you get nothing.

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Therefore you abandon the
notion that electrons have any

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trajectories.

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You don't want to abandon it,
but you have to,

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because that assumption,
which is very reasonable,

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just doesn't agree with
experiment.

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Then you say,
"Okay, what should I do?

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Newtonian mechanics is wrong.

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What's going to take its
place?"

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To find that,
you have to move away from this

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x and move up and down
this row here and see what you

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get, and I think I told you what
you get.

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You get a pattern that looks
like this.

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So the real I1 2 looks like
this: it's got ups and downs and

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ups and downs.

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And let's say these downs
really correspond to 0.

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That means nobody comes here,
a lot of them come here,

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no one comes here,
a few come here and so on.

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That's what you find.

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I'm just telling you what
happens when you do the

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experiment.

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So you put yourself in the
place of a person who did the

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experiment.

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You thought of moving away from
that point x and you plot

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it, it makes no sense in the
language of particles.

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But this is such a familiar
pattern.

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If you're a trained physicist,
which you guys are,

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you will say,
"Hey, this reminds me of

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this wave interference,
with water waves or sound waves

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or any waves.

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" Obviously there's some
wavelength.

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The minute you give me
wavelength and a slit

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separation, I can calculate this
pattern.

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dsinθ
= l and whatnot,

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and from that
sinθ,

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where you get a minimum or a
maximum,

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and if there's a certain
separation to the screen,

11:30.389 --> 11:32.899
you can find the precise
location of these maxima or

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minima.

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Or given the maxima and minima,
you can work back and find the

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wavelength.

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And the wavelength happens to
be some number called

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ℏ, which is 10^(-34)
joule seconds,

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divided by the momentum.

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p is the momentum.

11:53.960 --> 11:56.250
In other words,
you find that if you send more

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energetic electrons,
accelerate them through bigger

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voltage,
increase the p,

12:00.538 --> 12:04.148
l goes down,
the pattern gets squeezed.

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You slow down the electrons,
p reduces,

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l increases,
the pattern spreads out,

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and the dependence on momentum
is inversely proportional to

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momentum.

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And you fool around and find
out the coefficient of

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proportionality,
which people used to call

12:17.437 --> 12:20.097
h in the old days is now
written as 2p

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xℏ,
but it doesn't matter,

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it's some constant.

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And the number is 10^(-34).

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So you can successfully
reproduce this pattern,

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but what does it tell you about
what's going on?

12:38.720 --> 12:40.050
What good is that pattern?

12:40.048 --> 12:43.248
The pattern tells you that if
you repeat the experiment with

12:43.250 --> 12:46.450
this electron gun a million
times or a billion times and you

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plotted the histogram patiently,
the histogram will eventually

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fill out and take this shape.

12:55.440 --> 12:59.330
So this wave is not the wave
associated with a huge stream of

12:59.331 --> 13:00.111
electrons.

13:00.110 --> 13:03.630
A single electron in the lab is
controlled by this wave.

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You need a whole wave for 1
electron, so it's obviously not

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a wave of electrons.

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It's not a wave of charge,
like the wave of water.

13:10.460 --> 13:13.920
It's a mathematical function
and you are drawn to it,

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because the only way you know
how to get this wiggly graph is

13:16.923 --> 13:20.233
to take something with definite
wavelength and let it interfere.

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So you're forced to think about
this wave.

13:22.808 --> 13:26.958
And the intensity of the wave,
the brightness if you like,

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the square of its height,
gives you what?

13:30.090 --> 13:33.910
Gives you the graph you will
get if you repeat the experiment

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many times.

13:34.740 --> 13:36.770
And what does it mean for the
individual trial?

13:36.769 --> 13:41.219
What does it mean for the
millionth one electron?

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For the millionth one electron,
it gives you the odds of where

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it will land on that screen,
okay?

13:48.288 --> 13:50.048
You can never tell exactly
where it will be.

13:50.048 --> 13:52.608
You can tell what the odds are,
and the only way to test the

13:52.610 --> 13:55.040
statistical theory is to do the
experiment many times.

13:55.038 --> 13:57.558
And if you do it,
it works, and it seems to work

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for everything,
for electrons,

13:59.154 --> 14:00.364
for protons,
for photons,

14:00.360 --> 14:04.690
whatever it is,
the wavelength and momentum are

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connected in this fashion.

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So this wave is forced upon us,
and it gives you the odds of

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finding the electron somewhere.

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And we say that the
probability-- I'll be a little

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more precise in a minute on what
I mean by the probability to

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find it at a location x,
but let's just say,

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if you draw the graph of
Y^(2),

14:25.590 --> 14:27.670
wherever it is big,
the probability is larger;

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wherever it is small,
probability is small;

14:29.649 --> 14:33.479
wherever it is 0,
probability is 0.

14:33.480 --> 14:37.030
So there seems to be a function
whose amplitude or whose square

14:37.027 --> 14:38.627
gives you the probability.

14:38.629 --> 14:42.169
That function is called the
wave function,

14:42.168 --> 14:46.478
and we know it exists,
because it's the only way to

14:46.481 --> 14:50.281
calculate the result of this
experiment.

14:50.279 --> 14:55.329
Once you tell me that the fate
of a particle is controlled by a

14:55.327 --> 14:57.767
wave,
you're immediately led to some

14:57.768 --> 15:00.558
other conclusions,
so I'm going to tell you what

15:00.557 --> 15:01.107
they are.

15:01.110 --> 15:07.080
First conclusion is this:
if I make a single slit,

15:07.080 --> 15:09.720
let's call this the x
direction,

15:09.720 --> 15:12.140
let's call this the y
direction,

15:12.139 --> 15:17.589
and I'm sending a bunch of
particles in the x

15:17.591 --> 15:23.151
direction with some momentum
p_0.

15:23.149 --> 15:27.379
In Newtonian mechanics,
I can manufacture for you an

15:27.380 --> 15:31.120
electron of known momentum and
known position,

15:31.115 --> 15:34.015
or known to arbitrary accuracy.

15:34.019 --> 15:38.509
If Dx is the
uncertainty in my position,

15:38.509 --> 15:41.269
and Dp is the
uncertainty in momentum,

15:41.269 --> 15:44.029
I can make each of them as
small as I like.

15:44.029 --> 15:48.869
So here's an actual practical
way to prepare such a state.

15:48.870 --> 15:51.500
If I say, "Give me an
electron of known position,

15:51.498 --> 15:53.778
known momentum,"
here's what I will do.

15:53.779 --> 15:56.369
I will take a slit with a very
tiny hole in it.

15:56.370 --> 16:00.340
The width of that hole is
d, and whoever comes out

16:00.340 --> 16:03.390
on the other side,
what can I say about that

16:03.388 --> 16:04.308
particle?

16:04.308 --> 16:07.638
Its position has an uncertainty
of order d,

16:07.639 --> 16:11.309
because if it was not--I'm
sorry, Dy now,

16:11.307 --> 16:14.227
because this is the y
direction.

16:14.230 --> 16:15.380
Right?

16:15.379 --> 16:19.289
Anything who came out of hole
right after it had to have a y

16:19.294 --> 16:22.154
position, whose known to within
d.

16:22.149 --> 16:24.879
It's got to come anywhere
within the slit,

16:24.883 --> 16:25.953
but that's it.

16:25.950 --> 16:28.840
So that is how I have prepared
for you, that's how we filter

16:28.835 --> 16:30.445
electrons of definite position.

16:30.450 --> 16:32.590
And you can make
Dy as small as you

16:32.589 --> 16:34.639
like by making the slit as thin
as you like.

16:34.639 --> 16:36.289
What about its momentum?

16:36.288 --> 16:38.698
If you had a momentum
p_0 in the

16:38.696 --> 16:41.556
x direction,
no momentum in the y

16:41.557 --> 16:44.057
direction,
therefore py was

16:44.057 --> 16:46.307
strictly 0,
no uncertainty.

16:46.308 --> 16:49.438
Dpy is 0 and the
fellow I catch here is moving

16:49.442 --> 16:52.472
horizontally with that momentum
p_0 whose

16:52.469 --> 16:54.699
y position is within
d,

16:54.700 --> 16:56.570
and I can make the d as
small as I like.

16:56.570 --> 16:59.520
This is Newtonian physics.

16:59.519 --> 17:02.659
But we have now learned that
the fate of the particle is not

17:02.658 --> 17:03.668
in its own hands.

17:03.669 --> 17:06.999
It's contained in this wave.

17:07.000 --> 17:11.890
So what should I do in this
context to find out what it will

17:11.894 --> 17:12.314
do?

17:12.308 --> 17:17.138
Any idea what I should do to
find out what will happen in

17:17.142 --> 17:20.682
this experiment,
given what we learned?

17:20.680 --> 17:21.530
Yes?

17:21.528 --> 17:25.338
Student:
>

17:25.338 --> 17:28.528
Prof: Right,
but how will I calculate what

17:28.528 --> 17:30.718
will happen in this experiment?

17:30.720 --> 17:33.680
What will decide?

17:33.680 --> 17:34.430
Yes?

17:34.430 --> 17:40.550
Student:
>

17:40.549 --> 17:42.479
Prof: Yes,
this is one hole,

17:42.484 --> 17:45.664
and a light beam is coming from
the left--

17:45.660 --> 17:47.910
I mean, a wave is coming from
the left,

17:47.910 --> 17:50.180
and if the particle has
momentum p_0,

17:50.180 --> 17:54.620
it's got some wavelength,
which is 2p

17:54.615 --> 17:57.005
ℏ/p_0.

17:57.009 --> 18:00.859
But if you want to know what
will happen on the other side of

18:00.859 --> 18:04.069
this slit, I have to find the
fate of that wave.

18:04.068 --> 18:07.348
Yes, you can put a screen,
but what will I see on a

18:07.348 --> 18:07.938
screen?

18:07.940 --> 18:12.190
Will the wave just hit this
region?

18:12.190 --> 18:14.540
You know it will spread out
from diffraction.

18:14.538 --> 18:17.668
I've told you,
the light will spread out.

18:17.670 --> 18:21.630
There are tiny wiggles we
ignore, and this point,

18:21.630 --> 18:24.410
where you get most of the
action, that angle,

18:24.410 --> 18:27.790
θ,
satisfies dsinθ=

18:27.790 --> 18:29.060
l.

18:29.059 --> 18:30.429
This is just wave theory.

18:30.430 --> 18:33.400
That's when you can pair up the
points in the slit,

18:33.400 --> 18:35.500
in the hole,
to things shifting,

18:35.502 --> 18:39.222
differing by half a wavelength,
so for every one I can find a

18:39.222 --> 18:41.922
partner that cancels it,
you will get 0 here.

18:41.920 --> 18:44.880
Beyond that,
you may get a few more rises,

18:44.878 --> 18:48.198
but it's pretty much dead
outside this cone.

18:48.200 --> 18:54.690
That means you will observe
this particle anywhere in this

18:54.688 --> 18:56.508
angular width.

18:56.509 --> 19:01.739
Now a particle cannot go from
this slit to there unless it had

19:01.741 --> 19:05.861
a momentum, which had a
component in the y

19:05.858 --> 19:07.058
direction.

19:07.058 --> 19:09.848
You cannot get there from here,
unless you have y momentum.

19:09.848 --> 19:14.908
It's the y momentum is
uncertain to within that cone.

19:14.910 --> 19:18.030
So what's the uncertainty in
the y momentum?

19:18.028 --> 19:20.658
For a vector of length
p_0 when it

19:20.663 --> 19:23.513
gets shifted by an angle
Dθ or by an angle

19:23.506 --> 19:25.376
θ,
it is just

19:25.384 --> 19:27.864
p_0xθ.

19:27.858 --> 19:31.338
Or, if you like,
precisely, p_0

19:31.343 --> 19:33.273
times sinθ.

19:33.269 --> 19:37.509
But p_0
sinθ,

19:37.509 --> 19:39.709
sinθ
is controlled by

19:39.710 --> 19:41.910
dsinθ
= l,

19:41.910 --> 19:48.720
so this is l/d.

19:48.720 --> 19:54.530
But l is 2p
ℏ/p_0 x d.

19:54.529 --> 19:59.319
You cancel that,
you find D py X d =

19:59.324 --> 20:02.844
2pℏ.

20:02.838 --> 20:07.968
That means D py D y is
roughly--forget the

20:07.967 --> 20:12.967
2p's--of our
ℏ.

20:12.970 --> 20:17.200
So you should understand this
much completely without any

20:17.199 --> 20:20.219
doubt: if the future of the
particle,

20:20.220 --> 20:22.670
the fate of the particle,
is controlled by the wave,

20:22.670 --> 20:25.380
you try to narrow the location
of the particle by making the

20:25.382 --> 20:28.762
hole smaller and smaller,
the wave fans out more and more.

20:28.759 --> 20:30.449
That's just wave theory.

20:30.450 --> 20:32.900
People knew this about waves
hundreds of years back.

20:32.900 --> 20:36.440
What is novel is that this wave
is going to tell you where the

20:36.435 --> 20:37.765
particle will end up.

20:37.769 --> 20:40.289
This wave is going to control
the odds of where the particle

20:40.288 --> 20:43.058
will end up,
and the odds are pretty much

20:43.058 --> 20:46.098
concentrated in this cone,
not of 0 opening angle,

20:46.101 --> 20:48.481
but an angle θ,
so that dsinθ

20:48.476 --> 20:49.336
= l.

20:49.338 --> 20:54.648
l in turn is connected to the
momentum of the particle.

20:54.650 --> 20:59.030
This is where the uncertainty
principle comes in.

20:59.029 --> 21:02.489
What Heisenberg said is,
"You had in your mind the

21:02.490 --> 21:05.890
classical notion that you can
have a particle of known

21:05.887 --> 21:07.937
position and known momentum.

21:07.940 --> 21:10.960
Let me see you produce that
particle.

21:10.960 --> 21:13.650
So you try to do it by putting
a slit and catching guys with a

21:13.653 --> 21:17.003
very narrow range in y,
but now you find out that the

21:17.000 --> 21:19.350
momentum gets broader and
broader,

21:19.348 --> 21:22.188
and that's the result of the
wave associated with the

21:22.192 --> 21:22.742
project.

21:22.740 --> 21:24.160
" It was not in the
Newtonian picture.

21:24.160 --> 21:25.360
Yes?

21:25.358 --> 21:28.838
Student:  Why with the
single slit, like you've drawn

21:28.836 --> 21:32.076
there, do you get the little
wiggle at the outside--?

21:32.078 --> 21:33.728
Prof: You mean,
why would it have those

21:33.731 --> 21:34.101
wiggles?

21:34.098 --> 21:39.228
Okay, so if I look at the
single slit, I could think of

21:39.230 --> 21:41.890
many little point sources.

21:41.890 --> 21:44.490
In the forward direction,
if you go very far,

21:44.491 --> 21:48.161
so you treat them as roughly
parallel, they are all in step.

21:48.160 --> 21:50.220
You've got a big maximum.

21:50.220 --> 21:53.940
In another direction where this
difference is l,

21:53.942 --> 21:58.302
dsinθ = l,
this guy and this guy are

21:58.300 --> 22:00.520
differing by l/2.

22:00.519 --> 22:03.139
Student:  So you're
assuming the slit is large

22:03.136 --> 22:05.046
enough for them to do that
________.

22:05.048 --> 22:08.838
Prof: These are little
mathematical dots inside the

22:08.844 --> 22:09.314
slit.

22:09.308 --> 22:13.738
You know, when you make a slit,
every point on the slit looks

22:13.739 --> 22:17.429
like a source of light,
a point source of light.

22:17.430 --> 22:19.910
They are not really light
bulbs, but if you make any hole

22:19.913 --> 22:21.863
in the wall, a light comes
through that hole,

22:21.864 --> 22:23.554
looks like it's a source of
light.

22:23.548 --> 22:26.608
I'm taking every point on the
hole to be a source of light.

22:26.608 --> 22:30.558
And what I'm saying is,
there is a direction in which

22:30.561 --> 22:33.981
it will cancel,
but if you go a little further

22:33.982 --> 22:36.722
out, it won't cancel completely.

22:36.720 --> 22:38.560
This is the direction for
perfect cancelation,

22:38.555 --> 22:40.385
where I can pair them,
this one with this one,

22:40.391 --> 22:41.861
that one to that one,
and so on.

22:41.859 --> 22:43.539
They pair to give 0.

22:43.538 --> 22:46.468
But if you move further up,
you no longer cancel

22:46.467 --> 22:48.427
completely,
but you don't add perfectly

22:48.433 --> 22:49.743
either,
so things will get better,

22:49.739 --> 22:52.419
then they'll get worse again,
and better and worse and so on.

22:52.420 --> 22:54.360
So that's the origin of that
pattern.

22:54.359 --> 22:55.779
Yes?

22:55.779 --> 22:57.799
Student:  So you're
saying this is the first

22:57.796 --> 22:58.196
minimum.

22:58.200 --> 22:59.330
After that, it would increase
_______.

22:59.329 --> 23:00.049
Prof: Yes.

23:00.048 --> 23:03.638
For a single slit diffraction,
the big thing in the middle is

23:03.643 --> 23:05.263
pretty much all you have.

23:05.259 --> 23:10.519
It's not like double slit
experiment with two holes,

23:10.523 --> 23:14.553
where you get many times the
pattern.

23:14.548 --> 23:17.458
That's because--let's
understand why that is true.

23:17.460 --> 23:22.720
Here, if these two differ by 1
wavelength, they don't differ at

23:22.720 --> 23:23.230
all.

23:23.230 --> 23:25.890
I can find another angle where
they differ by 2 wavelengths.

23:25.890 --> 23:27.360
They don't differ at all.

23:27.358 --> 23:29.638
There's only 2 sources,
so you can engineer them to

23:29.642 --> 23:32.612
differ by either 1 wavelength or
2 wavelength or 3 wavelengths.

23:32.608 --> 23:36.658
Here in a single slit,
each point is like a source of

23:36.656 --> 23:37.276
light.

23:37.279 --> 23:39.229
You got them all to agree.

23:39.230 --> 23:41.310
You can get them to agree only
in the forward direction.

23:41.308 --> 23:44.208
In any other direction,
you can get them to neutralize

23:44.211 --> 23:46.951
each other, but never for
perfect reinforcement.

23:46.950 --> 23:49.740
So you cannot get it more
than--this is the only real

23:49.736 --> 23:50.536
maximum here.

23:50.539 --> 23:51.769
Everything else is tiny.

23:51.769 --> 23:52.859
Yes?

23:52.858 --> 23:55.578
Student:  Why can't you
do the same trick with the

23:55.582 --> 23:58.452
double slit, where each of the
one slits has its _______?

23:58.450 --> 23:59.510
Prof: No,
in double slit,

23:59.511 --> 24:01.911
what's happening is,
we take--in double slit,

24:01.911 --> 24:05.941
the number d I used in
double slit was not the size of

24:05.941 --> 24:09.001
a slit,
but the space in between the

24:09.002 --> 24:10.002
slit, okay?

24:10.000 --> 24:14.160
So there I took the size of
every slit to be vanishingly

24:14.163 --> 24:14.773
small.

24:14.769 --> 24:18.519
That means a light coming out
of this slit spreads out

24:18.520 --> 24:21.140
completely in all directions,
okay?

24:21.140 --> 24:26.110
It's as if this packet became
that broad.

24:26.108 --> 24:32.098
Likewise a light from this one,
we are still in the first

24:32.098 --> 24:34.558
maximum of that slit.

24:34.558 --> 24:38.398
So you've got to understand,
in a single slit experiment of

24:38.404 --> 24:41.924
diffraction, the slit size is
what you are varying.

24:41.920 --> 24:44.520
In a double slit experiment,
the slit is taken to be

24:44.518 --> 24:47.318
mathematically point like,
so it fans out completely.

24:47.318 --> 24:50.438
It's the interference between
those two point sources that

24:50.442 --> 24:51.322
you're adding.

24:51.318 --> 24:54.588
They can add and cancel,
add and cancel,

24:54.594 --> 24:58.544
many, many times as you move
along this line.

24:58.538 --> 25:01.408
Okay, so what we learn is it's
when you combine waves and

25:01.407 --> 25:04.787
particles and go back and forth
that you run into the situation.

25:04.788 --> 25:08.408
So you cannot make a state of
perfect momentum.

25:08.410 --> 25:10.480
By the way, I said one thing,
I thought about it,

25:10.480 --> 25:13.810
which is incorrect,
which is, in the microscope,

25:13.808 --> 25:16.598
I said if you want to locate
the position of an electron in a

25:16.598 --> 25:19.648
microscope,
take a microscope with an

25:19.654 --> 25:24.374
opening, and electron is
somewhere on this line.

25:24.368 --> 25:26.988
I said you're shining light
down here.

25:26.990 --> 25:33.080
It hits the electron,
but it goes in through the slit

25:33.083 --> 25:35.313
by spreading out.

25:35.308 --> 25:39.838
So the photon that came in goes
into the eyepiece with a certain

25:39.844 --> 25:42.224
uncertainty in its final angle.

25:42.220 --> 25:45.500
That means we know the incoming
momentum, but we don't know the

25:45.500 --> 25:47.300
outgoing momentum of the photon.

25:47.298 --> 25:50.118
The lens picks up everything
inside that cone.

25:50.118 --> 25:52.738
That means we don't know how
much momentum it gave to the

25:52.743 --> 25:53.263
electron.

25:53.259 --> 25:56.729
It gives an indefinite amount
of momentum to the electron.

25:56.730 --> 25:59.850
Therefore the x momentum
of the electron is uncertain by

25:59.845 --> 26:01.955
that little shape,
the conical shape of the

26:01.955 --> 26:02.605
momentum.

26:02.608 --> 26:05.508
And if you do the uncertainty
principle argument,

26:05.511 --> 26:08.411
you'll again find
DxDp is

26:08.413 --> 26:09.203
h, h.

26:09.200 --> 26:13.770
Now what I don't like about my
experimental setup is that I had

26:13.770 --> 26:18.270
the incoming light also coming
from inside the microscope,

26:18.269 --> 26:20.579
but that means incoming light,
when it comes through this hole

26:20.577 --> 26:24.167
here,
will itself spread.

26:24.170 --> 26:25.840
Then it will hit this guy,
and that will go to the

26:25.836 --> 26:26.206
aperture.

26:26.210 --> 26:27.930
That will spread some more.

26:27.930 --> 26:31.050
This uncertainty in incoming
momentum is unnecessary.

26:31.048 --> 26:33.908
We can do better than that,
because in other words,

26:33.905 --> 26:37.105
when the light is picked up,
it is picked up by this tiny

26:37.105 --> 26:37.615
hole.

26:37.618 --> 26:39.888
There's no reason it should
also come from the tiny hole.

26:39.890 --> 26:43.250
It can come from a source far
away, say on the other side,

26:43.247 --> 26:46.717
so that it is a well defined
direction, it's not diffracting

26:46.721 --> 26:47.371
at all.

26:47.368 --> 26:50.008
So I want it to come in through
a very broad hole,

26:50.009 --> 26:53.029
so it's got well defined
direction, so the light here has

26:53.026 --> 26:54.046
known momentum.

26:54.048 --> 26:57.158
It hits the electron and goes
into the microscope.

26:57.160 --> 27:01.550
It is the final momentum of the
photon I don't know.

27:01.549 --> 27:03.139
And I cannot make it better.

27:03.140 --> 27:05.930
If I make it better,
I've got to open this eyepiece

27:05.932 --> 27:06.382
a lot.

27:06.380 --> 27:09.190
If I open the eyepiece a lot,
I don't know where I caught

27:09.188 --> 27:09.738
this guy.

27:09.740 --> 27:12.880
So again the problem between
taking a very tiny eyepiece,

27:12.880 --> 27:15.130
so that if I see a flicker,
I know the electron was in

27:15.132 --> 27:18.042
front of it,
but the light coming from the

27:18.040 --> 27:21.410
reflected electron fans out more
and more.

27:21.410 --> 27:24.260
Okay, so anyway,
this is the uncertainty

27:24.263 --> 27:28.733
principle and the uncertainty
principle told us something very

27:28.728 --> 27:29.898
interesting.

27:29.900 --> 27:32.700
I asked you,
what can be the function here

27:32.696 --> 27:36.856
that produces this interference
pattern in the double slit?

27:36.859 --> 27:38.609
We know the wavelength.

27:38.608 --> 27:43.748
Wavelength was 2p
ℏ/p.

27:43.750 --> 27:51.030
And you know from basic physics
that a function like cosine

27:51.025 --> 27:56.415
2px/l has got
wavelength l.

27:56.420 --> 27:59.280
So let's put in the formula for
l here.

27:59.279 --> 28:02.939
You get cosine
2px.

28:02.940 --> 28:07.420
l is 2pℏ
/p.

28:07.420 --> 28:15.510
Cancel the 2 p's,
you get A cosine

28:15.505 --> 28:18.735
px/ℏ.

28:18.740 --> 28:21.440
That function,
when you throw it at a double

28:21.438 --> 28:23.488
slit,
will form two little wavelets,

28:23.492 --> 28:27.562
and they will interfere,
that produce an interference

28:27.561 --> 28:30.481
pattern of the type you want.

28:30.480 --> 28:33.280
Do you understand that the
experiment only showed you

28:33.280 --> 28:34.520
there's a wavelength.

28:34.519 --> 28:38.079
It did not tell you what the
actual function is?

28:38.079 --> 28:39.059
That's very, very important.

28:39.058 --> 28:41.478
When Young did the experiment
with the double slit,

28:41.480 --> 28:43.900
he found the oscillations and
he could read off the

28:43.903 --> 28:44.633
wavelength.

28:44.630 --> 28:46.290
It's just geometry.

28:46.288 --> 28:47.888
But he didn't know what was
oscillating.

28:47.890 --> 28:50.340
He didn't know there's an
electric and magnetic field

28:50.340 --> 28:51.520
underneath all of that.

28:51.519 --> 28:54.069
But you can always read out the
wavelength without knowing

28:54.066 --> 28:54.866
what's going on.

28:54.868 --> 28:57.068
Likewise, we have the
wavelength.

28:57.068 --> 29:00.138
We know it comes from a
function with a well defined

29:00.143 --> 29:03.703
wavelength, so I make my first
guess to be this function.

29:03.700 --> 29:07.040
But I told you what was wrong
with this choice.

29:07.039 --> 29:10.889
You guys remember that?

29:10.890 --> 29:14.810
I said this function violates
the uncertainty principle.

29:14.808 --> 29:18.388
The uncertainty principle says
if you know the position to an

29:18.387 --> 29:21.287
accuracy Dx,
and if you know the momentum to

29:21.285 --> 29:25.345
accuracy Dp,
the product must be at least as

29:25.346 --> 29:29.966
big as hx some
number of order 1.

29:29.970 --> 29:34.800
We have taken the particles to
have well defined momentum.

29:34.798 --> 29:38.578
If they have well defined
momentum, Dp is 0.

29:38.578 --> 29:42.208
Dp is 0,
Dx is infinity.

29:42.210 --> 29:47.240
Now I told the square of the
wave function is the probability

29:47.240 --> 29:51.430
to find it somewhere,
and you have no idea where it

29:51.432 --> 29:52.022
is.

29:52.019 --> 29:56.339
In other words,
a particle of perfectly known

29:56.340 --> 30:00.270
momentum has totally unknown
position.

30:00.269 --> 30:04.519
So the probability should look
flat, Y^(2) should

30:04.517 --> 30:05.337
look flat.

30:05.338 --> 30:09.278
But the Y^(2),
due to cosine,

30:09.277 --> 30:11.337
of course does this.

30:11.338 --> 30:14.918
It prefers some locations to
another, but you're not supposed

30:14.924 --> 30:18.754
to have any preference for any
x, so we have a problem.

30:18.750 --> 30:24.300
How do I put in a wavelength
into a function whose square is

30:24.300 --> 30:24.960
flat?

30:24.960 --> 30:27.360
That's the problem we have.

30:27.358 --> 30:30.178
When you think about it,
you realize your trigonometric

30:30.175 --> 30:31.945
functions will not do the trick.

30:31.950 --> 30:33.930
If they have a wavelength,
their square is not flat.

30:33.930 --> 30:36.450
The square is also oscillating.

30:36.450 --> 30:39.870
But then what came to the
rescue is the following

30:39.865 --> 30:41.705
function,
not a cosine but

30:41.707 --> 30:44.607
Ae^(ipx)
^(/ℏ),

30:44.608 --> 30:48.908
rather than cosine
xp/ℏ.

30:48.910 --> 30:51.360
Look at this function.

30:51.358 --> 30:52.848
This function,
I've told you many,

30:52.847 --> 30:55.097
many times, if you don't know
your complex numbers,

30:55.104 --> 30:57.094
you're definitely going to have
trouble.

30:57.088 --> 31:02.888
It looks like a vector of
length A and angle

31:02.893 --> 31:06.843
θ,
which is px/ℏ.

31:06.838 --> 31:10.408
As you vary x,
this x changes and this

31:10.410 --> 31:13.840
will rotate round,
but as it rotates the amplitude

31:13.838 --> 31:16.538
of this complex number,
absolute value of

31:16.539 --> 31:19.629
Y^(2),
which is Y times

31:19.634 --> 31:23.414
Y', which is A--
I'm taking A to be real

31:23.405 --> 31:26.345
here,
so A' is this,

31:26.346 --> 31:31.596
times e^(ipx/ℏ),
times e^( −ipx/ℏ).

31:31.598 --> 31:35.718
That cancels out,
you just get A^(2).

31:35.720 --> 31:38.080
In other words,
the complex number describing

31:38.075 --> 31:41.015
the wave function changes in
phase but not amplitude.

31:41.019 --> 31:44.609
It's the amplitude that gives
the probability.

31:44.608 --> 31:48.118
Now there is no problem with
this guy having a wavelength,

31:48.118 --> 31:50.458
because this oscillates in
x.

31:50.460 --> 31:52.970
Its real part and imaginary
part both oscillate,

31:52.974 --> 31:56.244
but the square of the real the
imaginary square is 1 [A^2].

31:56.240 --> 31:59.970
That's why the amplitude
doesn't change.

31:59.970 --> 32:04.380
So we are driven now to the
very interesting result that the

32:04.384 --> 32:08.884
wave function for a particle of
definite momentum p is

32:08.875 --> 32:09.545
this.

32:09.548 --> 32:12.068
So this is a very important
lesson.

32:12.068 --> 32:15.008
Let me label this function by
label p to tell you,

32:15.006 --> 32:17.306
"Hey, I'm not talking
about any old wave

32:17.313 --> 32:18.313
function."

32:18.308 --> 32:20.718
This guy has a definite
momentum p.

32:20.720 --> 32:28.150
Its wave function looks like
e^( ipx/ℏ) times

32:28.152 --> 32:33.852
any number A you want in
front of it.

32:33.848 --> 32:35.888
That's a very important thing
to know.

32:35.890 --> 32:38.660
This is called a plane wave,
and a plane wave with a

32:38.655 --> 32:41.795
p right where it is
describes a particle of momentum

32:41.800 --> 32:43.700
p in particle mechanics.

32:43.700 --> 32:46.530
And I told you particles of
momentum p are

32:46.531 --> 32:47.301
everywhere.

32:47.298 --> 32:49.908
Every machine produces them,
every accelerator produces

32:49.910 --> 32:52.230
them, and if you want to
describe them in quantum

32:52.230 --> 32:54.550
mechanics, you have to know
complex numbers.

32:54.548 --> 32:57.628
There's just no way you can get
a real answer to our

32:57.633 --> 32:58.483
predicament.

32:58.480 --> 33:01.180
It's complex.

33:01.180 --> 33:06.300
So that's roughly where I left
you, and I want to remind you of

33:06.304 --> 33:10.354
a few other things,
this further discussion of the

33:10.353 --> 33:12.423
result we have,
okay?

33:12.420 --> 33:16.040
The discussion is,
if the world is really this

33:16.040 --> 33:20.710
messed up at the microscopic
level, why do I think it's the

33:20.710 --> 33:24.010
world I see in the macroscopic
level?

33:24.009 --> 33:25.719
Where are all these
oscillations?

33:25.720 --> 33:28.030
Why is it that when there's a
concrete wall,

33:28.032 --> 33:30.672
making another hole is bad for
me and not good?

33:30.670 --> 33:32.390
Why do all these things happen?

33:32.390 --> 33:35.950
Why do I think particles have
definite momentum and position?

33:35.950 --> 33:38.790
Why do I think that if I make a
hole in the wall and I send a

33:38.790 --> 33:40.760
beam,
the beam will go on the other

33:40.761 --> 33:44.061
side of the wall with a shape
precisely like the shape of the

33:44.055 --> 33:46.605
hole,
no spreading out?

33:46.608 --> 33:50.738
It all has to do with the size
of the object.

33:50.740 --> 33:53.810
The laws of physics are always
quantum mechanical laws,

33:53.809 --> 33:56.989
but when you apply it to an
elephant, you get one kind of

33:56.993 --> 33:58.293
answer;
when you apply it to an

33:58.291 --> 33:59.521
electron, you get another kind
of answer.

33:59.519 --> 34:02.589
You don't have separate laws
for big and small things.

34:02.588 --> 34:05.068
The real question is,
how do these very same laws,

34:05.068 --> 34:06.958
when applied to big things--by
big things,

34:06.960 --> 34:09.180
I mean things you see in daily
life--

34:09.179 --> 34:12.829
give the impression that the
world is Newtonian?

34:12.829 --> 34:16.319
So let's look at the double
slit experiment.

34:16.320 --> 34:19.500
Here's a double slit and we are
told, "Send something.

34:19.500 --> 34:21.350
See what happens on the other
side."

34:21.349 --> 34:24.909
And the prediction is that you
get these oscillations,

34:24.913 --> 34:28.343
with the peculiar property that
with two holes open,

34:28.342 --> 34:30.832
you don't get anything
somewhere.

34:30.829 --> 34:33.289
We don't seem to see that in
daily life, and you can ask,

34:33.286 --> 34:34.556
"Why is that so?"

34:34.559 --> 34:39.659
Well, you remember that the
condition for the next minimum

34:39.663 --> 34:42.623
is like
dsinθ,

34:42.617 --> 34:44.227
is l/2.

34:44.230 --> 34:51.190
So if θ is very
small, it's like θ

34:51.186 --> 34:57.196
xd is l/2,
or θ = l/d.

34:57.199 --> 35:00.529
That's the angle you've got to
go through from the central

35:00.525 --> 35:01.105
maximum.

35:01.110 --> 35:04.710
That's the central maximum to
the first minimum here.

35:04.710 --> 35:10.870
That tiny angle is given by
l/d.

35:10.869 --> 35:15.009
The reason you don't see the
oscillations is when you put in

35:15.005 --> 35:17.735
the values for l and
d.

35:17.739 --> 35:21.089
Let's pick a reasonable value
for this angle θ.

35:21.090 --> 35:23.010
Do you understand what
θ is?

35:23.010 --> 35:26.190
In that maximum,
there are some oscillations.

35:26.190 --> 35:29.920
I want to go to the first
minimum near that.

35:29.920 --> 35:33.170
The distance between these two
is roughly the spacing between

35:33.168 --> 35:35.278
maxima and minima,
maxima and minima.

35:35.280 --> 35:37.540
That angle is l/2.

35:37.539 --> 35:41.489
l is
2pℏ over

35:41.492 --> 35:44.882
the momentum and that is
d.

35:44.880 --> 35:48.190
In a microscopic world,
p is mv,

35:48.192 --> 35:52.172
and let's take an object of
mass 1 kilogram moving at 1

35:52.166 --> 35:55.696
meter per second and a slit hole
is 1 meter.

35:55.699 --> 35:56.799
The size of the slit is 1 meter.

35:56.800 --> 36:01.430
Put everything equal to 1 for a
typical estimate.

36:01.429 --> 36:10.719
You find this number is
10^(-34) radians.

36:10.719 --> 36:14.869
That means that the angular
difference between the maximum

36:14.871 --> 36:19.091
and the minimum and the maximum
and the minimum is 10^(-34)

36:19.094 --> 36:19.974
radians.

36:19.969 --> 36:24.139
What that means is,
if you put a screen 1 meter

36:24.143 --> 36:25.993
away,
the distance between one

36:25.985 --> 36:28.865
maximum and the next maximum or
one minimum to the next minimum

36:28.869 --> 36:33.269
and so on,
that spacing will be 10^(-34)

36:33.266 --> 36:36.386
meters,
because that's how a radian is

36:36.391 --> 36:36.981
defined.

36:36.980 --> 36:39.190
If that angle is
θ, that distance is

36:39.193 --> 36:41.733
just the distance to the screen
times θ.

36:41.730 --> 36:44.630
That will be 10^(-34) meters.

36:44.630 --> 36:47.950
That means the wavelength of
the oscillation on your screen

36:47.954 --> 36:49.164
is 10^(-34) meters.

36:49.159 --> 36:50.839
Can you see it?

36:50.840 --> 36:54.020
Well, make the world's smallest
detector.

36:54.019 --> 36:56.799
It's as big as 1 proton, okay.

36:56.800 --> 36:57.620
Nothing can be smaller.

36:57.619 --> 37:00.139
That's your whole detector,
all the parts,

37:00.137 --> 37:01.547
everything, 1 proton.

37:01.550 --> 37:06.970
Size of a proton is 10^(-15)
meters.

37:06.969 --> 37:12.689
That means you will have
10^(19) oscillations inside your

37:12.693 --> 37:14.333
tiny detector.

37:14.329 --> 37:15.719
So don't be fooled by the 19.

37:15.719 --> 37:17.779
Let's take a minute to savor
this.

37:17.780 --> 37:23.540
That's how many oscillations
you have, okay?

37:23.539 --> 37:25.689
You've got enough now?

37:25.690 --> 37:29.260
3,6, 9,12, well,
I don't have enough time.

37:29.260 --> 37:32.040
That's a lot of oscillations.

37:32.039 --> 37:35.029
You should check the numbers
though, okay?

37:35.030 --> 37:39.620
I'm saying they typical angle
will be 10^(-34) radians and if

37:39.621 --> 37:43.971
you put the screen 1 meter away,
the spacing will be

37:43.972 --> 37:47.142
10^(-34)meters,
and you look at it with an

37:47.139 --> 37:50.459
object, a detector,
whose size is 10^(-15) meters,

37:50.460 --> 37:52.540
which looks very small,
size of a proton.

37:52.539 --> 37:57.399
But look, 10^(19) fit into that
length, so your proton detector

37:57.400 --> 37:58.420
looks huge.

37:58.420 --> 38:01.320
In fact, I cannot even show it
here.

38:01.320 --> 38:02.500
So you don't see the
oscillations;

38:02.500 --> 38:06.070
you see the average only.

38:06.070 --> 38:09.420
If you see only the average,
you can show that with 2 slits

38:09.423 --> 38:11.973
open, the sum is the sum of the
two averages,

38:11.969 --> 38:14.109
so you don't see the
oscillation.

38:14.110 --> 38:15.920
That's the first reason.

38:15.920 --> 38:17.670
Now you can say,
"You took a kilogram.

38:17.670 --> 38:19.090
Let me take a gram."

38:19.090 --> 38:20.010
I said, "Go ahead.

38:20.010 --> 38:22.110
Take a gram,
take a milligram and take a

38:22.105 --> 38:24.735
slit which is not 1 meter wide
but--1 meter apart,

38:24.737 --> 38:26.507
but 1 millimeter apart."

38:26.510 --> 38:27.730
It doesn't matter.

38:27.730 --> 38:30.280
You're playing around with
factors like 10 and 100 and

38:30.284 --> 38:30.674
1,000.

38:30.670 --> 38:33.900
I got 10^(19) here.

38:33.900 --> 38:36.620
So nothing you do will make any
dent on that.

38:36.619 --> 38:39.569
So in the macroscopic world,
you will not see this

38:39.574 --> 38:40.484
interference.

38:40.480 --> 38:45.340
Another reason you won't see it
is that the particle should have

38:45.342 --> 38:47.042
a definite momentum.

38:47.039 --> 38:49.339
It's got an indefinite
momentum, it's coming in with

38:49.338 --> 38:51.368
different momenta,
then each will have its own

38:51.365 --> 38:53.705
interference pattern and they'll
get washed out.

38:53.710 --> 38:57.260
Finally, I told you,
if you ever try to see which

38:57.260 --> 39:01.330
slit the particle took by
putting a light beam here,

39:01.329 --> 39:04.029
the minute you catch the
electron going through one slit

39:04.032 --> 39:07.662
or the other,
this pattern is gone.

39:07.659 --> 39:10.409
It will do this "I'm not
here and I'm not there"

39:10.409 --> 39:12.919
routine only if you never catch
it being anywhere.

39:12.920 --> 39:14.740
That's very interesting.

39:14.739 --> 39:17.659
The electron behaves like it
does not go through any one

39:17.657 --> 39:20.047
particular slit,
as long as you don't catch it

39:20.047 --> 39:21.477
going through one slit.

39:21.480 --> 39:23.770
You put enough light to catch
every electron,

39:23.768 --> 39:26.838
then you can add the numbers
and you must get the sum of the

39:26.836 --> 39:27.666
two numbers.

39:27.670 --> 39:30.470
Now for the atomic world,
it's possible for the electron

39:30.474 --> 39:33.284
to go for a long time without
encountering anything,

39:33.280 --> 39:36.220
and the interference effects
come into play.

39:36.219 --> 39:39.169
In a macroscopic world,
there is no way a macroscopic

39:39.170 --> 39:42.570
object can travel for any length
of time without running into

39:42.574 --> 39:43.374
something.

39:43.369 --> 39:44.879
It will run into other air
molecules.

39:44.880 --> 39:46.910
It will run into cosmic ray
radiation.

39:46.909 --> 39:49.939
It can collide with black body
radiation from the big bang,

39:49.936 --> 39:50.506
anything.

39:50.510 --> 39:53.720
The minute you have any contact
with it, this funny thing will

39:53.717 --> 39:54.347
disappear.

39:54.349 --> 39:56.099
So that's one reason you don't
see it.

39:56.099 --> 39:58.269
Now we can go on and on and
give other numbers.

39:58.268 --> 40:01.408
I've given examples in my
notes, which I will post later

40:01.414 --> 40:01.704
on.

40:01.699 --> 40:04.589
One of them is the uncertainty
principle.

40:04.590 --> 40:08.540
Why does it look like the
uncertainty principle is not

40:08.543 --> 40:09.443
important?

40:09.440 --> 40:13.990
Take again, this is 10^(-34).

40:13.989 --> 40:16.109
Everything is in MKS units.

40:16.110 --> 40:23.230
So take an object of mass 1
kilogram whose location is known

40:23.228 --> 40:26.848
to the accuracy of 1 proton.

40:26.849 --> 40:27.289
Okay?

40:27.289 --> 40:34.859
So this number is 10^(-15)
meters, do you understand?

40:34.860 --> 40:38.620
You take an object made of
10^(23) protons and you know its

40:38.623 --> 40:40.963
location to the width of 1
proton.

40:40.960 --> 40:43.150
That's all you don't know about
its location.

40:43.150 --> 40:44.000
That's your Dx.

40:44.000 --> 40:45.200
What's the Dp?

40:45.199 --> 40:48.589
Well, Dp is now
10 to the, what,

40:48.592 --> 40:48.972
19?

40:48.969 --> 40:52.199
10^(-19).

40:52.199 --> 40:56.009
Now Dp is
m times Dv.

40:56.010 --> 40:57.620
That's 10^(-19).

40:57.619 --> 41:01.659
If this is 1 kilogram,
Dv is

41:01.659 --> 41:04.419
10^(-19)meters per second.

41:04.420 --> 41:08.840
You don't know its velocity to
1 part in 10^(19).

41:08.840 --> 41:10.790
Now how bad is that?

41:10.789 --> 41:13.339
Well, suppose I start a
particle off exactly known

41:13.340 --> 41:15.630
velocity, I know where it will
be forever.

41:15.630 --> 41:19.740
But suppose I don't know the
velocity to this accuracy,

41:19.737 --> 41:22.017
and I let it run for 1 year.

41:22.018 --> 41:24.448
So I don't know precisely where
it is, but how bad is it?

41:24.449 --> 41:25.839
How badly do I not know?

41:25.840 --> 41:34.390
Well, 1 year is 10^(7) seconds,
so if it runs for 1 year,

41:34.391 --> 41:40.501
it will be unknown to 10^(-12
)meters.

41:40.500 --> 41:49.330
10^(-12)meters is what,
let's see?

41:49.329 --> 41:55.669
It's 1/100 of an atom size,
1/100 the size of an atom.

41:55.670 --> 41:59.850
So you see, these uncertainties
are not important in real life.

41:59.849 --> 42:02.459
So everything that you think
has a definite position and

42:02.463 --> 42:04.653
momentum actually has a slight
uncertainty,

42:04.650 --> 42:08.050
but the uncertainties don't
lead to any measurable

42:08.052 --> 42:12.292
consequences over any distances
that you can actually have.

42:12.289 --> 42:14.689
So what I'm trying to tell you
is, there are these waves.

42:14.690 --> 42:17.350
They do all kinds of things,
they do interference and

42:17.351 --> 42:19.961
everything, but the condition
for them is really the

42:19.963 --> 42:21.093
microscopic world.

42:21.090 --> 42:24.180
The minute the masses become
comparable to gram or kilogram

42:24.179 --> 42:26.149
and distances and slits and so
on,

42:26.150 --> 42:29.710
or like a meter or a
centimeter, these effects get

42:29.713 --> 42:30.663
washed out.

42:30.659 --> 42:34.939
But in the atomic scale,
they are seen.

42:34.940 --> 42:41.480
Now the final thing I want to
mention before moving on to a

42:41.481 --> 42:47.911
completely new topic is the role
of probability in quantum

42:47.909 --> 42:49.489
mechanics.

42:49.489 --> 42:53.229
We have seen that quantum
mechanics makes probabilistic

42:53.230 --> 42:54.200
predictions.

42:54.199 --> 42:56.729
It says if you do the double
slit experiment,

42:56.730 --> 43:00.120
I don't know where this guy
will land, but I'll give you the

43:00.121 --> 43:00.641
odds.

43:00.639 --> 43:04.449
Okay, now that looks like
something we have done many

43:04.449 --> 43:06.719
times in classical mechanics.

43:06.719 --> 43:08.989
For example,
if you have a coin and you

43:08.989 --> 43:11.319
throw the coin,
you flip it and you say,

43:11.320 --> 43:13.650
"Which way will it
land?"

43:13.650 --> 43:17.200
well, it's a very difficult
calculation to do,

43:17.199 --> 43:19.069
but it can be done in
principle, because a coin,

43:19.070 --> 43:23.430
once released from your hand,
can only land in one way.

43:23.429 --> 43:25.579
That's the determinism of
Newtonian mechanics.

43:25.579 --> 43:27.799
If you knew exactly how you
released it with what angle or

43:27.802 --> 43:29.972
momentum,
what's the viscosity of air,

43:29.967 --> 43:32.277
whatever you want,
if you give me all the numbers,

43:32.275 --> 43:33.475
I'll tell you it's head or
tails.

43:33.480 --> 43:35.220
There's no need to guess.

43:35.219 --> 43:38.089
In practice,
no one can do the calculation.

43:38.090 --> 43:40.480
What you do in practice is,
you throw the same coin 5,000

43:40.481 --> 43:43.261
times and you find out the odds
for head or tails and you say,

43:43.260 --> 43:45.410
"I predict that when you
throw it next time,

43:45.409 --> 43:50.389
it will be .51 chance that it
will be heads."

43:50.389 --> 43:52.599
That's how you give statistical
predictions.

43:52.599 --> 43:57.609
Now you did not have to use
statistics.

43:57.610 --> 44:00.600
You use it, because you cannot
really in practice do the hard

44:00.601 --> 44:01.301
calculation.

44:01.300 --> 44:03.740
In principle, you can.

44:03.739 --> 44:07.669
Secondly, if you toss a coin
and I hide it in my hand,

44:07.670 --> 44:10.050
I don't show it to you,
it's either head or tails,

44:10.050 --> 44:11.270
and I say, "What do you
think it is?"

44:11.268 --> 44:13.638
you'll say, "It's .1
chance that it's heads."

44:13.639 --> 44:16.089
And I look at it,
it may be head or it may be

44:16.090 --> 44:16.480
tail.

44:16.480 --> 44:18.350
Suppose I got head.

44:18.349 --> 44:22.399
It means that it was head even
before I opened my hand,

44:22.400 --> 44:23.000
right?

44:23.000 --> 44:25.180
The correct answer's already
inside my hand.

44:25.179 --> 44:26.149
I just didn't know it.

44:26.150 --> 44:28.060
I'm using odds,
but when I look at it,

44:28.059 --> 44:28.989
I get an answer.

44:28.989 --> 44:34.569
That's the answer it had even
before I looked.

44:34.570 --> 44:36.630
So I'll give another analogy
here.

44:36.630 --> 44:45.920
So this is a probability of
locating me somewhere.

44:45.920 --> 44:50.920
This is my home town,
Cheshire, this is Yale,

44:50.918 --> 44:55.008
and this is the infamous Route
10.

44:55.010 --> 44:57.400
So somebody has studied me for
a long time and said,

44:57.396 --> 45:00.106
"If you look for this guy,
here are the odds."

45:00.110 --> 45:04.500
Either he's working at home or
he's working at Yale,

45:04.503 --> 45:07.523
and sometimes he's driving,
okay?

45:07.519 --> 45:08.869
This is the probability.

45:08.869 --> 45:12.929
First thing to understand is
the spread out probability does

45:12.927 --> 45:15.677
not mean I am myself spread out,
okay?

45:15.679 --> 45:19.779
Unless I got into a terrible
accident on Route 10,

45:19.780 --> 45:21.790
I'm in only one place.

45:21.789 --> 45:24.219
So probability's being extended
doesn't mean the thing you're

45:24.217 --> 45:25.227
looking at is extended.

45:25.230 --> 45:27.840
I am in some sense a particle
which can be somewhere.

45:27.840 --> 45:29.540
These are the odds.

45:29.539 --> 45:34.429
Well, suppose you catch me here
on one of your many trials.

45:34.429 --> 45:36.809
If you catch me only once,
you don't know if the

45:36.811 --> 45:38.991
prediction's even good,
so you repeat it.

45:38.989 --> 45:41.369
You study me over many times
and you agree the person had got

45:41.367 --> 45:44.157
the right picture,
because after observing me

45:44.155 --> 45:48.665
many, many days you in fact get
the histogram that looks like

45:48.668 --> 45:49.268
this.

45:49.268 --> 45:52.638
The important thing is,
every time you catch me

45:52.635 --> 45:56.835
somewhere, I was already there;
you just happened to catch me

45:56.838 --> 45:57.268
there.

45:57.269 --> 46:00.179
I had a definite location.

46:00.179 --> 46:03.799
It was not known to you,
but I had it.

46:03.800 --> 46:07.500
I had a definite location
because in the macroscopic world

46:07.496 --> 46:10.416
I'm moving in,
my location is being constantly

46:10.416 --> 46:11.256
measured.

46:11.260 --> 46:14.560
You didn't ask or you didn't
find out, but I'm running

46:14.556 --> 46:16.046
through air molecules.

46:16.050 --> 46:16.900
I've slammed into them.

46:16.900 --> 46:18.510
They know that.

46:18.510 --> 46:20.520
I ran over this ant.

46:20.518 --> 46:22.778
That was the last thing the ant
knew, okay?

46:22.780 --> 46:26.700
So I'm leaving behind a trail
of destruction and they all keep

46:26.704 --> 46:28.124
track of where I am.

46:28.119 --> 46:29.729
My location is well known.

46:29.730 --> 46:32.110
You just happened to find out.

46:32.110 --> 46:37.170
But now let's change this
picture and say this is not me.

46:37.170 --> 46:40.720
This is an electron and it's
got two nuclei.

46:40.719 --> 46:43.629
This is nucleus 1 and this is
nucleus 2.

46:43.630 --> 46:46.770
It can be either near this
nucleus or that nucleus,

46:46.768 --> 46:49.398
and this is the
Y^(2) for the

46:49.404 --> 46:50.224
electron.

46:50.219 --> 46:53.619
That means it's the probability
you'll catch it here and you'll

46:53.617 --> 46:54.547
catch it there.

46:54.550 --> 46:58.500
Once again, if you catch the
electron, you will catch all of

46:58.496 --> 46:59.696
it in one place.

46:59.699 --> 47:02.269
It is wrong to think the
electronic charge is somehow

47:02.273 --> 47:03.713
spread out around the atom.

47:03.710 --> 47:04.580
It's not true.

47:04.579 --> 47:06.069
The charge is in one place.

47:06.070 --> 47:07.360
The odds are spread out.

47:07.360 --> 47:09.460
That looks just like my case.

47:09.460 --> 47:11.620
But the difference is,
if you catch the

47:11.623 --> 47:15.093
electron--let's in fact simplify
life and say there are only 2

47:15.094 --> 47:16.124
possibilities.

47:16.119 --> 47:19.549
Either it is near atom
1--nucleus 1 or nucleus 2,

47:19.550 --> 47:21.410
only 2 discrete choices.

47:21.409 --> 47:26.599
If you catch it near 2,
it is wrong to think that it

47:26.597 --> 47:29.647
was there before you got it.

47:29.650 --> 47:31.630
So where was it?

47:31.630 --> 47:34.940
It was not in any one place.

47:34.940 --> 47:39.720
It had no location till you
found its location.

47:39.719 --> 47:40.809
That's very strange.

47:40.809 --> 47:44.199
We think of measurement as
revealing a pre-existing

47:44.195 --> 47:45.885
property of the object.

47:45.889 --> 47:48.659
But in quantum theory,
it's not that you don't know

47:48.659 --> 47:49.989
where the electron is.

47:49.989 --> 47:50.839
It does not know.

47:50.840 --> 47:52.520
It is not anywhere.

47:52.518 --> 47:57.028
It's the act of measurement
that confers a location or

47:57.025 --> 47:59.315
position on the electron.

47:59.320 --> 48:02.760
That state of being,
where you can be either here or

48:02.764 --> 48:06.754
there, or simultaneously here or
there, has no analog in the

48:06.748 --> 48:08.098
classical world.

48:08.099 --> 48:11.509
If anybody tries to give you an
example, don't believe it,

48:11.514 --> 48:14.994
because there are no examples
in the macroscopic world that

48:14.990 --> 48:16.130
look like this.

48:16.130 --> 48:21.200
No analogies should satisfy
you, because this has no analog

48:21.197 --> 48:23.467
in the real world,
okay?

48:23.469 --> 48:26.179
So this is the interesting
thing in quantum mechanics.

48:26.179 --> 48:29.069
If this is a possible wave
function Y,

48:29.070 --> 48:32.180
electron near nucleus 1,
and that's a possible wave

48:32.175 --> 48:35.045
function Y,
electron near nucleus 2,

48:35.045 --> 48:37.065
you can add the two functions.

48:37.070 --> 48:39.830
That's another possible
function.

48:39.829 --> 48:41.809
But what does that describe?

48:41.809 --> 48:45.399
It describes an electron which
upon measurement could be found

48:45.396 --> 48:47.276
here and could be found there.

48:47.280 --> 48:50.380
It's not like finding me in
Cheshire or finding me in New

48:50.380 --> 48:52.070
Haven,
because in those cases,

48:52.070 --> 48:54.140
on a given day on a given
measurement,

48:54.139 --> 48:56.469
you can only get one answer,
depending on where I am.

48:56.469 --> 48:58.739
Right now if they look for me,
they can only find me here.

48:58.739 --> 49:01.039
They cannot find me anywhere
else.

49:01.039 --> 49:04.609
But in the case of the
electron, the one and the same

49:04.610 --> 49:07.770
electron, on a given trial,
at a given instant,

49:07.768 --> 49:10.788
is fully capable of being here
or there.

49:10.789 --> 49:15.749
It's like tossing a coin and
it's in my hand.

49:15.750 --> 49:18.910
We all know that when I reveal
it to you, you can only get one

49:18.907 --> 49:22.167
answer, now that the toss has
been done, it's got one answer.

49:22.170 --> 49:25.260
If it's a quantum mechanical
coin, you don't know,

49:25.257 --> 49:28.027
and it doesn't have a value
till you look.

49:28.030 --> 49:30.000
When you look,
it has a definite value.

49:30.000 --> 49:32.100
Before you look,
it doesn't have a definite

49:32.099 --> 49:32.499
value.

49:32.500 --> 49:34.510
That's exactly like saying,
when you looked,

49:34.514 --> 49:36.114
it goes through a definite
slit.

49:36.110 --> 49:39.800
When you don't look,
it's wrong to assume it went

49:39.800 --> 49:41.800
through a definite slit.

49:41.800 --> 49:42.880
Yes?

49:42.880 --> 49:45.920
Student:  Say you had a
double slit experiment but

49:45.918 --> 49:48.898
instead of having a screen that
went all the way in both

49:48.900 --> 49:52.070
directions,
you just sort of had _____

49:52.070 --> 49:52.780
screen.

49:52.780 --> 49:58.460
So then you would only be
looking at the final location of

49:58.458 --> 50:00.548
a _______ electron.

50:00.550 --> 50:02.860
The other half you would know.

50:02.860 --> 50:07.850
How would that work,
because location for some of

50:07.849 --> 50:10.969
them has to be ____________.

50:10.969 --> 50:13.029
Prof: The minute you
find the location to the

50:13.034 --> 50:15.064
accuracy of knowing which slit
it went through,

50:15.059 --> 50:17.599
you've got 2 slits or only 1 in
the experiment?

50:17.599 --> 50:21.089
Student:  You have 2
slits but only a half screen,

50:21.088 --> 50:23.018
and nothing on the other one.

50:23.018 --> 50:25.428
Prof: Oh,
you've got a screen that only

50:25.427 --> 50:26.817
comes to here,
you mean?

50:26.820 --> 50:27.440
Student:  Yes.

50:27.440 --> 50:30.140
Prof: Yes,
the real problem of location

50:30.143 --> 50:33.873
that I'm talking about here is
not when it hits the screen,

50:33.869 --> 50:36.099
but here, when you try to see
which hole it went through,

50:36.099 --> 50:37.559
by putting a light source here.

50:37.559 --> 50:40.289
I was referring to the fact
that if you have the right kind

50:40.293 --> 50:42.513
of light to know which hole it
went through,

50:42.510 --> 50:46.110
if you give it enough momentum
to wash out the pattern.

50:46.110 --> 50:48.630
As far as the screen goes,
once it comes out,

50:48.628 --> 50:51.658
it's the sum of these 2 waves
coming from the 2 holes,

50:51.664 --> 50:54.704
and it also doesn't have a well
defined position.

50:54.699 --> 50:58.039
The probability for finding it
may look like this.

50:58.039 --> 51:00.489
In fact, the probability will
look like this.

51:00.489 --> 51:01.209
Forget your screen.

51:01.210 --> 51:02.680
This is the probability.

51:02.679 --> 51:04.949
The minute you catch it,
it is found there,

51:04.952 --> 51:07.392
that's the location after that
measurement.

51:07.389 --> 51:11.569
Prior to the measurement,
it can really be anywhere where

51:11.568 --> 51:13.358
the function is not 0.

51:13.360 --> 51:14.770
There are many wave functions.

51:14.768 --> 51:18.108
There's the 1 to the left of
the slit and there is 1--then it

51:18.110 --> 51:20.950
becomes 2 wave functions coming
from the 2 slits.

51:20.949 --> 51:23.429
They form the interference
pattern and that gives you the

51:23.429 --> 51:25.689
odds that if you looked for it,
you will find it.

51:25.690 --> 51:28.540
Now till you find it,
it's not anywhere.

51:28.539 --> 51:31.559
It can be anywhere on this line
at that instant.

51:31.559 --> 51:34.239
It's only the act of
measurement, or hitting a

51:34.239 --> 51:36.919
detector that tells you that's
my location.

51:36.920 --> 51:39.290
So you will have to get used to
that.

51:39.289 --> 51:42.879
You'll have to get used to the
fact that things don't have

51:42.875 --> 51:45.135
position, momentum,
angular momentum,

51:45.139 --> 51:47.969
energy or anything,
until you measure it.

51:47.969 --> 51:54.459
Okay, so I've got to tell you a
little more now about just

51:54.456 --> 51:55.706
position.

51:55.710 --> 52:00.490
So let's take--by the way,
any questions so far?

52:00.489 --> 52:01.919
Yes?

52:01.920 --> 52:04.510
Student:  Can you
explain again how you can tell

52:04.514 --> 52:07.114
which hole the electron goes
through with the light?

52:07.110 --> 52:08.840
Prof: Well,
you just see it.

52:08.840 --> 52:11.090
You see a flicker and if it's
near this hole,

52:11.088 --> 52:13.028
you know this guy went through
that.

52:13.030 --> 52:15.600
And if it's near that one,
you know it went through that.

52:15.599 --> 52:17.779
But to have such good
resolution, the wavelength

52:17.775 --> 52:20.595
should be much smaller than the
space in between the slits.

52:20.599 --> 52:23.789
Otherwise you'll get a big blur
and you won't know which one it

52:23.788 --> 52:24.558
went through.

52:24.559 --> 52:27.559
That's a soft measurement that
doesn't do you any good.

52:27.559 --> 52:28.739
It won't destroy the pattern.

52:28.739 --> 52:30.999
That's because you don't know
which hole it went through.

52:31.000 --> 52:34.260
If you make it fine enough to
know which hole it went through,

52:34.262 --> 52:37.582
you will disrupt the electron's
momentum enough to wipe out the

52:37.577 --> 52:38.217
pattern.

52:38.219 --> 52:39.369
Yes?

52:39.369 --> 52:48.639
Student:
>

52:48.639 --> 52:50.229
Prof: Oh, here?

52:50.230 --> 52:52.280
You mean what happened to the 2?

52:52.280 --> 52:55.410
Oh yeah, forget the 2s.

52:55.409 --> 52:58.389
There are 2p's I forgot, right?

52:58.389 --> 53:01.359
There's a lot of 2p's too
you've got to put in.

53:01.360 --> 53:05.780
10^(-34 )is not exactly the
answer, because I got 2p there.

53:05.780 --> 53:08.350
I'm just saying,
look, if it's 10^(-34),

53:08.346 --> 53:12.356
suppose you're talking about
how much money Bill Gates has.

53:12.360 --> 53:14.480
It's 10^(9).

53:14.480 --> 53:17.630
Now is it 2 x 10^(9),
3 x 10^(9)?

53:17.630 --> 53:19.930
I don't know,
but I'm not worried about his

53:19.927 --> 53:22.547
financial wellbeing,
because it's up there in the

53:22.552 --> 53:23.212
10^(9)s.

53:23.210 --> 53:26.140
So whenever I do these
arguments, you should get used

53:26.143 --> 53:28.543
to this notion,
it's very common for physicist

53:28.541 --> 53:30.341
when they argue in quantum
mechanics,

53:30.340 --> 53:34.300
will use the symbol that's not
quite an =.

53:34.300 --> 53:36.710
It looks like a wiggle and =.

53:36.710 --> 53:39.390
Basically it means,
I'm not quite sure,

53:39.389 --> 53:41.999
but the number is in this
ballpark.

53:42.000 --> 53:44.680
And if this ballpark is 1,000
miles from that ballpark,

53:44.677 --> 53:46.907
you just have to know it's in
the ballpark.

53:46.909 --> 53:48.969
You don't have to know where it
is.

53:48.969 --> 53:50.479
Some things are definitely
macroscopic;

53:50.480 --> 53:52.150
some things are definitely
microscopic.

53:52.150 --> 53:56.120
But something very interesting
is happening at Yale right now.

53:56.119 --> 54:00.169
People are asking the following
question: how small an object

54:00.166 --> 54:04.546
has to be before I can see its
quantum mechanical fluctuations?

54:04.550 --> 54:06.650
We know that if it's like an
electron, it's completely

54:06.653 --> 54:07.213
fluctuating.

54:07.210 --> 54:09.080
You don't know anything.

54:09.079 --> 54:11.439
If it's like a bowling ball,
it seems to have a well defined

54:11.440 --> 54:11.880
position.

54:11.880 --> 54:14.290
Make the objects smaller and
smaller and smaller.

54:14.289 --> 54:19.529
How small can it be before it's
first beginning to show quantum

54:19.530 --> 54:23.250
effects,
like quantization of energy or

54:23.253 --> 54:27.953
quantization of momentum,
depending on the problem,

54:27.947 --> 54:31.167
or quantum fluctuations in
position?

54:31.170 --> 54:33.650
So those measurements are now
being carried out at Yale.

54:33.650 --> 54:36.770
It's a very exciting time and
it's so many years after the

54:36.766 --> 54:38.566
discovery of quantum mechanics.

54:38.570 --> 54:41.500
Because we knew the really big
world and we knew the really

54:41.501 --> 54:43.391
small world,
but now we're trying to go,

54:43.389 --> 54:47.039
because of nanotechnology,
continuously from big to that

54:47.043 --> 54:49.353
small,
and how small is small?

54:49.349 --> 54:50.619
That's the question?

54:50.619 --> 54:55.689
Can a little macroscopic object
simultaneously be in two places?

54:55.690 --> 54:57.850
Most of them seem to have a
well defined location.

54:57.849 --> 55:02.309
Can you create a situation when
it's capable of being found here

55:02.306 --> 55:03.576
and found there?

55:03.579 --> 55:05.629
It's very hard,
because you have to isolate the

55:05.626 --> 55:07.136
particle from the outside world.

55:07.139 --> 55:08.479
That's the first condition.

55:08.480 --> 55:10.640
That's what ruins everything.

55:10.639 --> 55:12.809
A quantum computer,
you must know,

55:12.809 --> 55:16.769
has got these bits called qbits
and unlike the bits in your

55:16.766 --> 55:19.316
laptop,
which are either 0 or 1,

55:19.324 --> 55:21.984
a qbit can also give you 0 or
1,

55:21.980 --> 55:26.810
but it can also be in a state
where it can give either 0 or 1

55:26.809 --> 55:28.339
on a given trial.

55:28.340 --> 55:32.030
The bits in your computer,
the particular bit right now is

55:32.032 --> 55:33.332
either a 0 or a 1.

55:33.329 --> 55:35.619
Maybe you don't know it,
but it can only give you that

55:35.617 --> 55:37.127
answer because that's what it
is.

55:37.130 --> 55:39.950
Because that bit is in contact
with the world and the world is

55:39.945 --> 55:41.465
constantly measuring its value.

55:41.469 --> 55:45.089
A qbit is a quantum system
which can do one of two things,

55:45.085 --> 55:48.315
but it's isolated and it's
neither this nor that.

55:48.320 --> 55:51.170
It's like the electron going
through both slits.

55:51.170 --> 55:55.910
So a quantum bit can explore
many possibilities.

55:55.909 --> 56:02.859
If you build a computer with 10
qbits it can be doing 2^(10)

56:02.858 --> 56:05.918
things at the same time.

56:05.920 --> 56:09.460
And if it's got a million bits,
it's 2^(million) operations,

56:09.460 --> 56:12.220
things it can be exploring at
the same time.

56:12.219 --> 56:15.949
That's why, as you know,
one of the ways to securely

56:15.954 --> 56:19.844
send your credit card
information is to use very large

56:19.836 --> 56:22.916
numbers,
on the assumption that no one

56:22.920 --> 56:24.390
can factorize them.

56:24.389 --> 56:27.419
You can always multiply a 100
digit number by a 100 digit

56:27.423 --> 56:29.703
number on your computer
instantaneously.

56:29.699 --> 56:32.809
But if I gave you the 200 digit
number and told you to find the

56:32.806 --> 56:34.406
two factors you won't find it.

56:34.409 --> 56:37.759
You won't find it in the age of
the universe.

56:37.760 --> 56:39.860
It's amazing,
but that simple problem of

56:39.860 --> 56:43.470
factorization cannot be done if
the numbers are 100 digits long,

56:43.469 --> 56:46.289
and that's the reason why
people openly broadcast the

56:46.291 --> 56:48.181
product,
they may broadcast one of the

56:48.182 --> 56:51.102
numbers,
and only the other person knows

56:51.103 --> 56:52.623
the second number.

56:52.619 --> 56:56.989
Now there is something somebody
called Shore,

56:56.989 --> 56:58.679
Peter Shore,
showed that if you have a

56:58.677 --> 57:01.297
quantum computer,
made up of these qbits,

57:01.302 --> 57:05.552
it can actually factor the
number exponentially faster,

57:05.550 --> 57:07.720
namely, instead of taking
10^(10 )seconds,

57:07.719 --> 57:10.759
it will take 10 seconds.

57:10.760 --> 57:14.650
So if you build a quantum
computer, you have two options.

57:14.650 --> 57:18.630
Either you can become famous,
or get tenure at Yale,

57:18.630 --> 57:22.430
maybe, or you can go on the
biggest shopping spree of your

57:22.434 --> 57:25.184
life,
because you can get anybody's

57:25.177 --> 57:26.697
credit card number.

57:26.699 --> 57:27.789
So that's the choice.

57:27.789 --> 57:30.459
When you come to that fork,
you decide which way you want

57:30.458 --> 57:30.838
to go.

57:30.840 --> 57:33.130
Maybe you can go through both
choices, I don't know.

57:33.130 --> 57:35.890
That's something in your future.

57:35.889 --> 57:38.029
So why is it so hard to build a
quantum computer?

57:38.030 --> 57:40.280
There are many,
many quantum systems which can

57:40.280 --> 57:42.380
do one or two things,
and can be the state,

57:42.380 --> 57:44.180
but they are both this and
that.

57:44.179 --> 57:47.579
The problem is,
they cannot be in contact with

57:47.579 --> 57:50.979
the outside world,
because single contact with

57:50.980 --> 57:52.870
them is like a dream.

57:52.869 --> 57:54.049
Think about it, it's gone.

57:54.050 --> 57:54.550
Same thing.

57:54.550 --> 57:56.310
Any measurement destroys it.

57:56.309 --> 57:59.799
Any unintended measurement also
destroys it, so you've got to

57:59.795 --> 58:01.765
keep your system fully isolated.

58:01.768 --> 58:04.688
A system that is not talking to
the outside world,

58:04.693 --> 58:07.323
unfortunately,
is also not talking to you.

58:07.320 --> 58:09.850
So you cannot ask it any
questions, and if it knows the

58:09.849 --> 58:11.209
answer, it cannot tell you.

58:11.210 --> 58:12.910
So sometimes you want it to
talk.

58:12.909 --> 58:13.859
It's like relationships.

58:13.860 --> 58:15.890
Sometimes you don't want it to
talk.

58:15.889 --> 58:17.069
So what do you do?

58:17.070 --> 58:19.360
You've got to build a system
where sometimes,

58:19.362 --> 58:21.972
in a controlled way,
you can make contact with your

58:21.967 --> 58:23.997
system, namely give it the
problem.

58:24.000 --> 58:26.560
Then it does its quantum thing,
then you've got to make a

58:26.556 --> 58:28.606
measurement to find out what the
answer is.

58:28.610 --> 58:31.550
Then you want to be able to get
into it again.

58:31.550 --> 58:35.020
So the challenge for quantum
computers is how to keep them

58:35.019 --> 58:37.759
isolated long enough to do the
calculation.

58:37.760 --> 58:40.260
That's a challenge,
how to keep it from--how to

58:40.257 --> 58:43.077
keep it in what's called a
quantum coherent state.

58:43.079 --> 58:46.879
A coherent state is really when
it's doing many things at the

58:46.880 --> 58:47.640
same time.

58:47.639 --> 58:52.159
All right, so I want to tell
you now more formally how to do

58:52.163 --> 58:54.083
more quantum mechanics.

58:54.079 --> 58:58.309
So let's take a simple example,
a particle living on a line.

58:58.309 --> 59:03.259
That's the function Y(x).

59:03.260 --> 59:07.240
Let's ask the following
question: how do we do business

59:07.244 --> 59:09.094
in Newtonian mechanics?

59:09.090 --> 59:10.310
We say, "Here's a particle.

59:10.309 --> 59:11.009
That's its x.

59:11.010 --> 59:11.680
Here's the momentum.

59:11.679 --> 59:13.299
That's its p."

59:13.300 --> 59:16.750
Given that, I know everything I
need to know right now.

59:16.750 --> 59:19.190
Angular momentum,
r x

59:19.193 --> 59:21.623
p, Kinetic energy,
p^(2)/2m.

59:21.619 --> 59:25.209
Everything is given in terms of
the coordinates and momentum.

59:25.210 --> 59:28.760
In quantum theory,
you don't even tell me where it

59:28.757 --> 59:29.117
is.

59:29.119 --> 59:32.789
For every possible x,
there is a function whose

59:32.786 --> 59:36.726
square, if you now square this
guy, everything will be now

59:36.728 --> 59:38.318
positive, definite.

59:38.320 --> 59:39.870
I don't know,
it's something like this.

59:39.869 --> 59:42.989
This is Y^(2).

59:42.989 --> 59:46.899
We say the height is
proportional to finding it

59:46.902 --> 59:48.012
everywhere.

59:48.010 --> 59:49.870
What is the condition on the
function side?

59:49.869 --> 59:52.669
The answer is,
whatever you like.

59:52.670 --> 59:56.260
Anything I can draw,
with no special effort,

59:56.257 --> 59:59.677
is a possible function for an
electron.

59:59.679 --> 1:00:00.779
There are no restrictions.

1:00:00.780 --> 1:00:03.360
It's like saying,
what should the position of the

1:00:03.358 --> 1:00:04.108
particle be?

1:00:04.110 --> 1:00:05.000
x.

1:00:05.000 --> 1:00:08.510
Any x you want is a
possible x.

1:00:08.510 --> 1:00:12.770
Likewise, any Y you draw is a
possible Y.

1:00:12.768 --> 1:00:15.978
That's a set of all possible
ampli--it's called wave

1:00:15.976 --> 1:00:19.056
functions--whose square is the
set of all possible

1:00:19.057 --> 1:00:20.187
probabilities.

1:00:20.190 --> 1:00:24.300
So I said Y^(2),
Y at the point x^(2),

1:00:24.302 --> 1:00:29.142
is connected to the probability
of finding it at x.

1:00:29.139 --> 1:00:32.059
That has to be in fact to be
refined.

1:00:32.059 --> 1:00:33.639
That's not precisely the story.

1:00:33.639 --> 1:00:35.369
I'll tell you why.

1:00:35.369 --> 1:00:41.319
If a statistical event has got
6 possible answers,

1:00:41.320 --> 1:00:43.990
like you throw the die,
you want to get any number from

1:00:43.994 --> 1:00:46.084
1 to 6,
you can give the probability

1:00:46.083 --> 1:00:52.053
for 1,
probability for 2,3, 4,5 and 6.

1:00:52.050 --> 1:00:57.510
These are all the odds for
getting any number from 1 to 6.

1:00:57.510 --> 1:00:59.760
Since there are only a finite
number of things that can

1:00:59.762 --> 1:01:02.052
happen,
I call them I = 1 to 6,

1:01:02.050 --> 1:01:04.900
there's a probability for each
I,

1:01:04.900 --> 1:01:07.460
and if you add all the
probabilities,

1:01:07.460 --> 1:01:13.430
you should get 1.

1:01:13.429 --> 1:01:17.229
But if the set of things that
can happen is a continuous

1:01:17.231 --> 1:01:19.791
variable like x,
in other words,

1:01:19.786 --> 1:01:22.966
the location of the electron is
not a discrete set of numbers.

1:01:22.969 --> 1:01:26.189
It's any real number is a
possible location.

1:01:26.190 --> 1:01:29.760
Then you cannot give a finite
probability for any one

1:01:29.755 --> 1:01:30.505
x.

1:01:30.510 --> 1:01:33.600
If that was finite,
since the number of points is

1:01:33.601 --> 1:01:36.951
infinite, you cannot make the
total probability 1.

1:01:36.949 --> 1:01:39.309
So what we really mean by
Y^(2) is called

1:01:39.311 --> 1:01:40.451
the probability density.

1:01:40.449 --> 1:01:42.679
That means draw the
Y^(2),

1:01:42.677 --> 1:01:45.417
let this be the height of
Y^(2).

1:01:45.420 --> 1:01:48.760
Take a little sliver of width
dx.

1:01:48.760 --> 1:01:53.160
The area under the graph,
P(x) dx,

1:01:53.157 --> 1:01:58.317
that is the probability of
finding the electron,

1:01:58.324 --> 1:02:03.384
or whatever particle,
between x and x

1:02:03.380 --> 1:02:04.810
dx.

1:02:04.809 --> 1:02:05.249
You understand?

1:02:05.250 --> 1:02:07.310
It's called a density.

1:02:07.309 --> 1:02:10.299
So you don't give a finite
probability to each point.

1:02:10.300 --> 1:02:14.910
For an infinitesimal region,
you give it infinitesimal

1:02:14.913 --> 1:02:19.443
probability, which is the
function P(x) dx.

1:02:19.440 --> 1:02:23.150
And the statement that the
particle has to be somewhere,

1:02:23.150 --> 1:02:25.800
namely, all the probabilities
add up to 0,

1:02:25.800 --> 1:02:30.970
is the statement that when you
integrate this probability

1:02:30.969 --> 1:02:38.379
density from - to infinity,
namely, Y^(2)dx,

1:02:38.380 --> 1:02:50.100
from - to infinity,
that should be 1.

1:02:50.099 --> 1:02:51.819
This is called normalization.

1:02:51.820 --> 1:02:53.120
It's a mathematical term.

1:02:53.119 --> 1:02:58.379
Norm is connected to length in
some way, and these can be

1:02:58.375 --> 1:03:00.905
viewed as length squared.

1:03:00.909 --> 1:03:07.039
Anyway, this is called the
normalization.

1:03:07.039 --> 1:03:13.449
Now if somebody gave you a Y,
which did not obey this

1:03:13.447 --> 1:03:16.527
condition, here is a Y.

1:03:16.530 --> 1:03:18.440
This already tells you a nice
story, right?

1:03:18.440 --> 1:03:20.880
It tells you the odds are
pretty big here,

1:03:20.876 --> 1:03:22.596
pretty small there,
0 here.

1:03:22.599 --> 1:03:28.679
Now take a function that's
twice as tall.

1:03:28.679 --> 1:03:31.639
That gives you the same
relative odds,

1:03:31.639 --> 1:03:32.999
you understand.

1:03:33.000 --> 1:03:35.830
So when you multiply Y by any
number, you don't change the

1:03:35.831 --> 1:03:37.771
basic predictive power of the
theory.

1:03:37.768 --> 1:03:41.068
It is just that if your
original Y had a square integral

1:03:41.065 --> 1:03:44.415
= 1, the new one may not have,
but the information is the

1:03:44.420 --> 1:03:44.960
same.

1:03:44.960 --> 1:03:47.710
It's really the relative height
of the function.

1:03:47.710 --> 1:03:52.570
That's another shocking thing
in quantum mechanics.

1:03:52.570 --> 1:03:59.330
If Y stood for a string
vibrating, 2 � Y (this is Y,

1:03:59.326 --> 1:04:06.846
this is 2 � Y) is a totally
different configuration of the

1:04:06.847 --> 1:04:08.247
string.

1:04:08.250 --> 1:04:13.070
But in quantum mechanics,
Y and any multiple of Y are

1:04:13.070 --> 1:04:16.950
physically equivalent,
because what we extract from

1:04:16.947 --> 1:04:19.947
Y is the relative
probability of finding it here

1:04:19.952 --> 1:04:21.792
and there and there and there.

1:04:21.789 --> 1:04:27.109
So scaling the whole thing by a
factor, 2 or 4 or any number,

1:04:27.106 --> 1:04:28.876
it doesn't matter.

1:04:28.880 --> 1:04:30.150
That's a very new thing.

1:04:30.150 --> 1:04:32.920
That's why the Y is not
very physical.

1:04:32.920 --> 1:04:35.920
If you took a string and you
pulled it by twice as much,

1:04:35.920 --> 1:04:37.940
it's a totally different
situation.

1:04:37.940 --> 1:04:40.480
If you took the electric field
and made it twice as big,

1:04:40.478 --> 1:04:41.908
that's a different situation.

1:04:41.909 --> 1:04:43.909
Forces on electrons are doubled
now.

1:04:43.909 --> 1:04:45.679
In the quantum mechanical
Y,

1:04:45.679 --> 1:04:48.569
when you double it,
it stands for the same physical

1:04:48.565 --> 1:04:51.775
condition of the electron,
because the odds of being here

1:04:51.775 --> 1:04:53.615
versus being there are not
altered.

1:04:53.619 --> 1:04:57.719
The only job of Y is to
give you the odds.

1:04:57.719 --> 1:05:03.319
Therefore it's like saying in 2
dimensions, that's a vector,

1:05:03.320 --> 1:05:05.980
that's a different vector.

1:05:05.980 --> 1:05:08.750
But suppose you only care about
the direction of the vector.

1:05:08.750 --> 1:05:10.980
For some reason,
you don't care how long it is,

1:05:10.981 --> 1:05:14.041
you just want to know which way
it points, then of course all of

1:05:14.036 --> 1:05:15.536
these are considered equal.

1:05:15.539 --> 1:05:17.319
And that's really how it is for
quantum mechanics.

1:05:17.320 --> 1:05:21.620
Every Y and every
multiple of it stands for one

1:05:21.619 --> 1:05:22.999
situation only.

1:05:23.000 --> 1:05:26.160
So what one normally does is
from all these vectors in that

1:05:26.161 --> 1:05:28.111
direction,
you may pick one whose length

1:05:28.108 --> 1:05:30.488
is 1 and say,
"Let me use that member of

1:05:30.490 --> 1:05:33.020
the family to stand for the
situation."

1:05:33.018 --> 1:05:37.328
That's like saying of all the
Y's obtained by scaling

1:05:37.333 --> 1:05:41.433
up and down, I'll pick one whose
square integral is 1.

1:05:41.429 --> 1:05:44.299
So let me do a concrete
example, so you know what I'm

1:05:44.297 --> 1:05:45.177
talking about.

1:05:45.179 --> 1:05:50.489
So let's take a function that
looks like this.

1:05:50.489 --> 1:05:57.499
It is 0 everywhere,
and it has a height A between a

1:05:57.498 --> 1:05:58.758
and -a.

1:05:58.760 --> 1:05:59.740
That's my Y.

1:05:59.739 --> 1:06:06.889
So Y(x) = A for
absolute value(x) less

1:06:06.887 --> 1:06:09.967
than a and = 0 outside.

1:06:09.969 --> 1:06:17.709
That's a possible wave function
Y.

1:06:17.710 --> 1:06:20.110
Now what does it tell you in
words?

1:06:20.110 --> 1:06:22.560
If it's a word problem,
what does it tell you about the

1:06:22.559 --> 1:06:23.059
electron?

1:06:23.059 --> 1:06:25.719
Can anybody tell me?

1:06:25.719 --> 1:06:28.819
What can you say about the
electron given by this function?

1:06:28.820 --> 1:06:31.530
What do you know about it?

1:06:31.530 --> 1:06:32.370
Yes?

1:06:32.369 --> 1:06:35.129
Student:  It must be
found within or -a.

1:06:35.130 --> 1:06:38.410
Prof: It must be found
within or -a and more than that.

1:06:38.409 --> 1:06:40.699
Student:
>

1:06:40.699 --> 1:06:41.209
Prof: That is correct.

1:06:41.210 --> 1:06:42.380
Student:  With the same
probability.

1:06:42.380 --> 1:06:43.800
Prof: With the same
probability, okay?

1:06:43.800 --> 1:06:46.870
The probability is it
restricted to -a to a,

1:06:46.869 --> 1:06:50.009
and it's the same throughout
the interval.

1:06:50.010 --> 1:06:52.590
After all, if you just set it
restricted to -a to a,

1:06:52.588 --> 1:06:55.368
it's true for this function
too, but that's not the same

1:06:55.371 --> 1:06:56.131
everywhere.

1:06:56.130 --> 1:06:58.040
I've got a guy who's same
everywhere.

1:06:58.039 --> 1:07:03.869
Do you agree that this function
has exactly the same property,

1:07:03.867 --> 1:07:09.217
restricted to -a to a,
and the probability's constant?

1:07:09.219 --> 1:07:11.319
So there are many,
many functions you can draw,

1:07:11.320 --> 1:07:15.280
all with the same statement
that this object has got equal

1:07:15.282 --> 1:07:18.762
likelihood to be in this
interval and 0 outside.

1:07:18.760 --> 1:07:21.550
Of this family,
we are going to pick one guy

1:07:21.548 --> 1:07:23.428
whose square integral is 1.

1:07:23.429 --> 1:07:25.919
So I'm going to keep this
number A,

1:07:25.920 --> 1:07:28.430
the height of the function,
as a free parameter,

1:07:28.429 --> 1:07:32.559
and I'm going to choose is so
that A^(2)--

1:07:32.559 --> 1:07:36.499
so that the
Y^(2)dx

1:07:36.503 --> 1:07:42.043
from - to infinity,
I want it to be 1.

1:07:42.039 --> 1:07:44.059
And I'll pick A so that
that is true.

1:07:44.059 --> 1:07:46.589
Well, we can do this integral
in our head.

1:07:46.590 --> 1:07:47.970
What is this integral?

1:07:47.969 --> 1:07:52.569
This is just the square of
A times the width of this

1:07:52.574 --> 1:07:53.294
region.

1:07:53.289 --> 1:07:56.209
That's got to be 1.

1:07:56.210 --> 1:07:59.760
That tells me that A
must be chosen to be

1:07:59.755 --> 1:08:03.295
1/(2a)^(1/2),
where this little a,

1:08:03.302 --> 1:08:06.852
2a, is now the width of
this region.

1:08:06.849 --> 1:08:11.949
Therefore from this whole
family, the normalized Y

1:08:11.952 --> 1:08:15.692
will look like
1/(2a)^(1/2 )for mod

1:08:15.688 --> 1:08:19.878
x <
or = a and 0 outside.

1:08:19.880 --> 1:08:23.140
And we can all see at a glance
that if you squared this

1:08:23.137 --> 1:08:26.457
normalized Y and
integrated from -a to a you will

1:08:26.456 --> 1:08:27.056
get 1.

1:08:27.060 --> 1:08:30.270
This is normalization.

1:08:30.270 --> 1:08:32.730
So sometimes,
people will give you a wave

1:08:32.731 --> 1:08:36.611
function and they will say as a
first step, "Normalize this

1:08:36.612 --> 1:08:38.092
wave function."

1:08:38.090 --> 1:08:40.920
What you have to do is,
you've got to square the wave

1:08:40.921 --> 1:08:43.591
function and then put a number
in front of it,

1:08:43.590 --> 1:08:47.540
and choose it so that the
number makes the square integral

1:08:47.542 --> 1:08:47.822
1.

1:08:47.819 --> 1:08:50.539
Let me give you another example.

1:08:50.538 --> 1:08:54.018
There's a very famous function,
called the Gaussian function.

1:08:54.020 --> 1:08:59.410
It looks like this.

1:08:59.408 --> 1:09:03.278
The function e^(-α(x
(squared)))dx from - to

1:09:03.282 --> 1:09:07.082
infinity happens to have an area
which is square root of

1:09:07.083 --> 1:09:09.093
p/a.

1:09:09.090 --> 1:09:13.150
That is just one of those
tabulated integrals.

1:09:13.149 --> 1:09:18.529
So here's a bell shaped
function with this property.

1:09:18.529 --> 1:09:24.639
Now I want to make a quantum
mechanical wave function that

1:09:24.640 --> 1:09:28.820
looks like the
Y(x) = A

1:09:28.823 --> 1:09:33.223
e^(- x) squared over 2 D
squared.

1:09:33.220 --> 1:09:35.530
That's a possible wave
function, right?

1:09:35.529 --> 1:09:38.039
Nothing funny about it,
but what do you know about the

1:09:38.041 --> 1:09:38.801
wave function?

1:09:38.800 --> 1:09:41.880
It's biggest at x = 0.

1:09:41.880 --> 1:09:44.960
It's symmetric between and
-x.

1:09:44.960 --> 1:09:48.960
And it dies off very quickly,
but how far should you go?

1:09:48.960 --> 1:09:52.190
You can easily guess that when
x is much bigger than D,

1:09:52.189 --> 1:09:53.769
this function is gone,
because

1:09:53.765 --> 1:09:56.805
x/D^(2) is
going on the exponent.

1:09:56.810 --> 1:09:59.400
So if that number's big,
it's e to the - big,

1:09:59.403 --> 1:10:00.703
which is very small.

1:10:00.698 --> 1:10:04.308
So roughly speaking the width
of this graph is of order

1:10:04.305 --> 1:10:05.905
D or 2D.

1:10:05.908 --> 1:10:09.008
I'm just going to call it
D, just to give you an

1:10:09.012 --> 1:10:10.222
order of magnitude.

1:10:10.220 --> 1:10:14.680
So that's an electron whose
location is roughly known to an

1:10:14.684 --> 1:10:16.074
amount D.

1:10:16.069 --> 1:10:18.849
But this is not normalized,
because if I take the square of

1:10:18.845 --> 1:10:19.895
this, I won't get 1.

1:10:19.899 --> 1:10:23.149
So I will choose A so
that 1 =

1:10:23.149 --> 1:10:26.039
Y^(2)xdx.

1:10:26.038 --> 1:10:28.948
This being real,
I don't need the absolute value

1:10:28.952 --> 1:10:29.822
of Y.

1:10:29.819 --> 1:10:35.539
That gives me A^(2)e
^(-x(squared))/

1:10:35.539 --> 1:10:38.809
D^(2)dx.

1:10:38.810 --> 1:10:41.810
The 2 went away,
because I squared the function

1:10:41.807 --> 1:10:43.957
Y, so don't forget that.

1:10:43.960 --> 1:10:47.710
Now I look at the table of
integrals and what is a?

1:10:47.710 --> 1:10:53.070
When I compare these 2,
a is just 1/D^(2).

1:10:53.069 --> 1:11:02.949
So it's square root of
pD^(2).

1:11:02.948 --> 1:11:05.358
This is an easy thing,
because I'm already giving you

1:11:05.356 --> 1:11:07.986
the integral you need to do,
but I want you to get used to

1:11:07.994 --> 1:11:08.324
it.

1:11:08.319 --> 1:11:10.739
So this whole thing should be 1.

1:11:10.738 --> 1:11:15.508
That means A is
1/pD^(2) to the fourth

1:11:18.029 --> 1:11:21.139
Therefore the normalized wave
function,

1:11:21.140 --> 1:11:27.340
Y normalized,
looks like 1/pD^(2) to

1:11:27.344 --> 1:11:33.304
the ¼,
e^( −x(squared

1:11:33.296 --> 1:11:36.826
))/2D^(2).

1:11:36.828 --> 1:11:38.348
Normalization is just a
discipline.

1:11:38.350 --> 1:11:41.930
You discipline yourself to take
all functions and normalize

1:11:41.934 --> 1:11:44.534
them, because why do you
normalize them?

1:11:44.529 --> 1:11:48.949
If you normalize them to 1,
then Y^(2) is

1:11:48.953 --> 1:11:52.563
directly the absolute
probability density.

1:11:52.560 --> 1:11:55.510
That means when you add it all
up, you'll get 1.

1:11:55.510 --> 1:11:58.270
If you don't normalize it to 1,
Y^(2) is the

1:11:58.274 --> 1:11:59.784
relative probability density.

1:11:59.779 --> 1:12:01.979
It will still tell you the
relative odds of this and that,

1:12:01.979 --> 1:12:05.099
but you cannot say this
interval from here to here,

1:12:05.100 --> 1:12:07.280
the chances are 30 percent for
catching it.

1:12:07.279 --> 1:12:09.879
You must take the region that
you're integrating,

1:12:09.881 --> 1:12:11.401
divide by the whole thing.

1:12:11.399 --> 1:12:15.629
But you don't have to divide by
the whole thing if you've

1:12:15.625 --> 1:12:17.205
normalized it to 1.

1:12:17.210 --> 1:12:20.250
Okay, this is just practice in
normalization.

1:12:20.250 --> 1:12:24.550
So I'm going to give you a
little hint on what is going to

1:12:24.545 --> 1:12:29.065
happen next, but I won't do it
now, so you guys don't have to

1:12:29.065 --> 1:12:30.795
take down anything.

1:12:30.800 --> 1:12:36.040
Just ask the following question
and we'll come back to it on

1:12:36.042 --> 1:12:37.112
Wednesday.

1:12:37.109 --> 1:12:42.389
I've told you that in Newtonian
mechanics, every particle has an

1:12:42.386 --> 1:12:45.146
x and it has a p.

1:12:45.149 --> 1:12:47.599
In quantum theory,
instead we traded for a

1:12:47.604 --> 1:12:51.204
function Y(x) and
we learned the meaning of the

1:12:51.198 --> 1:12:53.768
Y(x) is that
absolute value of

1:12:53.774 --> 1:12:57.014
Y^(2) is the
probability density,

1:12:57.010 --> 1:13:00.330
meaning P(x) dx is the
probability of finding it

1:13:00.328 --> 1:13:02.478
between x and x
dx.

1:13:02.479 --> 1:13:04.939
Now we can say,
"Okay, that's enough about

1:13:04.940 --> 1:13:05.530
position.

1:13:05.529 --> 1:13:08.769
What about momentum?"

1:13:08.770 --> 1:13:10.520
I can measure the momentum of a
particle.

1:13:10.520 --> 1:13:13.490
You talked about momentum on
and off in the lecture.

1:13:13.488 --> 1:13:18.168
If I measure momentum,
what answer will I get?"

1:13:18.170 --> 1:13:24.680
What are the odds for getting
this or that answer?

1:13:24.680 --> 1:13:28.160
So given Y(x)
that looks like this,

1:13:28.161 --> 1:13:31.571
you square it,
you get Y(P(x)).

1:13:31.569 --> 1:13:33.419
The question is,
x is not the only thing

1:13:33.416 --> 1:13:34.296
we're interested in.

1:13:34.300 --> 1:13:37.550
Even in Newtonian mechanics,
x and p were

1:13:37.546 --> 1:13:38.746
equally important.

1:13:38.750 --> 1:13:41.110
What do you think will happen
now?

1:13:41.109 --> 1:13:46.159
How do I find out what happens
if, instead of being interested

1:13:46.157 --> 1:13:50.207
in where I find it,
I ask, with what momentum will

1:13:50.211 --> 1:13:51.371
I find it?

1:13:51.368 --> 1:13:55.848
Can you imagine a guess on what
the answer might be or in what

1:13:55.853 --> 1:13:58.723
form the answer will be given to
you?

1:13:58.720 --> 1:13:59.880
This is a wild guess.

1:13:59.880 --> 1:14:04.090
Nobody expects you to invent
quantum mechanics in 30 seconds,

1:14:04.094 --> 1:14:05.714
so make a wild guess.

1:14:05.710 --> 1:14:06.520
Yes?

1:14:06.520 --> 1:14:09.940
Anybody there want to make a
wild guess?

1:14:09.939 --> 1:14:11.589
No?

1:14:11.590 --> 1:14:12.830
Go ahead, yes, you're smiling.

1:14:12.829 --> 1:14:15.979
Make a guess.

1:14:15.979 --> 1:14:20.229
I want the odds for different
values for momentum.

1:14:20.229 --> 1:14:22.739
How do you think that
information will be contained in

1:14:22.744 --> 1:14:23.414
this theory?

1:14:23.408 --> 1:14:31.418
Student:
>

1:14:31.420 --> 1:14:31.760
Prof: Pardon me?

1:14:31.760 --> 1:14:32.830
Student:
>

1:14:32.828 --> 1:14:35.168
Prof: Maybe,
based on the uncertainty

1:14:35.171 --> 1:14:38.601
principle, but I want for every
value of momentum a probability,

1:14:38.604 --> 1:14:39.154
right?

1:14:39.149 --> 1:14:40.749
I want the odds of getting this
p or that p or

1:14:40.750 --> 1:14:40.940
that.

1:14:40.939 --> 1:14:46.959
So what do you think we need to
get the odds for every momentum?

1:14:46.960 --> 1:14:47.610
Yes?

1:14:47.609 --> 1:14:50.549
Student:
>

1:14:50.550 --> 1:14:51.860
Prof: Pardon me?

1:14:51.859 --> 1:14:54.879
Student:  If you have a
more defined location _________

1:14:54.877 --> 1:14:57.947
calculate the probabilities of
momentum, same way we did it for

1:14:57.945 --> 1:14:58.585
location?

1:14:58.590 --> 1:14:58.960
Prof: Right.

1:14:58.960 --> 1:15:02.640
So what you will need,
it seems reasonable to think,

1:15:02.640 --> 1:15:05.200
that this guy contained all the
information on where you will

1:15:05.199 --> 1:15:08.309
find it,
maybe there's a different

1:15:08.310 --> 1:15:14.500
function of momentum,
whose square will give you the

1:15:14.496 --> 1:15:18.876
probability density that you
get--

1:15:18.880 --> 1:15:21.340
if that function looks like
this and you square that,

1:15:21.340 --> 1:15:24.910
that's the odds for getting one
momentum versus another

1:15:24.912 --> 1:15:25.642
momentum.

1:15:25.640 --> 1:15:28.390
After all, every variable in
classical mechanics you can

1:15:28.385 --> 1:15:31.275
measure in the quantum theory
and you can give the odds.

1:15:31.279 --> 1:15:33.779
And for every variable,
it looks like you need a

1:15:33.783 --> 1:15:34.373
function.

1:15:34.368 --> 1:15:37.438
What I will show you is that
you don't need that.

1:15:37.439 --> 1:15:40.339
Y(x) itself
contains information on what

1:15:40.342 --> 1:15:42.372
happens when you measure
momentum,

1:15:42.368 --> 1:15:43.868
what happens when you measure
energy,

1:15:43.868 --> 1:15:45.988
what happens when you measure
anything--

1:15:45.988 --> 1:15:48.298
and how do you extract it is
what we'll talk about.

1:15:48.300 --> 1:15:53.000