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All right.
Last time, what we had done is

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that we had looked at the first
evidence for the particle-like

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nature of radiation.
And that evidence was a

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photoelectric effect.
The evidence was that what you

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had to have was a photon or a
particle of energy,

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a quantum of energy,
a packet of energy,

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in order to get an electron
out.

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And that energy had to be at
least the energy of the work

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function of the metal.
And so for every packet you put

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in there, you got one electron
out.

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That is an example of the
particle-like nature of

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radiation.
But Einstein went on to show an

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even more convincing property of
the particle likeness of

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radiation or a photon.
And that is that what he did

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was showed that a photon has
momentum.

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It has momentum,
even though a photon does not

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have mass, although a photon
does not have rest mass,

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for those of you in the know in
this area.

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And having momentum is very
much a particle-like property,

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right?
Because you know how to write

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down momentum.
Momentum is mass times

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velocity.
You've got a mass in here.

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That is a particle-like
property.

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And, yes, I am starting out
with the lecture notes from

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number four, which I didn't
finish last time.

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That is a particle-like
property.

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But what Einstein showed was,
from the relativistic equations

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of motion, what drops out from
the relativistic equations of

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motion is the fact that a
photon, at a frequency nu,

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has a momentum h nu over c.

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And because we know the
relationship between nu and c,

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nu times lambda equals c,

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I can write the momentum of a
photon as h over lambda.

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If you have some radiation,
this is the photon momentum

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here.

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If you have some radiation,
at a wavelength lambda,

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that radiation or those photons
have this momentum p given by h

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over that wavelength.
Now, that was a prediction from

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the relativistic equations of
motion.

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And it took another eight,
ten years or so before there

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actually was an experiment that
demonstrated the momentum of a

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photon.
And that experiment was called

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the Compton experiment.
What went on in that experiment

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is that an X-ray beam came into
some material or some molecule,

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some atoms, and they could
actually see the transfer in the

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momentum from the photon to the
atom.

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Kind of like in this website
from the University of Colorado,

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here.
This is just a cartoon of what

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is happening,
but I got this photon done and

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I got this atom coming at me.
And I cannot move this fast

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enough.
I am going to get clobbered.

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You have a different computer
than I have.

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Oh, I have to push down.
Okay.

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Well, if I get aimed here,
these photons are coming at

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this atom, coming at me.
And, boy, if I do it fast

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enough, I can turn it around.
Hey, I did it.

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But now, of course,
if I go and lower the power.

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Come on.
Come on.

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Aah!
I got killed.

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[LAUGHTER] Christine,
I don't like your computer.

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Oh, wait.
I have got to get it.

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Get it.
Get it.

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Get it.
Please.

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Please.
Please.

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Aah.
All right.

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Well, you guys are going to be
a lot better at this than I am.

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You can go and play with this.
Christine is going to try now.

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Oh, look at that.
She is going to get it.

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She is going to get it.
She is going to get it.

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Yeah!
[APPLAUSE] Three cheers for

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Christine.
Oh, now it something else.

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This is going to keep going
here.

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You need more power there.
[LAUGHTER] Hey,

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not that guy.

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Fantastic.
All right.

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And it is actually just this
effect that was used by Steve

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Chu at Stanford and Bill
Phillips at NIST and Cohen and

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Tanugi who provided some of the
theory behind it.

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It is just that effect that
they used to literally trap an

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atom in space.
How do you do that?

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Well, you take an unsuspecting
atom, and you bring in a high

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power laser beam coming out in
this direction.

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And those photons transfer
momentum, and they push that

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atom this way.
But you are smarter than that,

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so you bring in a laser beam
this way.

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And so you have momentum
transfer this way and this way.

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You just trapped the atom in
this dimension.

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And then you bring in a laser
beam this way.

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Bring in a laser beam that way.
You have trapped the atom now

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in this dimension.
What is that?

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Your dog.
That is not part of my lecture.

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[LAUGHTER] And then you bring
in the laser beam this way.

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And so now you have constrained
in the three dimensions.

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And so the atom is trapped in
space by this photon pressure,

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by this momentum transfer.
And this is called laser

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trapping.
And these three gentlemen,

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whose names I gave you just a
moment ago, are laser atom

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trapping and received a Nobel
Prize in 1997 for this

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demonstration.
But the other reason why this

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laser atom trapping was really
so important is because it is

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actually the first step in
another experiment.

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It is the first step in
producing a Bose-Einstein

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condensate.
What this laser trapping does

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is literally to slow the atom
down or to cool the atom,

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because temperature and the
velocity of the atom,

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the speed of the atom are
related.

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The slower the speed,
the lower the temperature.

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And to produce a Bose-Einstein
condensate, you have to have

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bosons, which you lower in
temperature.

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And ultimately,
they condense.

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And the temperatures have to be
on the order of micro-Kelvin.

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And so this is the first step
in producing that Bose-Einstein

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condensate.
This will bring you down to

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temperatures of,
say, a Kelvin or so.

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And then there are lots of
other techniques,

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a couple of other steps that
bring you down to the

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microKelvin range.
And then, finally,

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you can get the bosons to
condense.

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And one of my colleagues in the
Physics Department,

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Wolfgang Ketterle,
also received the Nobel Prize

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for the formation of the
Bose-Einstein condensate.

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I actually think he is teaching
a recitation section in 8.01.

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Maybe some of you have him.
You do?

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No, you don't have him.
Okay.

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But you will be able to meet
him and talk to him.

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Question?
I'm sorry.

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Fantastic.
Has he told you about this yet?

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Oh, he went to a conference.
Okay.

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Well, you can imagine he is in
demand.

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But you will see him,
right?

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I hope.
Very important effect here.

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We have radiation that is
exhibiting both wave-like

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properties and particle-like
properties.

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And, just in general,
experiments where the radiation

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produces a change in the state
of the matter such as the

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photoelectron effect.
In photoelectron effect,

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the matter changes in the sense
that an electron is pulled off

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of it.
In those experiments,

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the radiation usually exhibits
the particle-like behavior.

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In experiments where there is a
change in the spatial

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distribution of the radiation,
or where the radiation

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interaction results in a change
in the spatial distribution of

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the radiation,
that is when the radiation

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exhibits its wave-like behavior.
And so it really is not

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appropriate to ask,
is light or radiation a

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particle or a wave?
The appropriate question to ask

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is, how does light behave?
Does it behave like a particle

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or does it behave like a wave
under particular experimental

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circumstances?
And having both behaviors,

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this wave-particle duality of
radiation is not a

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contradiction.
It just is the fundamental

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nature of radiation,
of light.

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You may think it is a
contradiction because in your

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everyday experience,
you either see a wave or you

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see a particle.
But that is your everyday

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experience.
And there are parts of nature

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that you cannot see every single
day.

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And those deeper parts of
nature have different rules.

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And you have to be accepting of
those different rules.

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And so it is not a
contradiction in terms.

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It just seems strange to you
just because that isn't your

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everyday experience.
It is the fundamental nature of

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radiation.
Well, not only is that the

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fundamental nature of radiation,
but the wave-particle duality

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of matter is also the
fundamental nature of matter.

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And that is what we are going
to talk about right now.

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We are going to move to matter,
particles.

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The particle-like nature of
matter is within your everyday

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experience, but it is the
wave-like nature of matter that

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is not within your everyday
experience.

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And so let's take a look at
that.

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Suppose we did this experiment.
That is, we had a nickel

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crystal.
And these two atoms here are

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just two of the atoms on the
surface of a nickel crystal.

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And we know the spacings
between these two atoms in the

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crystal because we know the
crystal structure of the nickel.

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That spacing is about 2x10^-10
meters.

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Naively, if you brought in a
beam of electrons,

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particles, and we know they
have mass.

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J.J.
Thompson taught us they were

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particles, they had mass.
But, naively,

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if you brought them in,
you might expect these

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electrons to scatter
isotropically.

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That is that they would scatter
equally in all directions so

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that when they ultimately hit
this screen here,

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this curved phosphor screen,
and I changed the geometry here

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to a curved screen just so that
it will be a little bit easier

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to analyze the geometry of this
problem, which we are going to

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do in a moment,
you might expect this screen to

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be lit up uniformly at all
angles.

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Well, this is exactly the
experiment that Davidson and

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Germer did in 1927,
along with this gentleman,

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G.
Thompson, George Thompson,

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son of J.J.
Thompson.

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And J.J.
Thompson actually did an

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experiment a little different
than Davidson and Germer.

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I am going to show you the
Davidson and Germer experiment

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here.
But here is the same diagram

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that I had before,
except that I made the nickel

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atoms a little bit smaller just
so that this diagram would be a

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little bit easier to understand.
I cleaned up the diagram,

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but kept the spacing between
the two nickel atoms the same.

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And so Davidson,
Germer and Thompson came in,

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scattered these electrons and
looked how they scattered back.

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And, lo and behold,
what they saw is that these

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electrons seemed to scatter back
at a preferential angle.

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The angular distribution was
not isotropic.

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Instead, it looked like the
electrons scattered back at a

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pretty well-defined angle here,
50.7 degrees.

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And not only did they scatter
back at that angle,

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they also scattered right back
at themselves.

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Backscattered this way,
so this scattering angle is

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zero degrees.
And under some particular

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conditions, the electrons also
scattered at a larger angle,

00:15:45.000 --> 00:15:48.000
here.
But the bottom line is that the

00:15:48.000 --> 00:15:51.000
scattering pattern was not
isotropic.

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There was a bright spot,
lots of electrons scattered at

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this angle, a dark spot,
no electrons scattered at this

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angle.
A bright spot,

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dark spot, bright spot.
This looks like interference

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phenomena, just like the two
slit experiment.

00:16:12.000 --> 00:16:15.000
Bright spot,
dark spot, bright spot,

00:16:15.000 --> 00:16:18.000
constructive,
destructive,

00:16:18.000 --> 00:16:22.000
constructive interference,
back and forth.

00:16:22.000 --> 00:16:27.000
That was their observation.
How do we understand that?

00:16:27.000 --> 00:16:33.000
Well, it is looking like these
electrons are behaving like

00:16:33.000 --> 00:16:38.000
waves.
Suppose these electrons are

00:16:38.000 --> 00:16:43.000
coming in, so we have this
constant stream of electrons

00:16:43.000 --> 00:16:48.000
impinging on our nickel crystal.
Well, what is happening here is

00:16:48.000 --> 00:16:53.000
that when these electrons are
reflecting back from the

00:16:53.000 --> 00:16:57.000
individual nickel atoms,
these individual nickel atoms

00:16:57.000 --> 00:17:03.000
are kind of functioning like
those little slits we saw in the

00:17:03.000 --> 00:17:08.000
two slit experiment.
That is, they are scattering

00:17:08.000 --> 00:17:11.000
back as a wave.
These electrons seem to be

00:17:11.000 --> 00:17:15.000
scattering as a wave,
so isotropically in all

00:17:15.000 --> 00:17:19.000
directions.
This semicircle around each one

00:17:19.000 --> 00:17:22.000
of the atoms,
and I only show you two atoms

00:17:22.000 --> 00:17:27.000
here, each semicircle is the
maximum of the wave front.

00:17:27.000 --> 00:17:31.000
It is the crest of the wave
front.

00:17:31.000 --> 00:17:35.000
And then, as time goes by,
of course, these waves

00:17:35.000 --> 00:17:39.000
propagate out.
And then another wave front,

00:17:39.000 --> 00:17:45.000
another wave maximum appears
and a distance between these two

00:17:45.000 --> 00:17:49.000
maxima is, of course,
the wavelength.

00:17:49.000 --> 00:17:53.000
And as time goes on,
they scatter further.

00:17:53.000 --> 00:17:58.000
And as time goes on,
they still scatter.

00:17:58.000 --> 00:18:02.000
And they keep the propagating
out until they reach the screen.

00:18:02.000 --> 00:18:06.000
And, lo and behold,
on the screen you see a bright

00:18:06.000 --> 00:18:08.000
spot, dark spot,
bright spot,

00:18:08.000 --> 00:18:10.000
dark spot.
Interference pattern.

00:18:10.000 --> 00:18:14.000
Let's analyze this.
Here is the diagram again.

00:18:14.000 --> 00:18:17.000
I just moved it over and
cleaned it up again.

00:18:17.000 --> 00:18:21.000
I want you to look at this spot
right in there.

00:18:21.000 --> 00:18:25.000
That is where we have the
maximum of a wave scattered from

00:18:25.000 --> 00:18:29.000
atom one at the same point in
space as the maximum of waves

00:18:29.000 --> 00:18:34.000
scattered from atom two.
Constructive interference.

00:18:34.000 --> 00:18:38.000
Here is another point of
constructive interference.

00:18:38.000 --> 00:18:42.000
Here is another point of
constructive interference.

00:18:42.000 --> 00:18:45.000
Everywhere along this line,
we have constructive

00:18:45.000 --> 00:18:48.000
interference,
which results in a large

00:18:48.000 --> 00:18:51.000
intensity right at this
scattering angle here,

00:18:51.000 --> 00:18:54.000
a bright spot.
And we already know the

00:18:54.000 --> 00:18:58.000
condition for constructive
interference.

00:18:58.000 --> 00:19:01.000
That is, in order to get this
constructive interference,

00:19:01.000 --> 00:19:05.000
the difference in the distance
traveled by the two waves that

00:19:05.000 --> 00:19:09.000
are interfering has to be an
integral multiple of the

00:19:09.000 --> 00:19:12.000
wavelength lambda.
Now, I use the term d instead

00:19:12.000 --> 00:19:16.000
of r, but it is the same thing
for the condition for

00:19:16.000 --> 00:19:18.000
constructive interference,
here.

00:19:18.000 --> 00:19:22.000
And if you went and analyzed
what the difference in the

00:19:22.000 --> 00:19:25.000
distance was for this
constructive interference along

00:19:25.000 --> 00:19:30.000
this line, you would find it was
n equals 1.

00:19:30.000 --> 00:19:33.000
The difference in the distance
traveled is one lambda.

00:19:33.000 --> 00:19:36.000
And, if you looked at the
points of constructive

00:19:36.000 --> 00:19:40.000
interference along this line
that led to this bright spot,

00:19:40.000 --> 00:19:44.000
the difference in the distance
traveled will be two lambda.

00:19:44.000 --> 00:19:47.000
This is our second-order
interference feature,

00:19:47.000 --> 00:19:49.000
our second-order diffraction
spot.

00:19:49.000 --> 00:19:52.000
And, if you look at it along
the center here,

00:19:52.000 --> 00:19:55.000
normal to the crystal,
that would be the zero-order

00:19:55.000 --> 00:19:57.000
spot.
d2 minus d1 is equal to zero

00:19:57.000 --> 00:20:01.000
lambda.

00:20:01.000 --> 00:20:04.000
All right.
That is what looks like is

00:20:04.000 --> 00:20:08.000
happening.
Now, what we are going to do is

00:20:08.000 --> 00:20:13.000
we are going to actually analyze
this geometry a bit more.

00:20:13.000 --> 00:20:17.000
We didn't do so in the two slit
experiment.

00:20:17.000 --> 00:20:19.000
We could have.
We didn't.

00:20:19.000 --> 00:20:24.000
We are going to do it here.
And what we are going to be

00:20:24.000 --> 00:20:29.000
after is if these electrons are
acting like waves,

00:20:29.000 --> 00:20:35.000
then they have a wavelength.
And we want to know what the

00:20:35.000 --> 00:20:38.000
wavelength is.
Davidson and Germer wanted to

00:20:38.000 --> 00:20:42.000
know what the wavelength was.
And we are going to use this

00:20:42.000 --> 00:20:44.000
scattering angle here,
theta.

00:20:44.000 --> 00:20:47.000
This angle from the normal to
where the electrons are

00:20:47.000 --> 00:20:51.000
scattering, that angle theta,
we are going to use that

00:20:51.000 --> 00:20:55.000
information, theta equals 50.7
degrees, to back out the

00:20:55.000 --> 00:20:58.000
wavelength.
And we know what the condition

00:20:58.000 --> 00:21:02.000
is for constructive
interference.

00:21:02.000 --> 00:21:07.000
We just talked about it.
Here it is, d2 minus d1.

00:21:07.000 --> 00:21:10.000
That is the wavelength we are

00:21:10.000 --> 00:21:14.000
after in this analysis.
Now, what is d2 here?

00:21:14.000 --> 00:21:19.000
Well, the length of this line,
d2, is the distance that the

00:21:19.000 --> 00:21:26.000
wave that scatters from electron
two travels from electron two to

00:21:26.000 --> 00:21:29.000
the screen.
d1 is the distance that the

00:21:29.000 --> 00:21:36.000
wave that scatters from atom one
travels to the screen.

00:21:36.000 --> 00:21:39.000
That is what d2 and d1 are,
here.

00:21:39.000 --> 00:21:44.000
Now, I am going to draw a
perpendicular from atom one to

00:21:44.000 --> 00:21:47.000
this line d2.
There is my right angle.

00:21:47.000 --> 00:21:51.000
Now, you can see,
then, that this leg of the

00:21:51.000 --> 00:21:57.000
triangle is d2 minus d1,
the difference in the

00:21:57.000 --> 00:22:02.000
distance traveled.
That is this quantity here.

00:22:02.000 --> 00:22:06.000
That is going to be important.
Now, you have got to convince

00:22:06.000 --> 00:22:11.000
yourself that this angle right
here in the triangle is the same

00:22:11.000 --> 00:22:15.000
as this scattering angle.
You can convince yourself of

00:22:15.000 --> 00:22:18.000
that pretty easily.
Now we have a well-defined

00:22:18.000 --> 00:22:21.000
triangle.
We know one length of it,

00:22:21.000 --> 00:22:25.000
we have measured the angle
theta, and d2 minus d1

00:22:25.000 --> 00:22:30.000
is something that we would
like to know.

00:22:30.000 --> 00:22:33.000
Let's do a little geometry.
The sine of theta,

00:22:33.000 --> 00:22:37.000
the sign of this angle is equal
to the opposite length,

00:22:37.000 --> 00:22:40.000
which is d2 minus d1,
divided by A,

00:22:40.000 --> 00:22:44.000
this distance between the two
atoms in the nickel crystal.

00:22:44.000 --> 00:22:48.000
We have two equations.

00:22:48.000 --> 00:22:52.000
This is the equation that in
theory should obtain for

00:22:52.000 --> 00:22:57.000
constructive interference.
This is the equation that we

00:22:57.000 --> 00:23:01.000
set up given the particular
physical geometry of our

00:23:01.000 --> 00:23:05.000
problem.
These two (d2 minus d1)'s

00:23:05.000 --> 00:23:11.000
better be equal to each other.
We have n lambda is A sine

00:23:11.000 --> 00:23:14.000
theta.

00:23:14.000 --> 00:23:18.000
Let's solve for lambda.
We can do that.

00:23:18.000 --> 00:23:23.000
That is A sine theta over n.
We already know what theta is,

00:23:23.000 --> 00:23:28.000
we know what A is,
so what is n?

00:23:28.000 --> 00:23:33.000
Well, n is going to be one
because this is the bright spot

00:23:33.000 --> 00:23:36.000
that is closest to the
zero-order spot,

00:23:36.000 --> 00:23:42.000
which is always present at the
normal angle there if you are

00:23:42.000 --> 00:23:47.000
coming in at normal orientation.
So n is equal 1 so I can plug

00:23:47.000 --> 00:23:50.000
things in.
And, when I do that,

00:23:50.000 --> 00:23:56.000
I find that the wavelength that
I predict is 1.66x10^-10 meters.

00:23:56.000 --> 00:24:01.000
We've got this wavelength.
Now, before I go on,

00:24:01.000 --> 00:24:07.000
I just want to point out that
this geometry of the problem

00:24:07.000 --> 00:24:13.000
that I set up here is identical
to the geometry in a technique

00:24:13.000 --> 00:24:18.000
known as X-ray diffraction.
X-ray diffraction does not use

00:24:18.000 --> 00:24:21.000
electrons coming in,
but uses X-rays,

00:24:21.000 --> 00:24:25.000
photons.
And it is a technique that is

00:24:25.000 --> 00:24:30.000
going to be important to you if
you do any kind of science

00:24:30.000 --> 00:24:36.000
involving materials or
biological systems.

00:24:36.000 --> 00:24:40.000
And it is important because
X-ray diffraction allows you to

00:24:40.000 --> 00:24:44.000
determine the structure,
in particular of proteins,

00:24:44.000 --> 00:24:48.000
crystal proteins.
You crystallize the protein,

00:24:48.000 --> 00:24:52.000
and you use this X-ray
diffraction to get out the

00:24:52.000 --> 00:24:55.000
structure.
And the reason why you want the

00:24:55.000 --> 00:25:00.000
structure of the proteins is
because the structure gives you

00:25:00.000 --> 00:25:06.000
a hint as to what the function
of the proteins are.

00:25:06.000 --> 00:25:11.000
And so in the use of X-ray
diffraction, we don't go and

00:25:11.000 --> 00:25:16.000
calculate what lambda is.
We already know what lambda is

00:25:16.000 --> 00:25:21.000
in X-ray diffraction.
We know the wavelength of the

00:25:21.000 --> 00:25:25.000
incident photons,
the X-rays.

00:25:25.000 --> 00:25:30.000
What we don't know in the
technique of X-ray diffraction

00:25:30.000 --> 00:25:36.000
is the distance between the
atoms in an unknown structure.

00:25:36.000 --> 00:25:41.000
And so in X-ray diffraction,
we know the wavelength,

00:25:41.000 --> 00:25:46.000
and we can figure out what
order it is and we measure the

00:25:46.000 --> 00:25:51.000
scattering angle.
And we use that to determine

00:25:51.000 --> 00:25:55.000
the distance between the atoms.
And, in that way,

00:25:55.000 --> 00:26:01.000
we back out the structure of
the sample.

00:26:07.000 --> 00:26:10.000
Yes.
We are just going to get there.

00:26:10.000 --> 00:26:13.000
We are going to do that.
All right.

00:26:13.000 --> 00:26:17.000
Same geometry here.
Now, this experiment of

00:26:17.000 --> 00:26:23.000
Davidson and Germer was really
an important one because just

00:26:23.000 --> 00:26:28.000
three years before this,
there was a prediction for what

00:26:28.000 --> 00:26:33.000
the wavelength of particles
ought to be.

00:26:33.000 --> 00:26:38.000
And that prediction was made by
this gentleman,

00:26:38.000 --> 00:26:42.000
Louis de Broglie.
In his Ph.D.

00:26:42.000 --> 00:26:45.000
thesis, no less,
what Mr.

00:26:45.000 --> 00:26:51.000
de Broglie did was that he
looked at the relativistic

00:26:51.000 --> 00:26:58.000
equations of motion that
Einstein wrote down and used to

00:26:58.000 --> 00:27:04.000
propose that a photon or
radiation with a wavelength

00:27:04.000 --> 00:27:11.000
lambda had momentum p.
Well, he took those same

00:27:11.000 --> 00:27:16.000
equations and said,
well, these relativistic

00:27:16.000 --> 00:27:22.000
equations of motion apply to
matter just as well as they

00:27:22.000 --> 00:27:27.000
apply to radiation.
Therefore, if you have

00:27:27.000 --> 00:27:35.000
radiation with a wavelength
lambda, you then have this

00:27:35.000 --> 00:27:42.000
momentum p for the radiation.
This is what Einstein said.

00:27:42.000 --> 00:27:50.000
But he turned it around and
said, if you have matter with a

00:27:50.000 --> 00:27:56.000
momentum p, well,
that matter ought to have a

00:27:56.000 --> 00:28:03.000
wavelength lambda.
He turned around Einstein's

00:28:03.000 --> 00:28:09.000
equations of motion and proposed
that the wavelength of a

00:28:09.000 --> 00:28:15.000
particle be given by h over p
where, of course,

00:28:15.000 --> 00:28:19.000
p here is the mass times the
velocity.

00:28:19.000 --> 00:28:24.000
Fantastic.

00:28:24.000 --> 00:28:27.000
What a great Ph.D.
thesis.

00:28:27.000 --> 00:28:32.000
I'm impressed.
Now, let's see how well,

00:28:32.000 --> 00:28:38.000
as you can imagine,
that predicts the wavelength

00:28:38.000 --> 00:28:43.000
that Davidson and Germer
actually measured.

00:28:43.000 --> 00:28:50.000
We know we have 54 electrons
coming into this nickel crystal.

00:28:50.000 --> 00:28:57.000
That is their kinetic energy,
one-half m v squared.

00:28:57.000 --> 00:29:02.000
Kinetic energy can also be

00:29:02.000 --> 00:29:05.000
written in terms of the
momentum.

00:29:05.000 --> 00:29:08.000
The momentum is p square over
2m.

00:29:08.000 --> 00:29:13.000
You can convince
yourself of this.

00:29:13.000 --> 00:29:18.000
This is a good thing to know
for doing these problems,

00:29:18.000 --> 00:29:23.000
that the kinetic energy is p
squared over 2 times the mass of

00:29:23.000 --> 00:29:27.000
the electron.
And so if you solve that,

00:29:27.000 --> 00:29:32.000
what you get for the momentum
of the electrons is 4.0x10^-24

00:29:32.000 --> 00:29:40.000
kilograms meters per second.
And now I can take this

00:29:40.000 --> 00:29:47.000
momentum and plug it into the
expression for de Broglie's

00:29:47.000 --> 00:29:55.000
wavelength, 6.6x10^-34 joule
seconds, over the momentum,

00:29:55.000 --> 00:29:58.000
4x10^-24.
What do I get?

00:29:58.000 --> 00:30:05.000
1.7x10^-10 meters.
Absolutely the same as the

00:30:05.000 --> 00:30:10.000
experiment.
De Broglie made a prediction.

00:30:10.000 --> 00:30:17.000
A few years after that,
experiments demonstrated that

00:30:17.000 --> 00:30:24.000
de Broglie was absolutely
correct in his prediction.

00:30:24.000 --> 00:30:29.000
What do we have here?
We have matter,

00:30:29.000 --> 00:30:33.000
particles, exhibiting wave-like
behavior.

00:30:33.000 --> 00:30:38.000
And, those particles can be
measured to have a wavelength

00:30:38.000 --> 00:30:43.000
that actually agrees with a
prediction, some theory,

00:30:43.000 --> 00:30:48.000
the de Broglie wavelength.
And we also have another

00:30:48.000 --> 00:30:51.000
phenomena here,
which I really enjoy,

00:30:51.000 --> 00:30:56.000
and that is Davidson and Germer
and George Thompson.

00:30:56.000 --> 00:31:03.000
They demonstrated that
electrons behave like waves.

00:31:03.000 --> 00:31:06.000
And what did J.J.
Thompson do,

00:31:06.000 --> 00:31:11.000
father of George Thompson,
well, he demonstrated that an

00:31:11.000 --> 00:31:17.000
electron was a particle.
Here, we have both the father

00:31:17.000 --> 00:31:23.000
and the son talking about
seemingly opposite behavior,

00:31:23.000 --> 00:31:28.000
but they are both right.
How often does that happen?

00:31:28.000 --> 00:31:33.000
That, I think,
is really amazing.

00:31:33.000 --> 00:31:40.000
But if matter is wave-like and
if electrons can be represented

00:31:40.000 --> 00:31:45.000
by a wavelength,
then what about your

00:31:45.000 --> 00:31:53.000
wavelengths and my wavelengths?
We should have a wavelength.

00:31:53.000 --> 00:31:56.000
And we do.
And just briefly,

00:31:56.000 --> 00:32:04.000
here, let's talk about what the
wavelength is of a baseball

00:32:04.000 --> 00:32:10.000
pitched by Curt Shilling at 90
mph.

00:32:10.000 --> 00:32:15.000
What is that wavelength?
Well, a baseball is five

00:32:15.000 --> 00:32:17.000
ounces.
90 mph.

00:32:17.000 --> 00:32:23.000
You can calculate the momentum.
It is in your notes there.

00:32:23.000 --> 00:32:29.000
We will calculate,
here, the wavelength.

00:32:29.000 --> 00:32:35.000
And what you are going to find
is that it is 1.2x10^-34 meters.

00:32:35.000 --> 00:32:39.000
That is pretty small.
What is the diameter of a

00:32:39.000 --> 00:32:42.000
nucleus?
10^-14, right.

00:32:42.000 --> 00:32:48.000
That is a good number to know.
Here, we have a wavelength that

00:32:48.000 --> 00:32:52.000
is 10^-34 meters.
Is that wavelength of a

00:32:52.000 --> 00:32:58.000
macroscopic size object going to
have any consequence in our

00:32:58.000 --> 00:33:01.000
world?
No, absolutely not.

00:33:01.000 --> 00:33:04.000
Why?
Because in order to see any

00:33:04.000 --> 00:33:10.000
effects from this small
wavelength, we are going to have

00:33:10.000 --> 00:33:16.000
to have slits or atoms that are
going to be on the order of this

00:33:16.000 --> 00:33:20.000
close together.
But there is no way that we are

00:33:20.000 --> 00:33:26.000
going to have two nickel atoms
that are this close together or

00:33:26.000 --> 00:33:33.000
two slits in a two slit
experiment this close together.

00:33:33.000 --> 00:33:40.000
And so for macroscopic objects,
the wavelike properties have no

00:33:40.000 --> 00:33:47.000
consequence in this world.
And that is simply because the

00:33:47.000 --> 00:33:53.000
mass is too large.
It makes the wavelength too

00:33:53.000 --> 00:33:59.000
small to have any effects in our
everyday lives.

00:33:59.000 --> 00:34:04.000
And it is actually --
Yes?

00:34:04.000 --> 00:34:07.000
Okay.

00:34:15.000 --> 00:34:19.000
Well, they are actually coming
in as waves.

00:34:19.000 --> 00:34:25.000
They are behaving as waves.
Remember my beach picture?

00:34:25.000 --> 00:34:30.000
I drew them coming in like a
circle.

00:34:30.000 --> 00:34:36.000
But remember my picture of this
barrier here on the beach,

00:34:36.000 --> 00:34:42.000
and I am laying here on the
sand, and then the waves are

00:34:42.000 --> 00:34:44.000
coming in?
Here is blue.

00:34:44.000 --> 00:34:48.000
These electrons,
as they are coming in,

00:34:48.000 --> 00:34:54.000
really need to be thought of as
these kind of plane waves.

00:34:54.000 --> 00:35:00.000
I drew them as kind of just
particles.

00:35:00.000 --> 00:35:05.000
But you have to really think of
them as plane waves and that

00:35:05.000 --> 00:35:11.000
they are reflecting off of these
two atoms in the way that I just

00:35:11.000 --> 00:35:15.000
explained.
Another question?

00:35:25.000 --> 00:35:29.000
The thing is that you cannot
get that velocity slow enough to

00:35:29.000 --> 00:35:33.000
make the wavelength large enough
to be of consequence.

00:35:33.000 --> 00:35:36.000
If you could then you would,
right?

00:35:36.000 --> 00:35:38.000
You would see the wave-like
behavior.

00:35:38.000 --> 00:35:41.000
But you cannot,
practically speaking,

00:35:41.000 --> 00:35:44.000
get it to that extent.

00:36:18.000 --> 00:36:20.000
No.
In this particular case,

00:36:20.000 --> 00:36:24.000
anything that is so massive,
any smaller effects,

00:36:24.000 --> 00:36:28.000
like what you are talking
about, exactly the point of

00:36:28.000 --> 00:36:33.000
observation is not going to have
an effect on anything that is so

00:36:33.000 --> 00:36:35.000
massive.
Pardon?

00:36:35.000 --> 00:36:41.000
Yes, your point of observation
will have an effect on your

00:36:41.000 --> 00:36:46.000
interpretation of the
experiment, if you are talking

00:36:46.000 --> 00:36:51.000
about something that has a much
larger wavelength.

00:36:51.000 --> 00:36:55.000
Absolutely.
In high energy physics

00:36:55.000 --> 00:37:00.000
experiments, for example.
Good question.

00:37:05.000 --> 00:37:14.000
It is just this observation,
here, of the wave-like behavior

00:37:14.000 --> 00:37:19.000
of electrons,
of particles,

00:37:19.000 --> 00:37:29.000
that led to the interpretation,
then, or led to the realization

00:37:29.000 --> 00:37:39.000
that maybe, you have to treat
electrons as waves.

00:37:39.000 --> 00:37:48.000
Or maybe you have to treat the
behavior of electrons as

00:37:48.000 --> 00:37:56.000
wave-like behavior.
And that is what this gentleman

00:37:56.000 --> 00:38:01.000
Schrˆdinger did.
He said, well,

00:38:01.000 --> 00:38:05.000
you know what?
This gave him an idea.

00:38:05.000 --> 00:38:10.000
Maybe what is wrong is that an
electron in an atom,

00:38:10.000 --> 00:38:14.000
I cannot treat that electron as
a particle.

00:38:14.000 --> 00:38:19.000
Instead, what I have to do is
treat is as a wave.

00:38:19.000 --> 00:38:23.000
I have to treat its wave-like
properties.

00:38:23.000 --> 00:38:28.000
And it was that impetus that
led him to write down a wave

00:38:28.000 --> 00:38:34.000
equation of motion.
An equation of motion for

00:38:34.000 --> 00:38:36.000
waves.
He realized,

00:38:36.000 --> 00:38:40.000
or he was guessing at the
moment, well,

00:38:40.000 --> 00:38:46.000
maybe in the case when a
microscopic particle has a

00:38:46.000 --> 00:38:53.000
wavelength that is on the order
of the size of its environment,

00:38:53.000 --> 00:38:57.000
in that case,
maybe the wavelength has an

00:38:57.000 --> 00:39:02.000
effect, makes a difference.
For example,

00:39:02.000 --> 00:39:07.000
in the case of the electrons,
we had a wavelength calculated,

00:39:07.000 --> 00:39:12.000
there, of 1.7x10^-10 meters.
That is a wavelength that is on

00:39:12.000 --> 00:39:16.000
the order of the size of the
environment of the electron,

00:39:16.000 --> 00:39:20.000
which is on the order of the
size of an atom.

00:39:20.000 --> 00:39:24.000
Maybe in that case I have to
pay attention to the wavelength.

00:39:24.000 --> 00:39:29.000
The reason you and I don't have
to pay any attention to our

00:39:29.000 --> 00:39:33.000
wavelength is because our
wavelength is 10^-30 meters or

00:39:33.000 --> 00:39:36.000
so.
And that is much,

00:39:36.000 --> 00:39:40.000
much larger than the size of
the environment.

00:39:40.000 --> 00:39:44.000
In this case,
we don't have to pay any

00:39:44.000 --> 00:39:49.000
attention to our wavelength.
But, for an electron in an

00:39:49.000 --> 00:39:51.000
atom, we have a problem,
here.

00:39:51.000 --> 00:39:55.000
What did Schrˆdinger do?
Schrˆdinger said,

00:39:55.000 --> 00:40:00.000
I have to write down a wave
equation.

00:40:00.000 --> 00:40:05.000
An equation of motion for
matter waves.

00:40:05.000 --> 00:40:10.000
And what is that equation of
motion?

00:40:10.000 --> 00:40:19.000
Well, that equation of motion
is H hat operating on Psi,

00:40:19.000 --> 00:40:25.000
and it gives us back an energy,
E, times a Psi.

00:40:25.000 --> 00:40:30.000
What is this?
Well, Psi, here,

00:40:30.000 --> 00:40:34.000
is a wave.
I am somehow going to let my

00:40:34.000 --> 00:40:37.000
electron in an atom be
represented by Psi.

00:40:37.000 --> 00:40:40.000
This is going to be a wave
form.

00:40:40.000 --> 00:40:43.000
This is going to be a wave
function.

00:40:43.000 --> 00:40:49.000
I am going to let my electron
be represented by the wave

00:40:49.000 --> 00:40:52.000
function.
Exactly how it is going to be

00:40:52.000 --> 00:40:58.000
represented by the wave function
is something that I am not going

00:40:58.000 --> 00:41:06.000
to tell you quite yet.
But it is going to represent

00:41:06.000 --> 00:41:10.000
the electron.
This energy,

00:41:10.000 --> 00:41:18.000
here, that energy is going to
turn out to be the binding

00:41:18.000 --> 00:41:23.000
energy of the electron in the
atom.

00:41:23.000 --> 00:41:30.000
This thing, here,
is called the Hamiltonian

00:41:30.000 --> 00:41:35.000
Operator.
That Hamiltonian Operator is

00:41:35.000 --> 00:41:41.000
specific to a particular
problem, and we will look at the

00:41:41.000 --> 00:41:45.000
Hamiltonian Operator for a
hydrogen atom.

00:41:45.000 --> 00:41:50.000
But this operator is operating
on Psi, and it gives you back a

00:41:50.000 --> 00:41:55.000
Psi, the same function,
multiplied by a constant.

00:41:55.000 --> 00:42:00.000
That constant is the binding
energy.

00:42:00.000 --> 00:42:04.000
Now, you think,
well, let me just cancel this

00:42:04.000 --> 00:42:08.000
and this.
But you cannot do that because

00:42:08.000 --> 00:42:13.000
this is an operator.
This has some derivatives in

00:42:13.000 --> 00:42:17.000
it.
Its operating on Psi gives you

00:42:17.000 --> 00:42:21.000
back the same function times the
constant.

00:42:21.000 --> 00:42:27.000
Now, how did Schrˆdinger
actually derive this equation,

00:42:27.000 --> 00:42:34.000
so to speak?
Well, what he did was to just

00:42:34.000 --> 00:42:40.000
guess at a wave function.
We are going to use a

00:42:40.000 --> 00:42:47.000
one-dimensional wave function.
He is going to say,

00:42:47.000 --> 00:42:55.000
let me represent my electron by
Psi of x equal to 2 a times

00:42:55.000 --> 00:43:02.000
cosine 2 pi x over lambda.

00:43:02.000 --> 00:43:09.000
That is going to be my wave

00:43:09.000 --> 00:43:11.000
function.
Why not?

00:43:11.000 --> 00:43:19.000
And then he said,
well, what I really need here

00:43:19.000 --> 00:43:27.000
is an equation of motion.
I need to know how Psi changes

00:43:27.000 --> 00:43:32.000
with x.
If you wanted an equation of

00:43:32.000 --> 00:43:37.000
motion, if you wanted to know
how Psi changes with x,

00:43:37.000 --> 00:43:41.000
what would you do to Psi?
Take the derivative.

00:43:41.000 --> 00:43:46.000
Let's take a derivative of Psi
of x, with respect to x.

00:43:46.000 --> 00:43:51.000
That is going to be
equal to minus 2 a

00:43:51.000 --> 00:43:56.000
times 2 pi over lambda times
sine of 2pi x over lambda.

00:44:03.000 --> 00:44:07.000
That is an equation of motion.
Now, this is actually kind of

00:44:07.000 --> 00:44:10.000
an equation of position.
But it is telling us how Psi

00:44:10.000 --> 00:44:12.000
changes with x.

00:44:20.000 --> 00:44:28.000
But now, since that gave us
some information about how Psi

00:44:28.000 --> 00:44:34.000
changes with x,
how would we get the rate of

00:44:34.000 --> 00:44:39.000
change of Psi with x?
Second derivative.

00:44:39.000 --> 00:44:43.000
Let's take the second
derivative of that.

00:44:43.000 --> 00:44:47.000
Second derivative of Psi of x
with respect to x,

00:44:47.000 --> 00:44:52.000
that is minus 2a times 4 pi
squared over lambda squared

00:44:52.000 --> 00:44:57.000
times cosine 2pi x
over lambda.

00:45:02.000 --> 00:45:07.000
That's pretty good,
but now, what do you see in

00:45:07.000 --> 00:45:09.000
this equation?
Psi.

00:45:09.000 --> 00:45:14.000
You see what I started with.
It is recursive,

00:45:14.000 --> 00:45:19.000
right.
You see the Psi of x here.

00:45:19.000 --> 00:45:23.000
Let me write that equation in

00:45:23.000 --> 00:45:31.000
terms of the original function.
Second derivative of Psi of x

00:45:31.000 --> 00:45:36.000
with respect to x,
minus 2pi over lambda squared

00:45:36.000 --> 00:45:41.000
times Psi of x.

00:45:41.000 --> 00:45:45.000
lambda)^2 Psi(x)** That is
pretty good.

00:45:45.000 --> 00:45:49.000
You know where I am trying to
go?

00:45:49.000 --> 00:45:53.000
I am trying to derive,
so to speak,

00:45:53.000 --> 00:46:00.000
Schrˆdinger's equation.
See, it is not very hard.

00:46:00.000 --> 00:46:03.000
Yes.
Now, this equation right here,

00:46:03.000 --> 00:46:08.000
this equation is just a
classical wave equation.

00:46:08.000 --> 00:46:14.000
The only thing I have done so
far is take derivatives.

00:46:14.000 --> 00:46:19.000
I have done nothing else.
I just took derivatives.

00:46:19.000 --> 00:46:23.000
It could represent any kind of
wave.

00:46:23.000 --> 00:46:29.000
There is nothing quantum
mechanical about it.

00:46:29.000 --> 00:46:36.000
But here comes the big leap
that Schrˆdinger made.

00:46:36.000 --> 00:46:43.000
He substituted in here for
lambda the momentum of the

00:46:43.000 --> 00:46:47.000
particle.
In other words,

00:46:47.000 --> 00:46:55.000
if this is a wave equation and
that wave has some wavelength,

00:46:55.000 --> 00:47:00.000
here, lambda.
He said, well,

00:47:00.000 --> 00:47:04.000
if this is a wave equation for
a matter wave,

00:47:04.000 --> 00:47:10.000
well, then I better get the
momentum of that particle in

00:47:10.000 --> 00:47:14.000
this wave.
And he knew how to do that

00:47:14.000 --> 00:47:18.000
because de Broglie told him how
to do that.

00:47:18.000 --> 00:47:22.000
De Broglie said,
lambda is equal to h over p.

00:47:22.000 --> 00:47:26.000
That is pretty good.

00:47:26.000 --> 00:47:32.000
What can I do here?
Well, what I can do is I can

00:47:32.000 --> 00:47:38.000
rearrange this and write this in
terms of the momentum of the

00:47:38.000 --> 00:47:42.000
particle.
Second derivative of Psi of x

00:47:42.000 --> 00:47:48.000
with respect to x is going to be
minus p squared over h bar

00:47:48.000 --> 00:47:53.000
squared times Psi of x.

00:47:53.000 --> 00:47:58.000
Now, let me explain
what h bar here is

00:47:58.000 --> 00:48:02.000
h bar is a shorthand way of

00:48:02.000 --> 00:48:07.000
writing h over 2pi. If you

00:48:07.000 --> 00:48:11.000
don't know it already,
you should know it.

00:48:11.000 --> 00:48:15.000
You are going to need it.
That is h over 2pi,

00:48:15.000 --> 00:48:19.000
so this is h bar squared.
Now, what do I have?

00:48:19.000 --> 00:48:24.000
Now I have a matter wave.
I have an equation of motion.

00:48:24.000 --> 00:48:32.000
I have something that tells me
how Psi moves with respect to X.

00:48:32.000 --> 00:48:35.000
The rate of change of Psi with
X.

00:48:35.000 --> 00:48:40.000
And I had the momentum of the
particle buried in here.

00:48:40.000 --> 00:48:44.000
From this form,
I am going to get to that.

00:48:44.115 --> 00:48:47.000
And I guess I am going to get
to that next Wednesday.