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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,

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here.
But the bottom line is that the

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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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00:16:02 --> 00:16:07
dark spot, bright spot.
This looks like interference

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phenomena, just like the two
slit experiment.

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Bright spot,
dark spot, bright spot,

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constructive,
destructive,

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constructive interference,
back and forth.

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That was their observation.
How do we understand that?

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Well, it is looking like these
electrons are behaving like

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waves.
Suppose these electrons are

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coming in, so we have this
constant stream of electrons

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impinging on our nickel crystal.
Well, what is happening here is

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that when these electrons are
reflecting back from the

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individual nickel atoms,
these individual nickel atoms

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are kind of functioning like
those little slits we saw in the

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two slit experiment.
That is, they are scattering

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00:17:08 --> 00:17:11
back as a wave.
These electrons seem to be

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scattering as a wave,
so isotropically in all

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directions.
This semicircle around each one

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of the atoms,
and I only show you two atoms

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00:17:22 --> 00:17:27
here, each semicircle is the
maximum of the wave front.

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00:17:27 --> 00:17:31
It is the crest of the wave
front.

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00:17:31 --> 00:17:35
And then, as time goes by,
of course, these waves

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propagate out.
And then another wave front,

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00:17:39 --> 00:17:45
another wave maximum appears
and a distance between these two

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00:17:45 --> 00:17:49
maxima is, of course,
the wavelength.

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00:17:49 --> 00:17:53
And as time goes on,
they scatter further.

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00:17:53 --> 00:17:58
And as time goes on,
they still scatter.

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00:17:58 --> 00:18:02
And they keep the propagating
out until they reach the screen.

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00:18:02 --> 00:18:06
And, lo and behold,
on the screen you see a bright

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00:18:06 --> 00:18:08
spot, dark spot,
bright spot,

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00:18:08 --> 00:18:10
dark spot.
Interference pattern.

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00:18:10 --> 00:18:14
Let's analyze this.
Here is the diagram again.

258
00:18:14 --> 00:18:17
I just moved it over and
cleaned it up again.

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00:18:17 --> 00:18:21
I want you to look at this spot
right in there.

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00:18:21 --> 00:18:25
That is where we have the
maximum of a wave scattered from

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00:18:25 --> 00:18:29
atom one at the same point in
space as the maximum of waves

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00:18:29 --> 00:18:34
scattered from atom two.
Constructive interference.

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00:18:34 --> 00:18:38
Here is another point of
constructive interference.

264
00:18:38 --> 00:18:42
Here is another point of
constructive interference.

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00:18:42 --> 00:18:45
Everywhere along this line,
we have constructive

266
00:18:45 --> 00:18:48
interference,
which results in a large

267
00:18:48 --> 00:18:51
intensity right at this
scattering angle here,

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00:18:51 --> 00:18:54
a bright spot.
And we already know the

269
00:18:54 --> 00:18:58
condition for constructive
interference.

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00:18:58 --> 00:19:01
That is, in order to get this
constructive interference,

271
00:19:01 --> 00:19:05
the difference in the distance
traveled by the two waves that

272
00:19:05 --> 00:19:09
are interfering has to be an
integral multiple of the

273
00:19:09 --> 00:19:12
wavelength lambda.
Now, I use the term d instead

274
00:19:12 --> 00:19:16
of r, but it is the same thing
for the condition for

275
00:19:16 --> 00:19:18
constructive interference,
here.

276
00:19:18 --> 00:19:22
And if you went and analyzed
what the difference in the

277
00:19:22 --> 00:19:25
distance was for this
constructive interference along

278
00:19:25 --> 00:19:30
this line, you would find it was
n equals 1.

279
00:19:30 --> 00:19:33
The difference in the distance
traveled is one lambda.

280
00:19:33 --> 00:19:36
And, if you looked at the
points of constructive

281
00:19:36 --> 00:19:40
interference along this line
that led to this bright spot,

282
00:19:40 --> 00:19:44
the difference in the distance
traveled will be two lambda.

283
00:19:44 --> 00:19:47
This is our second-order
interference feature,

284
00:19:47 --> 00:19:49
our second-order diffraction
spot.

285
00:19:49 --> 00:19:52
And, if you look at it along
the center here,

286
00:19:52 --> 00:19:55
normal to the crystal,
that would be the zero-order

287
00:19:55 --> 00:19:57
spot.
d2 minus d1 is equal to zero

288
00:19:57 --> 00:20:01
lambda.

289
00:20:01 --> 00:20:04
All right.
That is what looks like is

290
00:20:04 --> 00:20:08
happening.
Now, what we are going to do is

291
00:20:08 --> 00:20:13
we are going to actually analyze
this geometry a bit more.

292
00:20:13 --> 00:20:17
We didn't do so in the two slit
experiment.

293
00:20:17 --> 00:20:19
We could have.
We didn't.

294
00:20:19 --> 00:20:24
We are going to do it here.
And what we are going to be

295
00:20:24 --> 00:20:29
after is if these electrons are
acting like waves,

296
00:20:29 --> 00:20:35
then they have a wavelength.
And we want to know what the

297
00:20:35 --> 00:20:38
wavelength is.
Davidson and Germer wanted to

298
00:20:38 --> 00:20:42
know what the wavelength was.
And we are going to use this

299
00:20:42 --> 00:20:44
scattering angle here,
theta.

300
00:20:44 --> 00:20:47
This angle from the normal to
where the electrons are

301
00:20:47 --> 00:20:51
scattering, that angle theta,
we are going to use that

302
00:20:51 --> 00:20:55
information, theta equals 50.7
degrees, to back out the

303
00:20:55 --> 00:20:58
wavelength.
And we know what the condition

304
00:20:58 --> 00:21:02
is for constructive
interference.

305
00:21:02 --> 00:21:07
We just talked about it.
Here it is, d2 minus d1.

306
00:21:07 --> 00:21:10
That is the wavelength we are

307
00:21:10 --> 00:21:14
after in this analysis.
Now, what is d2 here?

308
00:21:14 --> 00:21:19
Well, the length of this line,
d2, is the distance that the

309
00:21:19 --> 00:21:26
wave that scatters from electron
two travels from electron two to

310
00:21:26 --> 00:21:29
the screen.
d1 is the distance that the

311
00:21:29 --> 00:21:36
wave that scatters from atom one
travels to the screen.

312
00:21:36 --> 00:21:39
That is what d2 and d1 are,
here.

313
00:21:39 --> 00:21:44
Now, I am going to draw a
perpendicular from atom one to

314
00:21:44 --> 00:21:47
this line d2.
There is my right angle.

315
00:21:47 --> 00:21:51
Now, you can see,
then, that this leg of the

316
00:21:51 --> 00:21:57
triangle is d2 minus d1,
the difference in the

317
00:21:57 --> 00:22:02
distance traveled.
That is this quantity here.

318
00:22:02 --> 00:22:06
That is going to be important.
Now, you have got to convince

319
00:22:06 --> 00:22:11
yourself that this angle right
here in the triangle is the same

320
00:22:11 --> 00:22:15
as this scattering angle.
You can convince yourself of

321
00:22:15 --> 00:22:18
that pretty easily.
Now we have a well-defined

322
00:22:18 --> 00:22:21
triangle.
We know one length of it,

323
00:22:21 --> 00:22:25
we have measured the angle
theta, and d2 minus d1

324
00:22:25 --> 00:22:30
is something that we would
like to know.

325
00:22:30 --> 00:22:33
Let's do a little geometry.
The sine of theta,

326
00:22:33 --> 00:22:37
the sign of this angle is equal
to the opposite length,

327
00:22:37 --> 00:22:40
which is d2 minus d1,
divided by A,

328
00:22:40 --> 00:22:44
this distance between the two
atoms in the nickel crystal.

329
00:22:44 --> 00:22:48
We have two equations.

330
00:22:48 --> 00:22:52
This is the equation that in
theory should obtain for

331
00:22:52 --> 00:22:57
constructive interference.
This is the equation that we

332
00:22:57 --> 00:23:01
set up given the particular
physical geometry of our

333
00:23:01 --> 00:23:05
problem.
These two (d2 minus d1)'s

334
00:23:05 --> 00:23:11
better be equal to each other.
We have n lambda is A sine

335
00:23:11 --> 00:23:14
theta.

336
00:23:14 --> 00:23:18
Let's solve for lambda.
We can do that.

337
00:23:18 --> 00:23:23
That is A sine theta over n.
We already know what theta is,

338
00:23:23 --> 00:23:28
we know what A is,
so what is n?

339
00:23:28 --> 00:23:33
Well, n is going to be one
because this is the bright spot

340
00:23:33 --> 00:23:36
that is closest to the
zero-order spot,

341
00:23:36 --> 00:23:42
which is always present at the
normal angle there if you are

342
00:23:42 --> 00:23:47
coming in at normal orientation.
So n is equal 1 so I can plug

343
00:23:47 --> 00:23:50
things in.
And, when I do that,

344
00:23:50 --> 00:23:56
I find that the wavelength that
I predict is 1.66x10^-10 meters.

345
00:23:56 --> 00:24:01
We've got this wavelength.
Now, before I go on,

346
00:24:01 --> 00:24:07
I just want to point out that
this geometry of the problem

347
00:24:07 --> 00:24:13
that I set up here is identical
to the geometry in a technique

348
00:24:13 --> 00:24:18
known as X-ray diffraction.
X-ray diffraction does not use

349
00:24:18 --> 00:24:21
electrons coming in,
but uses X-rays,

350
00:24:21 --> 00:24:25
photons.
And it is a technique that is

351
00:24:25 --> 00:24:30
going to be important to you if
you do any kind of science

352
00:24:30 --> 00:24:36
involving materials or
biological systems.

353
00:24:36 --> 00:24:40
And it is important because
X-ray diffraction allows you to

354
00:24:40 --> 00:24:44
determine the structure,
in particular of proteins,

355
00:24:44 --> 00:24:48
crystal proteins.
You crystallize the protein,

356
00:24:48 --> 00:24:52
and you use this X-ray
diffraction to get out the

357
00:24:52 --> 00:24:55
structure.
And the reason why you want the

358
00:24:55 --> 00:25:00
structure of the proteins is
because the structure gives you

359
00:25:00 --> 00:25:06
a hint as to what the function
of the proteins are.

360
00:25:06 --> 00:25:11
And so in the use of X-ray
diffraction, we don't go and

361
00:25:11 --> 00:25:16
calculate what lambda is.
We already know what lambda is

362
00:25:16 --> 00:25:21
in X-ray diffraction.
We know the wavelength of the

363
00:25:21 --> 00:25:25
incident photons,
the X-rays.

364
00:25:25 --> 00:25:30
What we don't know in the
technique of X-ray diffraction

365
00:25:30 --> 00:25:36
is the distance between the
atoms in an unknown structure.

366
00:25:36 --> 00:25:41
And so in X-ray diffraction,
we know the wavelength,

367
00:25:41 --> 00:25:46
and we can figure out what
order it is and we measure the

368
00:25:46 --> 00:25:51
scattering angle.
And we use that to determine

369
00:25:51 --> 00:25:55
the distance between the atoms.
And, in that way,

370
00:25:55 --> 00:26:01
we back out the structure of
the sample.

371
00:26:01 --> 00:26:07


372
00:26:07 --> 00:26:10
Yes.
We are just going to get there.

373
00:26:10 --> 00:26:13
We are going to do that.
All right.

374
00:26:13 --> 00:26:17
Same geometry here.
Now, this experiment of

375
00:26:17 --> 00:26:23
Davidson and Germer was really
an important one because just

376
00:26:23 --> 00:26:28
three years before this,
there was a prediction for what

377
00:26:28 --> 00:26:33
the wavelength of particles
ought to be.

378
00:26:33 --> 00:26:38
And that prediction was made by
this gentleman,

379
00:26:38 --> 00:26:42
Louis de Broglie.
In his Ph.D.

380
00:26:42 --> 00:26:45
thesis, no less,
what Mr.

381
00:26:45 --> 00:26:51
de Broglie did was that he
looked at the relativistic

382
00:26:51 --> 00:26:58
equations of motion that
Einstein wrote down and used to

383
00:26:58 --> 00:27:04
propose that a photon or
radiation with a wavelength

384
00:27:04 --> 00:27:11
lambda had momentum p.
Well, he took those same

385
00:27:11 --> 00:27:16
equations and said,
well, these relativistic

386
00:27:16 --> 00:27:22
equations of motion apply to
matter just as well as they

387
00:27:22 --> 00:27:27
apply to radiation.
Therefore, if you have

388
00:27:27 --> 00:27:35
radiation with a wavelength
lambda, you then have this

389
00:27:35 --> 00:27:42
momentum p for the radiation.
This is what Einstein said.

390
00:27:42 --> 00:27:50
But he turned it around and
said, if you have matter with a

391
00:27:50 --> 00:27:56
momentum p, well,
that matter ought to have a

392
00:27:56 --> 00:28:03
wavelength lambda.
He turned around Einstein's

393
00:28:03 --> 00:28:09
equations of motion and proposed
that the wavelength of a

394
00:28:09 --> 00:28:15
particle be given by h over p
where, of course,

395
00:28:15 --> 00:28:19
p here is the mass times the
velocity.

396
00:28:19 --> 00:28:24
Fantastic.

397
00:28:24 --> 00:28:27
What a great Ph.D.
thesis.

398
00:28:27 --> 00:28:32
I'm impressed.
Now, let's see how well,

399
00:28:32 --> 00:28:38
as you can imagine,
that predicts the wavelength

400
00:28:38 --> 00:28:43
that Davidson and Germer
actually measured.

401
00:28:43 --> 00:28:50
We know we have 54 electrons
coming into this nickel crystal.

402
00:28:50 --> 00:28:57
That is their kinetic energy,
one-half m v squared.

403
00:28:57 --> 00:29:02
Kinetic energy can also be

404
00:29:02 --> 00:29:05
written in terms of the
momentum.

405
00:29:05 --> 00:29:08
The momentum is p square over
2m.

406
00:29:08 --> 00:29:13
You can convince
yourself of this.

407
00:29:13 --> 00:29:18
This is a good thing to know
for doing these problems,

408
00:29:18 --> 00:29:23
that the kinetic energy is p
squared over 2 times the mass of

409
00:29:23 --> 00:29:27
the electron.
And so if you solve that,

410
00:29:27 --> 00:29:32
what you get for the momentum
of the electrons is 4.0x10^-24

411
00:29:32 --> 00:29:40
kilograms meters per second.
And now I can take this

412
00:29:40 --> 00:29:47
momentum and plug it into the
expression for de Broglie's

413
00:29:47 --> 00:29:55
wavelength, 6.6x10^-34 joule
seconds, over the momentum,

414
00:29:55 --> 00:29:58
4x10^-24.
What do I get?

415
00:29:58 --> 00:30:05
1.7x10^-10 meters.
Absolutely the same as the

416
00:30:05 --> 00:30:10
experiment.
De Broglie made a prediction.

417
00:30:10 --> 00:30:17
A few years after that,
experiments demonstrated that

418
00:30:17 --> 00:30:24
de Broglie was absolutely
correct in his prediction.

419
00:30:24 --> 00:30:29
What do we have here?
We have matter,

420
00:30:29 --> 00:30:33
particles, exhibiting wave-like
behavior.

421
00:30:33 --> 00:30:38
And, those particles can be
measured to have a wavelength

422
00:30:38 --> 00:30:43
that actually agrees with a
prediction, some theory,

423
00:30:43 --> 00:30:48
the de Broglie wavelength.
And we also have another

424
00:30:48 --> 00:30:51
phenomena here,
which I really enjoy,

425
00:30:51 --> 00:30:56
and that is Davidson and Germer
and George Thompson.

426
00:30:56 --> 00:31:03
They demonstrated that
electrons behave like waves.

427
00:31:03 --> 00:31:06
And what did J.J.
Thompson do,

428
00:31:06 --> 00:31:11
father of George Thompson,
well, he demonstrated that an

429
00:31:11 --> 00:31:17
electron was a particle.
Here, we have both the father

430
00:31:17 --> 00:31:23
and the son talking about
seemingly opposite behavior,

431
00:31:23 --> 00:31:28
but they are both right.
How often does that happen?

432
00:31:28 --> 00:31:33
That, I think,
is really amazing.

433
00:31:33 --> 00:31:40
But if matter is wave-like and
if electrons can be represented

434
00:31:40 --> 00:31:45
by a wavelength,
then what about your

435
00:31:45 --> 00:31:53
wavelengths and my wavelengths?
We should have a wavelength.

436
00:31:53 --> 00:31:56
And we do.
And just briefly,

437
00:31:56 --> 00:32:04
here, let's talk about what the
wavelength is of a baseball

438
00:32:04 --> 00:32:10
pitched by Curt Shilling at 90
mph.

439
00:32:10 --> 00:32:15
What is that wavelength?
Well, a baseball is five

440
00:32:15 --> 00:32:17
ounces.
90 mph.

441
00:32:17 --> 00:32:23
You can calculate the momentum.
It is in your notes there.

442
00:32:23 --> 00:32:29
We will calculate,
here, the wavelength.

443
00:32:29 --> 00:32:35
And what you are going to find
is that it is 1.2x10^-34 meters.

444
00:32:35 --> 00:32:39
That is pretty small.
What is the diameter of a

445
00:32:39 --> 00:32:42
nucleus?
10^-14, right.

446
00:32:42 --> 00:32:48
That is a good number to know.
Here, we have a wavelength that

447
00:32:48 --> 00:32:52
is 10^-34 meters.
Is that wavelength of a

448
00:32:52 --> 00:32:58
macroscopic size object going to
have any consequence in our

449
00:32:58 --> 00:33:01
world?
No, absolutely not.

450
00:33:01 --> 00:33:04
Why?
Because in order to see any

451
00:33:04 --> 00:33:10
effects from this small
wavelength, we are going to have

452
00:33:10 --> 00:33:16
to have slits or atoms that are
going to be on the order of this

453
00:33:16 --> 00:33:20
close together.
But there is no way that we are

454
00:33:20 --> 00:33:26
going to have two nickel atoms
that are this close together or

455
00:33:26 --> 00:33:33
two slits in a two slit
experiment this close together.

456
00:33:33 --> 00:33:40
And so for macroscopic objects,
the wavelike properties have no

457
00:33:40 --> 00:33:47
consequence in this world.
And that is simply because the

458
00:33:47 --> 00:33:53
mass is too large.
It makes the wavelength too

459
00:33:53 --> 00:33:59
small to have any effects in our
everyday lives.

460
00:33:59 --> 00:34:04
And it is actually --
Yes?

461
00:34:04 --> 00:34:07
Okay.

462
00:34:07 --> 00:34:15


463
00:34:15 --> 00:34:19
Well, they are actually coming
in as waves.

464
00:34:19 --> 00:34:25
They are behaving as waves.
Remember my beach picture?

465
00:34:25 --> 00:34:30
I drew them coming in like a
circle.

466
00:34:30 --> 00:34:36
But remember my picture of this
barrier here on the beach,

467
00:34:36 --> 00:34:42
and I am laying here on the
sand, and then the waves are

468
00:34:42 --> 00:34:44
coming in?
Here is blue.

469
00:34:44 --> 00:34:48
These electrons,
as they are coming in,

470
00:34:48 --> 00:34:54
really need to be thought of as
these kind of plane waves.

471
00:34:54 --> 00:35:00
I drew them as kind of just
particles.

472
00:35:00 --> 00:35:05
But you have to really think of
them as plane waves and that

473
00:35:05 --> 00:35:11
they are reflecting off of these
two atoms in the way that I just

474
00:35:11 --> 00:35:15
explained.
Another question?

475
00:35:15 --> 00:35:25


476
00:35:25 --> 00:35:29
The thing is that you cannot
get that velocity slow enough to

477
00:35:29 --> 00:35:33
make the wavelength large enough
to be of consequence.

478
00:35:33 --> 00:35:36
If you could then you would,
right?

479
00:35:36 --> 00:35:38
You would see the wave-like
behavior.

480
00:35:38 --> 00:35:41
But you cannot,
practically speaking,

481
00:35:41 --> 00:35:44
get it to that extent.

482
00:35:44 --> 00:36:18


483
00:36:18 --> 00:36:20
No.
In this particular case,

484
00:36:20 --> 00:36:24
anything that is so massive,
any smaller effects,

485
00:36:24 --> 00:36:28
like what you are talking
about, exactly the point of

486
00:36:28 --> 00:36:33
observation is not going to have
an effect on anything that is so

487
00:36:33 --> 00:36:35
massive.
Pardon?

488
00:36:35 --> 00:36:41
Yes, your point of observation
will have an effect on your

489
00:36:41 --> 00:36:46
interpretation of the
experiment, if you are talking

490
00:36:46 --> 00:36:51
about something that has a much
larger wavelength.

491
00:36:51 --> 00:36:55
Absolutely.
In high energy physics

492
00:36:55 --> 00:37:00
experiments, for example.
Good question.

493
00:37:00 --> 00:37:05


494
00:37:05 --> 00:37:14
It is just this observation,
here, of the wave-like behavior

495
00:37:14 --> 00:37:19
of electrons,
of particles,

496
00:37:19 --> 00:37:29
that led to the interpretation,
then, or led to the realization

497
00:37:29 --> 00:37:39
that maybe, you have to treat
electrons as waves.

498
00:37:39 --> 00:37:48
Or maybe you have to treat the
behavior of electrons as

499
00:37:48 --> 00:37:56
wave-like behavior.
And that is what this gentleman

500
00:37:56 --> 00:38:01
Schrödinger did.
He said, well,

501
00:38:01 --> 00:38:05
you know what?
This gave him an idea.

502
00:38:05 --> 00:38:10
Maybe what is wrong is that an
electron in an atom,

503
00:38:10 --> 00:38:14
I cannot treat that electron as
a particle.

504
00:38:14 --> 00:38:19
Instead, what I have to do is
treat is as a wave.

505
00:38:19 --> 00:38:23
I have to treat its wave-like
properties.

506
00:38:23 --> 00:38:28
And it was that impetus that
led him to write down a wave

507
00:38:28 --> 00:38:34
equation of motion.
An equation of motion for

508
00:38:34 --> 00:38:36
waves.
He realized,

509
00:38:36 --> 00:38:40
or he was guessing at the
moment, well,

510
00:38:40 --> 00:38:46
maybe in the case when a
microscopic particle has a

511
00:38:46 --> 00:38:53
wavelength that is on the order
of the size of its environment,

512
00:38:53 --> 00:38:57
in that case,
maybe the wavelength has an

513
00:38:57 --> 00:39:02
effect, makes a difference.
For example,

514
00:39:02 --> 00:39:07
in the case of the electrons,
we had a wavelength calculated,

515
00:39:07 --> 00:39:12
there, of 1.7x10^-10 meters.
That is a wavelength that is on

516
00:39:12 --> 00:39:16
the order of the size of the
environment of the electron,

517
00:39:16 --> 00:39:20
which is on the order of the
size of an atom.

518
00:39:20 --> 00:39:24
Maybe in that case I have to
pay attention to the wavelength.

519
00:39:24 --> 00:39:29
The reason you and I don't have
to pay any attention to our

520
00:39:29 --> 00:39:33
wavelength is because our
wavelength is 10^-30 meters or

521
00:39:33 --> 00:39:36
so.
And that is much,

522
00:39:36 --> 00:39:40
much larger than the size of
the environment.

523
00:39:40 --> 00:39:44
In this case,
we don't have to pay any

524
00:39:44 --> 00:39:49
attention to our wavelength.
But, for an electron in an

525
00:39:49 --> 00:39:51
atom, we have a problem,
here.

526
00:39:51 --> 00:39:55
What did Schrödinger do?
Schrödinger said,

527
00:39:55 --> 00:40:00
I have to write down a wave
equation.

528
00:40:00 --> 00:40:05
An equation of motion for
matter waves.

529
00:40:05 --> 00:40:10
And what is that equation of
motion?

530
00:40:10 --> 00:40:19
Well, that equation of motion
is H hat operating on Psi,

531
00:40:19 --> 00:40:25
and it gives us back an energy,
E, times a Psi.

532
00:40:25 --> 00:40:30
What is this?
Well, Psi, here,

533
00:40:30 --> 00:40:34
is a wave.
I am somehow going to let my

534
00:40:34 --> 00:40:37
electron in an atom be
represented by Psi.

535
00:40:37 --> 00:40:40
This is going to be a wave
form.

536
00:40:40 --> 00:40:43
This is going to be a wave
function.

537
00:40:43 --> 00:40:49
I am going to let my electron
be represented by the wave

538
00:40:49 --> 00:40:52
function.
Exactly how it is going to be

539
00:40:52 --> 00:40:58
represented by the wave function
is something that I am not going

540
00:40:58 --> 00:41:06
to tell you quite yet.
But it is going to represent

541
00:41:06 --> 00:41:10
the electron.
This energy,

542
00:41:10 --> 00:41:18
here, that energy is going to
turn out to be the binding

543
00:41:18 --> 00:41:23
energy of the electron in the
atom.

544
00:41:23 --> 00:41:30
This thing, here,
is called the Hamiltonian

545
00:41:30 --> 00:41:35
Operator.
That Hamiltonian Operator is

546
00:41:35 --> 00:41:41
specific to a particular
problem, and we will look at the

547
00:41:41 --> 00:41:45
Hamiltonian Operator for a
hydrogen atom.

548
00:41:45 --> 00:41:50
But this operator is operating
on Psi, and it gives you back a

549
00:41:50 --> 00:41:55
Psi, the same function,
multiplied by a constant.

550
00:41:55 --> 00:42:00
That constant is the binding
energy.

551
00:42:00 --> 00:42:04
Now, you think,
well, let me just cancel this

552
00:42:04 --> 00:42:08
and this.
But you cannot do that because

553
00:42:08 --> 00:42:13
this is an operator.
This has some derivatives in

554
00:42:13 --> 00:42:17
it.
Its operating on Psi gives you

555
00:42:17 --> 00:42:21
back the same function times the
constant.

556
00:42:21 --> 00:42:27
Now, how did Schrödinger
actually derive this equation,

557
00:42:27 --> 00:42:34
so to speak?
Well, what he did was to just

558
00:42:34 --> 00:42:40
guess at a wave function.
We are going to use a

559
00:42:40 --> 00:42:47
one-dimensional wave function.
He is going to say,

560
00:42:47 --> 00:42:55
let me represent my electron by
Psi of x equal to 2 a times

561
00:42:55 --> 00:43:02
cosine 2 pi x over lambda.

562
00:43:02 --> 00:43:09
That is going to be my wave

563
00:43:09 --> 00:43:11
function.
Why not?

564
00:43:11 --> 00:43:19
And then he said,
well, what I really need here

565
00:43:19 --> 00:43:27
is an equation of motion.
I need to know how Psi changes

566
00:43:27 --> 00:43:32
with x.
If you wanted an equation of

567
00:43:32 --> 00:43:37
motion, if you wanted to know
how Psi changes with x,

568
00:43:37 --> 00:43:41
what would you do to Psi?
Take the derivative.

569
00:43:41 --> 00:43:46
Let's take a derivative of Psi
of x, with respect to x.

570
00:43:46 --> 00:43:51
That is going to be
equal to minus 2 a

571
00:43:51 --> 00:43:56
times 2 pi over lambda times
sine of 2pi x over lambda.

572
00:43:56 --> 00:44:03


573
00:44:03 --> 00:44:07
That is an equation of motion.
Now, this is actually kind of

574
00:44:07 --> 00:44:10
an equation of position.
But it is telling us how Psi

575
00:44:10 --> 00:44:12
changes with x.

576
00:44:12 --> 00:44:20


577
00:44:20 --> 00:44:28
But now, since that gave us
some information about how Psi

578
00:44:28 --> 00:44:34
changes with x,
how would we get the rate of

579
00:44:34 --> 00:44:39
change of Psi with x?
Second derivative.

580
00:44:39 --> 00:44:43
Let's take the second
derivative of that.

581
00:44:43 --> 00:44:47
Second derivative of Psi of x
with respect to x,

582
00:44:47 --> 00:44:52
that is minus 2a times 4 pi
squared over lambda squared

583
00:44:52 --> 00:44:57
times cosine 2pi x
over lambda.

584
00:44:57 --> 00:45:02


585
00:45:02 --> 00:45:07
That's pretty good,
but now, what do you see in

586
00:45:07 --> 00:45:09
this equation?
Psi.

587
00:45:09 --> 00:45:14
You see what I started with.
It is recursive,

588
00:45:14 --> 00:45:19
right.
You see the Psi of x here.

589
00:45:19 --> 00:45:23
Let me write that equation in

590
00:45:23 --> 00:45:31
terms of the original function.
Second derivative of Psi of x

591
00:45:31 --> 00:45:36
with respect to x,
minus 2pi over lambda squared

592
00:45:36 --> 00:45:41
times Psi of x.

593
00:45:41 --> 00:45:45
lambda)^2 Psi(x)** That is
pretty good.

594
00:45:45 --> 00:45:49
You know where I am trying to
go?

595
00:45:49 --> 00:45:53
I am trying to derive,
so to speak,

596
00:45:53 --> 00:46:00
Schrödinger's equation.
See, it is not very hard.

597
00:46:00 --> 00:46:03
Yes.
Now, this equation right here,

598
00:46:03 --> 00:46:08
this equation is just a
classical wave equation.

599
00:46:08 --> 00:46:14
The only thing I have done so
far is take derivatives.

600
00:46:14 --> 00:46:19
I have done nothing else.
I just took derivatives.

601
00:46:19 --> 00:46:23
It could represent any kind of
wave.

602
00:46:23 --> 00:46:29
There is nothing quantum
mechanical about it.

603
00:46:29 --> 00:46:36
But here comes the big leap
that Schrödinger made.

604
00:46:36 --> 00:46:43
He substituted in here for
lambda the momentum of the

605
00:46:43 --> 00:46:47
particle.
In other words,

606
00:46:47 --> 00:46:55
if this is a wave equation and
that wave has some wavelength,

607
00:46:55 --> 00:47:00
here, lambda.
He said, well,

608
00:47:00 --> 00:47:04
if this is a wave equation for
a matter wave,

609
00:47:04 --> 00:47:10
well, then I better get the
momentum of that particle in

610
00:47:10 --> 00:47:14
this wave.
And he knew how to do that

611
00:47:14 --> 00:47:18
because de Broglie told him how
to do that.

612
00:47:18 --> 00:47:22
De Broglie said,
lambda is equal to h over p.

613
00:47:22 --> 00:47:26
That is pretty good.

614
00:47:26 --> 00:47:32
What can I do here?
Well, what I can do is I can

615
00:47:32 --> 00:47:38
rearrange this and write this in
terms of the momentum of the

616
00:47:38 --> 00:47:42
particle.
Second derivative of Psi of x

617
00:47:42 --> 00:47:48
with respect to x is going to be
minus p squared over h bar

618
00:47:48 --> 00:47:53
squared times Psi of x.

619
00:47:53 --> 00:47:58
Now, let me explain
what h bar here is

620
00:47:58 --> 00:48:02
h bar is a shorthand way of

621
00:48:02 --> 00:48:07
writing h over 2pi. If you

622
00:48:07 --> 00:48:11
don't know it already,
you should know it.

623
00:48:11 --> 00:48:15
You are going to need it.
That is h over 2pi,

624
00:48:15 --> 00:48:19
so this is h bar squared.
Now, what do I have?

625
00:48:19 --> 00:48:24
Now I have a matter wave.
I have an equation of motion.

626
00:48:24 --> 00:48:32
I have something that tells me
how Psi moves with respect to X.

627
00:48:32 --> 00:48:35
The rate of change of Psi with
X.

628
00:48:35 --> 00:48:40
And I had the momentum of the
particle buried in here.

629
00:48:40 --> 00:48:44
From this form,
I am going to get to that.

630
00:48:44.115 --> 48:47
And I guess I am going to get
to that next Wednesday.