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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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00:13:22,000 --> 00:13:25,000
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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00:14:15,000 --> 00:14:17,000
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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00:14:31,000 --> 00:14:35,000
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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00:14:48,000 --> 00:14:52,000
but kept the spacing between
the two nickel atoms the same.

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00:14:52,000 --> 00:14:56,000
And so Davidson,
Germer and Thompson came in,

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00:14:56,000 --> 00:15:02,000
scattered these electrons and
looked how they scattered back.

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00:15:02,000 --> 00:15:06,000
And, lo and behold,
what they saw is that these

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00:15:06,000 --> 00:15:11,000
electrons seemed to scatter back
at a preferential angle.

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00:15:11,000 --> 00:15:15,000
The angular distribution was
not isotropic.

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00:15:15,000 --> 00:15:20,000
Instead, it looked like the
electrons scattered back at a

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00:15:20,000 --> 00:15:24,000
pretty well-defined angle here,
50.7 degrees.

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00:15:24,000 --> 00:15:28,000
And not only did they scatter
back at that angle,

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00:15:28,000 --> 00:15:34,000
they also scattered right back
at themselves.

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00:15:34,000 --> 00:15:37,000
Backscattered this way,
so this scattering angle is

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00:15:37,000 --> 00:15:40,000
zero degrees.
And under some particular

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00:15:40,000 --> 00:15:45,000
conditions, the electrons also
scattered at a larger angle,

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00:15:45,000 --> 00:15:48,000
here.
But the bottom line is that the

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00:15:48,000 --> 00:15:51,000
scattering pattern was not
isotropic.

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00:15:51,000 --> 00:15:55,000
There was a bright spot,
lots of electrons scattered at

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00:15:55,000 --> 00:15:59,000
this angle, a dark spot,
no electrons scattered at this

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00:15:59,000 --> 00:16:02,000
angle.
A bright spot,

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00:16:02,000 --> 00:16:07,000
dark spot, bright spot.
This looks like interference

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00:16:07,000 --> 00:16:12,000
phenomena, just like the two
slit experiment.

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00:16:12,000 --> 00:16:15,000
Bright spot,
dark spot, bright spot,

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00:16:15,000 --> 00:16:18,000
constructive,
destructive,

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00:16:18,000 --> 00:16:22,000
constructive interference,
back and forth.

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00:16:22,000 --> 00:16:27,000
That was their observation.
How do we understand that?

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00:16:27,000 --> 00:16:33,000
Well, it is looking like these
electrons are behaving like

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00:16:33,000 --> 00:16:38,000
waves.
Suppose these electrons are

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00:16:38,000 --> 00:16:43,000
coming in, so we have this
constant stream of electrons

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00:16:43,000 --> 00:16:48,000
impinging on our nickel crystal.
Well, what is happening here is

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00:16:48,000 --> 00:16:53,000
that when these electrons are
reflecting back from the

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00:16:53,000 --> 00:16:57,000
individual nickel atoms,
these individual nickel atoms

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00:16:57,000 --> 00:17:03,000
are kind of functioning like
those little slits we saw in the

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00:17:03,000 --> 00:17:08,000
two slit experiment.
That is, they are scattering

238
00:17:08,000 --> 00:17:11,000
back as a wave.
These electrons seem to be

239
00:17:11,000 --> 00:17:15,000
scattering as a wave,
so isotropically in all

240
00:17:15,000 --> 00:17:19,000
directions.
This semicircle around each one

241
00:17:19,000 --> 00:17:22,000
of the atoms,
and I only show you two atoms

242
00:17:22,000 --> 00:17:27,000
here, each semicircle is the
maximum of the wave front.

243
00:17:27,000 --> 00:17:31,000
It is the crest of the wave
front.

244
00:17:31,000 --> 00:17:35,000
And then, as time goes by,
of course, these waves

245
00:17:35,000 --> 00:17:39,000
propagate out.
And then another wave front,

246
00:17:39,000 --> 00:17:45,000
another wave maximum appears
and a distance between these two

247
00:17:45,000 --> 00:17:49,000
maxima is, of course,
the wavelength.

248
00:17:49,000 --> 00:17:53,000
And as time goes on,
they scatter further.

249
00:17:53,000 --> 00:17:58,000
And as time goes on,
they still scatter.

250
00:17:58,000 --> 00:18:02,000
And they keep the propagating
out until they reach the screen.

251
00:18:02,000 --> 00:18:06,000
And, lo and behold,
on the screen you see a bright

252
00:18:06,000 --> 00:18:08,000
spot, dark spot,
bright spot,

253
00:18:08,000 --> 00:18:10,000
dark spot.
Interference pattern.

254
00:18:10,000 --> 00:18:14,000
Let's analyze this.
Here is the diagram again.

255
00:18:14,000 --> 00:18:17,000
I just moved it over and
cleaned it up again.

256
00:18:17,000 --> 00:18:21,000
I want you to look at this spot
right in there.

257
00:18:21,000 --> 00:18:25,000
That is where we have the
maximum of a wave scattered from

258
00:18:25,000 --> 00:18:29,000
atom one at the same point in
space as the maximum of waves

259
00:18:29,000 --> 00:18:34,000
scattered from atom two.
Constructive interference.

260
00:18:34,000 --> 00:18:38,000
Here is another point of
constructive interference.

261
00:18:38,000 --> 00:18:42,000
Here is another point of
constructive interference.

262
00:18:42,000 --> 00:18:45,000
Everywhere along this line,
we have constructive

263
00:18:45,000 --> 00:18:48,000
interference,
which results in a large

264
00:18:48,000 --> 00:18:51,000
intensity right at this
scattering angle here,

265
00:18:51,000 --> 00:18:54,000
a bright spot.
And we already know the

266
00:18:54,000 --> 00:18:58,000
condition for constructive
interference.

267
00:18:58,000 --> 00:19:01,000
That is, in order to get this
constructive interference,

268
00:19:01,000 --> 00:19:05,000
the difference in the distance
traveled by the two waves that

269
00:19:05,000 --> 00:19:09,000
are interfering has to be an
integral multiple of the

270
00:19:09,000 --> 00:19:12,000
wavelength lambda.
Now, I use the term d instead

271
00:19:12,000 --> 00:19:16,000
of r, but it is the same thing
for the condition for

272
00:19:16,000 --> 00:19:18,000
constructive interference,
here.

273
00:19:18,000 --> 00:19:22,000
And if you went and analyzed
what the difference in the

274
00:19:22,000 --> 00:19:25,000
distance was for this
constructive interference along

275
00:19:25,000 --> 00:19:30,000
this line, you would find it was
n equals 1.

276
00:19:30,000 --> 00:19:33,000
The difference in the distance
traveled is one lambda.

277
00:19:33,000 --> 00:19:36,000
And, if you looked at the
points of constructive

278
00:19:36,000 --> 00:19:40,000
interference along this line
that led to this bright spot,

279
00:19:40,000 --> 00:19:44,000
the difference in the distance
traveled will be two lambda.

280
00:19:44,000 --> 00:19:47,000
This is our second-order
interference feature,

281
00:19:47,000 --> 00:19:49,000
our second-order diffraction
spot.

282
00:19:49,000 --> 00:19:52,000
And, if you look at it along
the center here,

283
00:19:52,000 --> 00:19:55,000
normal to the crystal,
that would be the zero-order

284
00:19:55,000 --> 00:19:57,000
spot.
d2 minus d1 is equal to zero

285
00:19:57,000 --> 00:20:01,000
lambda.

286
00:20:01,000 --> 00:20:04,000
All right.
That is what looks like is

287
00:20:04,000 --> 00:20:08,000
happening.
Now, what we are going to do is

288
00:20:08,000 --> 00:20:13,000
we are going to actually analyze
this geometry a bit more.

289
00:20:13,000 --> 00:20:17,000
We didn't do so in the two slit
experiment.

290
00:20:17,000 --> 00:20:19,000
We could have.
We didn't.

291
00:20:19,000 --> 00:20:24,000
We are going to do it here.
And what we are going to be

292
00:20:24,000 --> 00:20:29,000
after is if these electrons are
acting like waves,

293
00:20:29,000 --> 00:20:35,000
then they have a wavelength.
And we want to know what the

294
00:20:35,000 --> 00:20:38,000
wavelength is.
Davidson and Germer wanted to

295
00:20:38,000 --> 00:20:42,000
know what the wavelength was.
And we are going to use this

296
00:20:42,000 --> 00:20:44,000
scattering angle here,
theta.

297
00:20:44,000 --> 00:20:47,000
This angle from the normal to
where the electrons are

298
00:20:47,000 --> 00:20:51,000
scattering, that angle theta,
we are going to use that

299
00:20:51,000 --> 00:20:55,000
information, theta equals 50.7
degrees, to back out the

300
00:20:55,000 --> 00:20:58,000
wavelength.
And we know what the condition

301
00:20:58,000 --> 00:21:02,000
is for constructive
interference.

302
00:21:02,000 --> 00:21:07,000
We just talked about it.
Here it is, d2 minus d1.

303
00:21:07,000 --> 00:21:10,000
That is the wavelength we are

304
00:21:10,000 --> 00:21:14,000
after in this analysis.
Now, what is d2 here?

305
00:21:14,000 --> 00:21:19,000
Well, the length of this line,
d2, is the distance that the

306
00:21:19,000 --> 00:21:26,000
wave that scatters from electron
two travels from electron two to

307
00:21:26,000 --> 00:21:29,000
the screen.
d1 is the distance that the

308
00:21:29,000 --> 00:21:36,000
wave that scatters from atom one
travels to the screen.

309
00:21:36,000 --> 00:21:39,000
That is what d2 and d1 are,
here.

310
00:21:39,000 --> 00:21:44,000
Now, I am going to draw a
perpendicular from atom one to

311
00:21:44,000 --> 00:21:47,000
this line d2.
There is my right angle.

312
00:21:47,000 --> 00:21:51,000
Now, you can see,
then, that this leg of the

313
00:21:51,000 --> 00:21:57,000
triangle is d2 minus d1,
the difference in the

314
00:21:57,000 --> 00:22:02,000
distance traveled.
That is this quantity here.

315
00:22:02,000 --> 00:22:06,000
That is going to be important.
Now, you have got to convince

316
00:22:06,000 --> 00:22:11,000
yourself that this angle right
here in the triangle is the same

317
00:22:11,000 --> 00:22:15,000
as this scattering angle.
You can convince yourself of

318
00:22:15,000 --> 00:22:18,000
that pretty easily.
Now we have a well-defined

319
00:22:18,000 --> 00:22:21,000
triangle.
We know one length of it,

320
00:22:21,000 --> 00:22:25,000
we have measured the angle
theta, and d2 minus d1

321
00:22:25,000 --> 00:22:30,000
is something that we would
like to know.

322
00:22:30,000 --> 00:22:33,000
Let's do a little geometry.
The sine of theta,

323
00:22:33,000 --> 00:22:37,000
the sign of this angle is equal
to the opposite length,

324
00:22:37,000 --> 00:22:40,000
which is d2 minus d1,
divided by A,

325
00:22:40,000 --> 00:22:44,000
this distance between the two
atoms in the nickel crystal.

326
00:22:44,000 --> 00:22:48,000
We have two equations.

327
00:22:48,000 --> 00:22:52,000
This is the equation that in
theory should obtain for

328
00:22:52,000 --> 00:22:57,000
constructive interference.
This is the equation that we

329
00:22:57,000 --> 00:23:01,000
set up given the particular
physical geometry of our

330
00:23:01,000 --> 00:23:05,000
problem.
These two (d2 minus d1)'s

331
00:23:05,000 --> 00:23:11,000
better be equal to each other.
We have n lambda is A sine

332
00:23:11,000 --> 00:23:14,000
theta.

333
00:23:14,000 --> 00:23:18,000
Let's solve for lambda.
We can do that.

334
00:23:18,000 --> 00:23:23,000
That is A sine theta over n.
We already know what theta is,

335
00:23:23,000 --> 00:23:28,000
we know what A is,
so what is n?

336
00:23:28,000 --> 00:23:33,000
Well, n is going to be one
because this is the bright spot

337
00:23:33,000 --> 00:23:36,000
that is closest to the
zero-order spot,

338
00:23:36,000 --> 00:23:42,000
which is always present at the
normal angle there if you are

339
00:23:42,000 --> 00:23:47,000
coming in at normal orientation.
So n is equal 1 so I can plug

340
00:23:47,000 --> 00:23:50,000
things in.
And, when I do that,

341
00:23:50,000 --> 00:23:56,000
I find that the wavelength that
I predict is 1.66x10^-10 meters.

342
00:23:56,000 --> 00:24:01,000
We've got this wavelength.
Now, before I go on,

343
00:24:01,000 --> 00:24:07,000
I just want to point out that
this geometry of the problem

344
00:24:07,000 --> 00:24:13,000
that I set up here is identical
to the geometry in a technique

345
00:24:13,000 --> 00:24:18,000
known as X-ray diffraction.
X-ray diffraction does not use

346
00:24:18,000 --> 00:24:21,000
electrons coming in,
but uses X-rays,

347
00:24:21,000 --> 00:24:25,000
photons.
And it is a technique that is

348
00:24:25,000 --> 00:24:30,000
going to be important to you if
you do any kind of science

349
00:24:30,000 --> 00:24:36,000
involving materials or
biological systems.

350
00:24:36,000 --> 00:24:40,000
And it is important because
X-ray diffraction allows you to

351
00:24:40,000 --> 00:24:44,000
determine the structure,
in particular of proteins,

352
00:24:44,000 --> 00:24:48,000
crystal proteins.
You crystallize the protein,

353
00:24:48,000 --> 00:24:52,000
and you use this X-ray
diffraction to get out the

354
00:24:52,000 --> 00:24:55,000
structure.
And the reason why you want the

355
00:24:55,000 --> 00:25:00,000
structure of the proteins is
because the structure gives you

356
00:25:00,000 --> 00:25:06,000
a hint as to what the function
of the proteins are.

357
00:25:06,000 --> 00:25:11,000
And so in the use of X-ray
diffraction, we don't go and

358
00:25:11,000 --> 00:25:16,000
calculate what lambda is.
We already know what lambda is

359
00:25:16,000 --> 00:25:21,000
in X-ray diffraction.
We know the wavelength of the

360
00:25:21,000 --> 00:25:25,000
incident photons,
the X-rays.

361
00:25:25,000 --> 00:25:30,000
What we don't know in the
technique of X-ray diffraction

362
00:25:30,000 --> 00:25:36,000
is the distance between the
atoms in an unknown structure.

363
00:25:36,000 --> 00:25:41,000
And so in X-ray diffraction,
we know the wavelength,

364
00:25:41,000 --> 00:25:46,000
and we can figure out what
order it is and we measure the

365
00:25:46,000 --> 00:25:51,000
scattering angle.
And we use that to determine

366
00:25:51,000 --> 00:25:55,000
the distance between the atoms.
And, in that way,

367
00:25:55,000 --> 00:26:01,000
we back out the structure of
the sample.

368
00:26:07,000 --> 00:26:10,000
Yes.
We are just going to get there.

369
00:26:10,000 --> 00:26:13,000
We are going to do that.
All right.

370
00:26:13,000 --> 00:26:17,000
Same geometry here.
Now, this experiment of

371
00:26:17,000 --> 00:26:23,000
Davidson and Germer was really
an important one because just

372
00:26:23,000 --> 00:26:28,000
three years before this,
there was a prediction for what

373
00:26:28,000 --> 00:26:33,000
the wavelength of particles
ought to be.

374
00:26:33,000 --> 00:26:38,000
And that prediction was made by
this gentleman,

375
00:26:38,000 --> 00:26:42,000
Louis de Broglie.
In his Ph.D.

376
00:26:42,000 --> 00:26:45,000
thesis, no less,
what Mr.

377
00:26:45,000 --> 00:26:51,000
de Broglie did was that he
looked at the relativistic

378
00:26:51,000 --> 00:26:58,000
equations of motion that
Einstein wrote down and used to

379
00:26:58,000 --> 00:27:04,000
propose that a photon or
radiation with a wavelength

380
00:27:04,000 --> 00:27:11,000
lambda had momentum p.
Well, he took those same

381
00:27:11,000 --> 00:27:16,000
equations and said,
well, these relativistic

382
00:27:16,000 --> 00:27:22,000
equations of motion apply to
matter just as well as they

383
00:27:22,000 --> 00:27:27,000
apply to radiation.
Therefore, if you have

384
00:27:27,000 --> 00:27:35,000
radiation with a wavelength
lambda, you then have this

385
00:27:35,000 --> 00:27:42,000
momentum p for the radiation.
This is what Einstein said.

386
00:27:42,000 --> 00:27:50,000
But he turned it around and
said, if you have matter with a

387
00:27:50,000 --> 00:27:56,000
momentum p, well,
that matter ought to have a

388
00:27:56,000 --> 00:28:03,000
wavelength lambda.
He turned around Einstein's

389
00:28:03,000 --> 00:28:09,000
equations of motion and proposed
that the wavelength of a

390
00:28:09,000 --> 00:28:15,000
particle be given by h over p
where, of course,

391
00:28:15,000 --> 00:28:19,000
p here is the mass times the
velocity.

392
00:28:19,000 --> 00:28:24,000
Fantastic.

393
00:28:24,000 --> 00:28:27,000
What a great Ph.D.
thesis.

394
00:28:27,000 --> 00:28:32,000
I'm impressed.
Now, let's see how well,

395
00:28:32,000 --> 00:28:38,000
as you can imagine,
that predicts the wavelength

396
00:28:38,000 --> 00:28:43,000
that Davidson and Germer
actually measured.

397
00:28:43,000 --> 00:28:50,000
We know we have 54 electrons
coming into this nickel crystal.

398
00:28:50,000 --> 00:28:57,000
That is their kinetic energy,
one-half m v squared.

399
00:28:57,000 --> 00:29:02,000
Kinetic energy can also be

400
00:29:02,000 --> 00:29:05,000
written in terms of the
momentum.

401
00:29:05,000 --> 00:29:08,000
The momentum is p square over
2m.

402
00:29:08,000 --> 00:29:13,000
You can convince
yourself of this.

403
00:29:13,000 --> 00:29:18,000
This is a good thing to know
for doing these problems,

404
00:29:18,000 --> 00:29:23,000
that the kinetic energy is p
squared over 2 times the mass of

405
00:29:23,000 --> 00:29:27,000
the electron.
And so if you solve that,

406
00:29:27,000 --> 00:29:32,000
what you get for the momentum
of the electrons is 4.0x10^-24

407
00:29:32,000 --> 00:29:40,000
kilograms meters per second.
And now I can take this

408
00:29:40,000 --> 00:29:47,000
momentum and plug it into the
expression for de Broglie's

409
00:29:47,000 --> 00:29:55,000
wavelength, 6.6x10^-34 joule
seconds, over the momentum,

410
00:29:55,000 --> 00:29:58,000
4x10^-24.
What do I get?

411
00:29:58,000 --> 00:30:05,000
1.7x10^-10 meters.
Absolutely the same as the

412
00:30:05,000 --> 00:30:10,000
experiment.
De Broglie made a prediction.

413
00:30:10,000 --> 00:30:17,000
A few years after that,
experiments demonstrated that

414
00:30:17,000 --> 00:30:24,000
de Broglie was absolutely
correct in his prediction.

415
00:30:24,000 --> 00:30:29,000
What do we have here?
We have matter,

416
00:30:29,000 --> 00:30:33,000
particles, exhibiting wave-like
behavior.

417
00:30:33,000 --> 00:30:38,000
And, those particles can be
measured to have a wavelength

418
00:30:38,000 --> 00:30:43,000
that actually agrees with a
prediction, some theory,

419
00:30:43,000 --> 00:30:48,000
the de Broglie wavelength.
And we also have another

420
00:30:48,000 --> 00:30:51,000
phenomena here,
which I really enjoy,

421
00:30:51,000 --> 00:30:56,000
and that is Davidson and Germer
and George Thompson.

422
00:30:56,000 --> 00:31:03,000
They demonstrated that
electrons behave like waves.

423
00:31:03,000 --> 00:31:06,000
And what did J.J.
Thompson do,

424
00:31:06,000 --> 00:31:11,000
father of George Thompson,
well, he demonstrated that an

425
00:31:11,000 --> 00:31:17,000
electron was a particle.
Here, we have both the father

426
00:31:17,000 --> 00:31:23,000
and the son talking about
seemingly opposite behavior,

427
00:31:23,000 --> 00:31:28,000
but they are both right.
How often does that happen?

428
00:31:28,000 --> 00:31:33,000
That, I think,
is really amazing.

429
00:31:33,000 --> 00:31:40,000
But if matter is wave-like and
if electrons can be represented

430
00:31:40,000 --> 00:31:45,000
by a wavelength,
then what about your

431
00:31:45,000 --> 00:31:53,000
wavelengths and my wavelengths?
We should have a wavelength.

432
00:31:53,000 --> 00:31:56,000
And we do.
And just briefly,

433
00:31:56,000 --> 00:32:04,000
here, let's talk about what the
wavelength is of a baseball

434
00:32:04,000 --> 00:32:10,000
pitched by Curt Shilling at 90
mph.

435
00:32:10,000 --> 00:32:15,000
What is that wavelength?
Well, a baseball is five

436
00:32:15,000 --> 00:32:17,000
ounces.
90 mph.

437
00:32:17,000 --> 00:32:23,000
You can calculate the momentum.
It is in your notes there.

438
00:32:23,000 --> 00:32:29,000
We will calculate,
here, the wavelength.

439
00:32:29,000 --> 00:32:35,000
And what you are going to find
is that it is 1.2x10^-34 meters.

440
00:32:35,000 --> 00:32:39,000
That is pretty small.
What is the diameter of a

441
00:32:39,000 --> 00:32:42,000
nucleus?
10^-14, right.

442
00:32:42,000 --> 00:32:48,000
That is a good number to know.
Here, we have a wavelength that

443
00:32:48,000 --> 00:32:52,000
is 10^-34 meters.
Is that wavelength of a

444
00:32:52,000 --> 00:32:58,000
macroscopic size object going to
have any consequence in our

445
00:32:58,000 --> 00:33:01,000
world?
No, absolutely not.

446
00:33:01,000 --> 00:33:04,000
Why?
Because in order to see any

447
00:33:04,000 --> 00:33:10,000
effects from this small
wavelength, we are going to have

448
00:33:10,000 --> 00:33:16,000
to have slits or atoms that are
going to be on the order of this

449
00:33:16,000 --> 00:33:20,000
close together.
But there is no way that we are

450
00:33:20,000 --> 00:33:26,000
going to have two nickel atoms
that are this close together or

451
00:33:26,000 --> 00:33:33,000
two slits in a two slit
experiment this close together.

452
00:33:33,000 --> 00:33:40,000
And so for macroscopic objects,
the wavelike properties have no

453
00:33:40,000 --> 00:33:47,000
consequence in this world.
And that is simply because the

454
00:33:47,000 --> 00:33:53,000
mass is too large.
It makes the wavelength too

455
00:33:53,000 --> 00:33:59,000
small to have any effects in our
everyday lives.

456
00:33:59,000 --> 00:34:04,000
And it is actually --
Yes?

457
00:34:04,000 --> 00:34:07,000
Okay.

458
00:34:15,000 --> 00:34:19,000
Well, they are actually coming
in as waves.

459
00:34:19,000 --> 00:34:25,000
They are behaving as waves.
Remember my beach picture?

460
00:34:25,000 --> 00:34:30,000
I drew them coming in like a
circle.

461
00:34:30,000 --> 00:34:36,000
But remember my picture of this
barrier here on the beach,

462
00:34:36,000 --> 00:34:42,000
and I am laying here on the
sand, and then the waves are

463
00:34:42,000 --> 00:34:44,000
coming in?
Here is blue.

464
00:34:44,000 --> 00:34:48,000
These electrons,
as they are coming in,

465
00:34:48,000 --> 00:34:54,000
really need to be thought of as
these kind of plane waves.

466
00:34:54,000 --> 00:35:00,000
I drew them as kind of just
particles.

467
00:35:00,000 --> 00:35:05,000
But you have to really think of
them as plane waves and that

468
00:35:05,000 --> 00:35:11,000
they are reflecting off of these
two atoms in the way that I just

469
00:35:11,000 --> 00:35:15,000
explained.
Another question?

470
00:35:25,000 --> 00:35:29,000
The thing is that you cannot
get that velocity slow enough to

471
00:35:29,000 --> 00:35:33,000
make the wavelength large enough
to be of consequence.

472
00:35:33,000 --> 00:35:36,000
If you could then you would,
right?

473
00:35:36,000 --> 00:35:38,000
You would see the wave-like
behavior.

474
00:35:38,000 --> 00:35:41,000
But you cannot,
practically speaking,

475
00:35:41,000 --> 00:35:44,000
get it to that extent.

476
00:36:18,000 --> 00:36:20,000
No.
In this particular case,

477
00:36:20,000 --> 00:36:24,000
anything that is so massive,
any smaller effects,

478
00:36:24,000 --> 00:36:28,000
like what you are talking
about, exactly the point of

479
00:36:28,000 --> 00:36:33,000
observation is not going to have
an effect on anything that is so

480
00:36:33,000 --> 00:36:35,000
massive.
Pardon?

481
00:36:35,000 --> 00:36:41,000
Yes, your point of observation
will have an effect on your

482
00:36:41,000 --> 00:36:46,000
interpretation of the
experiment, if you are talking

483
00:36:46,000 --> 00:36:51,000
about something that has a much
larger wavelength.

484
00:36:51,000 --> 00:36:55,000
Absolutely.
In high energy physics

485
00:36:55,000 --> 00:37:00,000
experiments, for example.
Good question.

486
00:37:05,000 --> 00:37:14,000
It is just this observation,
here, of the wave-like behavior

487
00:37:14,000 --> 00:37:19,000
of electrons,
of particles,

488
00:37:19,000 --> 00:37:29,000
that led to the interpretation,
then, or led to the realization

489
00:37:29,000 --> 00:37:39,000
that maybe, you have to treat
electrons as waves.

490
00:37:39,000 --> 00:37:48,000
Or maybe you have to treat the
behavior of electrons as

491
00:37:48,000 --> 00:37:56,000
wave-like behavior.
And that is what this gentleman

492
00:37:56,000 --> 00:38:01,000
Schrˆdinger did.
He said, well,

493
00:38:01,000 --> 00:38:05,000
you know what?
This gave him an idea.

494
00:38:05,000 --> 00:38:10,000
Maybe what is wrong is that an
electron in an atom,

495
00:38:10,000 --> 00:38:14,000
I cannot treat that electron as
a particle.

496
00:38:14,000 --> 00:38:19,000
Instead, what I have to do is
treat is as a wave.

497
00:38:19,000 --> 00:38:23,000
I have to treat its wave-like
properties.

498
00:38:23,000 --> 00:38:28,000
And it was that impetus that
led him to write down a wave

499
00:38:28,000 --> 00:38:34,000
equation of motion.
An equation of motion for

500
00:38:34,000 --> 00:38:36,000
waves.
He realized,

501
00:38:36,000 --> 00:38:40,000
or he was guessing at the
moment, well,

502
00:38:40,000 --> 00:38:46,000
maybe in the case when a
microscopic particle has a

503
00:38:46,000 --> 00:38:53,000
wavelength that is on the order
of the size of its environment,

504
00:38:53,000 --> 00:38:57,000
in that case,
maybe the wavelength has an

505
00:38:57,000 --> 00:39:02,000
effect, makes a difference.
For example,

506
00:39:02,000 --> 00:39:07,000
in the case of the electrons,
we had a wavelength calculated,

507
00:39:07,000 --> 00:39:12,000
there, of 1.7x10^-10 meters.
That is a wavelength that is on

508
00:39:12,000 --> 00:39:16,000
the order of the size of the
environment of the electron,

509
00:39:16,000 --> 00:39:20,000
which is on the order of the
size of an atom.

510
00:39:20,000 --> 00:39:24,000
Maybe in that case I have to
pay attention to the wavelength.

511
00:39:24,000 --> 00:39:29,000
The reason you and I don't have
to pay any attention to our

512
00:39:29,000 --> 00:39:33,000
wavelength is because our
wavelength is 10^-30 meters or

513
00:39:33,000 --> 00:39:36,000
so.
And that is much,

514
00:39:36,000 --> 00:39:40,000
much larger than the size of
the environment.

515
00:39:40,000 --> 00:39:44,000
In this case,
we don't have to pay any

516
00:39:44,000 --> 00:39:49,000
attention to our wavelength.
But, for an electron in an

517
00:39:49,000 --> 00:39:51,000
atom, we have a problem,
here.

518
00:39:51,000 --> 00:39:55,000
What did Schrˆdinger do?
Schrˆdinger said,

519
00:39:55,000 --> 00:40:00,000
I have to write down a wave
equation.

520
00:40:00,000 --> 00:40:05,000
An equation of motion for
matter waves.

521
00:40:05,000 --> 00:40:10,000
And what is that equation of
motion?

522
00:40:10,000 --> 00:40:19,000
Well, that equation of motion
is H hat operating on Psi,

523
00:40:19,000 --> 00:40:25,000
and it gives us back an energy,
E, times a Psi.

524
00:40:25,000 --> 00:40:30,000
What is this?
Well, Psi, here,

525
00:40:30,000 --> 00:40:34,000
is a wave.
I am somehow going to let my

526
00:40:34,000 --> 00:40:37,000
electron in an atom be
represented by Psi.

527
00:40:37,000 --> 00:40:40,000
This is going to be a wave
form.

528
00:40:40,000 --> 00:40:43,000
This is going to be a wave
function.

529
00:40:43,000 --> 00:40:49,000
I am going to let my electron
be represented by the wave

530
00:40:49,000 --> 00:40:52,000
function.
Exactly how it is going to be

531
00:40:52,000 --> 00:40:58,000
represented by the wave function
is something that I am not going

532
00:40:58,000 --> 00:41:06,000
to tell you quite yet.
But it is going to represent

533
00:41:06,000 --> 00:41:10,000
the electron.
This energy,

534
00:41:10,000 --> 00:41:18,000
here, that energy is going to
turn out to be the binding

535
00:41:18,000 --> 00:41:23,000
energy of the electron in the
atom.

536
00:41:23,000 --> 00:41:30,000
This thing, here,
is called the Hamiltonian

537
00:41:30,000 --> 00:41:35,000
Operator.
That Hamiltonian Operator is

538
00:41:35,000 --> 00:41:41,000
specific to a particular
problem, and we will look at the

539
00:41:41,000 --> 00:41:45,000
Hamiltonian Operator for a
hydrogen atom.

540
00:41:45,000 --> 00:41:50,000
But this operator is operating
on Psi, and it gives you back a

541
00:41:50,000 --> 00:41:55,000
Psi, the same function,
multiplied by a constant.

542
00:41:55,000 --> 00:42:00,000
That constant is the binding
energy.

543
00:42:00,000 --> 00:42:04,000
Now, you think,
well, let me just cancel this

544
00:42:04,000 --> 00:42:08,000
and this.
But you cannot do that because

545
00:42:08,000 --> 00:42:13,000
this is an operator.
This has some derivatives in

546
00:42:13,000 --> 00:42:17,000
it.
Its operating on Psi gives you

547
00:42:17,000 --> 00:42:21,000
back the same function times the
constant.

548
00:42:21,000 --> 00:42:27,000
Now, how did Schrˆdinger
actually derive this equation,

549
00:42:27,000 --> 00:42:34,000
so to speak?
Well, what he did was to just

550
00:42:34,000 --> 00:42:40,000
guess at a wave function.
We are going to use a

551
00:42:40,000 --> 00:42:47,000
one-dimensional wave function.
He is going to say,

552
00:42:47,000 --> 00:42:55,000
let me represent my electron by
Psi of x equal to 2 a times

553
00:42:55,000 --> 00:43:02,000
cosine 2 pi x over lambda.

554
00:43:02,000 --> 00:43:09,000
That is going to be my wave

555
00:43:09,000 --> 00:43:11,000
function.
Why not?

556
00:43:11,000 --> 00:43:19,000
And then he said,
well, what I really need here

557
00:43:19,000 --> 00:43:27,000
is an equation of motion.
I need to know how Psi changes

558
00:43:27,000 --> 00:43:32,000
with x.
If you wanted an equation of

559
00:43:32,000 --> 00:43:37,000
motion, if you wanted to know
how Psi changes with x,

560
00:43:37,000 --> 00:43:41,000
what would you do to Psi?
Take the derivative.

561
00:43:41,000 --> 00:43:46,000
Let's take a derivative of Psi
of x, with respect to x.

562
00:43:46,000 --> 00:43:51,000
That is going to be
equal to minus 2 a

563
00:43:51,000 --> 00:43:56,000
times 2 pi over lambda times
sine of 2pi x over lambda.

564
00:44:03,000 --> 00:44:07,000
That is an equation of motion.
Now, this is actually kind of

565
00:44:07,000 --> 00:44:10,000
an equation of position.
But it is telling us how Psi

566
00:44:10,000 --> 00:44:12,000
changes with x.

567
00:44:20,000 --> 00:44:28,000
But now, since that gave us
some information about how Psi

568
00:44:28,000 --> 00:44:34,000
changes with x,
how would we get the rate of

569
00:44:34,000 --> 00:44:39,000
change of Psi with x?
Second derivative.

570
00:44:39,000 --> 00:44:43,000
Let's take the second
derivative of that.

571
00:44:43,000 --> 00:44:47,000
Second derivative of Psi of x
with respect to x,

572
00:44:47,000 --> 00:44:52,000
that is minus 2a times 4 pi
squared over lambda squared

573
00:44:52,000 --> 00:44:57,000
times cosine 2pi x
over lambda.

574
00:45:02,000 --> 00:45:07,000
That's pretty good,
but now, what do you see in

575
00:45:07,000 --> 00:45:09,000
this equation?
Psi.

576
00:45:09,000 --> 00:45:14,000
You see what I started with.
It is recursive,

577
00:45:14,000 --> 00:45:19,000
right.
You see the Psi of x here.

578
00:45:19,000 --> 00:45:23,000
Let me write that equation in

579
00:45:23,000 --> 00:45:31,000
terms of the original function.
Second derivative of Psi of x

580
00:45:31,000 --> 00:45:36,000
with respect to x,
minus 2pi over lambda squared

581
00:45:36,000 --> 00:45:41,000
times Psi of x.

582
00:45:41,000 --> 00:45:45,000
lambda)^2 Psi(x)** That is
pretty good.

583
00:45:45,000 --> 00:45:49,000
You know where I am trying to
go?

584
00:45:49,000 --> 00:45:53,000
I am trying to derive,
so to speak,

585
00:45:53,000 --> 00:46:00,000
Schrˆdinger's equation.
See, it is not very hard.

586
00:46:00,000 --> 00:46:03,000
Yes.
Now, this equation right here,

587
00:46:03,000 --> 00:46:08,000
this equation is just a
classical wave equation.

588
00:46:08,000 --> 00:46:14,000
The only thing I have done so
far is take derivatives.

589
00:46:14,000 --> 00:46:19,000
I have done nothing else.
I just took derivatives.

590
00:46:19,000 --> 00:46:23,000
It could represent any kind of
wave.

591
00:46:23,000 --> 00:46:29,000
There is nothing quantum
mechanical about it.

592
00:46:29,000 --> 00:46:36,000
But here comes the big leap
that Schrˆdinger made.

593
00:46:36,000 --> 00:46:43,000
He substituted in here for
lambda the momentum of the

594
00:46:43,000 --> 00:46:47,000
particle.
In other words,

595
00:46:47,000 --> 00:46:55,000
if this is a wave equation and
that wave has some wavelength,

596
00:46:55,000 --> 00:47:00,000
here, lambda.
He said, well,

597
00:47:00,000 --> 00:47:04,000
if this is a wave equation for
a matter wave,

598
00:47:04,000 --> 00:47:10,000
well, then I better get the
momentum of that particle in

599
00:47:10,000 --> 00:47:14,000
this wave.
And he knew how to do that

600
00:47:14,000 --> 00:47:18,000
because de Broglie told him how
to do that.

601
00:47:18,000 --> 00:47:22,000
De Broglie said,
lambda is equal to h over p.

602
00:47:22,000 --> 00:47:26,000
That is pretty good.

603
00:47:26,000 --> 00:47:32,000
What can I do here?
Well, what I can do is I can

604
00:47:32,000 --> 00:47:38,000
rearrange this and write this in
terms of the momentum of the

605
00:47:38,000 --> 00:47:42,000
particle.
Second derivative of Psi of x

606
00:47:42,000 --> 00:47:48,000
with respect to x is going to be
minus p squared over h bar

607
00:47:48,000 --> 00:47:53,000
squared times Psi of x.

608
00:47:53,000 --> 00:47:58,000
Now, let me explain
what h bar here is

609
00:47:58,000 --> 00:48:02,000
h bar is a shorthand way of

610
00:48:02,000 --> 00:48:07,000
writing h over 2pi. If you

611
00:48:07,000 --> 00:48:11,000
don't know it already,
you should know it.

612
00:48:11,000 --> 00:48:15,000
You are going to need it.
That is h over 2pi,

613
00:48:15,000 --> 00:48:19,000
so this is h bar squared.
Now, what do I have?

614
00:48:19,000 --> 00:48:24,000
Now I have a matter wave.
I have an equation of motion.

615
00:48:24,000 --> 00:48:32,000
I have something that tells me
how Psi moves with respect to X.

616
00:48:32,000 --> 00:48:35,000
The rate of change of Psi with
X.

617
00:48:35,000 --> 00:48:40,000
And I had the momentum of the
particle buried in here.

618
00:48:40,000 --> 00:48:44,000
From this form,
I am going to get to that.

619
00:48:44,115 --> 00:48:47,000
And I guess I am going to get
to that next Wednesday.