All right. So by last Friday we discovered
the electron, discovered the nucleus.
And the problem then was how to describe the structure of the
atom. They knew there was an
electron, they knew there was a nucleus, but exactly how did the
electron and the nucleus hang together?
And we talked about how the scientists at that time proposed
a classical model where the interaction between the electron
and the nucleus was, of course, the Coulomb
interaction, but where the equations of motion were
Newton's equations of motion to describe this electron going
around the nucleus, F equals ma.
And, although we didn't show it, what we said was that if we
used that force field, the Coulomb force,
coupled with Newton's equations of motion.
And, actually, you also have to couple them
with some classical electromagnetism,
which I may not have said but which isn't important.
But if you use those classical ideas to describe how that
electron goes around that nucleus, well,
you would find that in 10^-10 seconds, that electron would be
on top of the nucleus. And you would no longer have
what we know to be the structure of the atom, in the sense that
you would no longer have some volume where the diameter of
that volume is on the order of an angstrom or so.
The bottom line is that these classical ideas,
classical mechanics and classical electromagnetism just
totally fail to explain the structure of the atom.
And that was the conundrum that the scientists faced in 1911.
This was quite a big problem for the scientific community and
was so for about 10, 12, 13 years.
And, of course, what fails are the classical
equations of motion, Newton's equations.
A new kind of mechanics is going to have to be invented in
order to understand how that atom is bound or how the
electron and the nucleus are bond, how it's stable.
But now, historically, the actual clues for why the
electron didn't collapse into the nucleus.
Those clues actually came from a very different area of
discussion, and that area of discussion is this wave particle
duality of light and matter. It was certainly long believed
that matter, with its particle-like behavior,
was distinct from light, which was this kind of
transmission of energy through space.
But in the late 1800s and early 1900s, there began to be made
some observations that seemed to suggest that well,
you know, maybe this boundary between matter and its
particle-like behavior and light and its wave-like behavior,
well, maybe that boundary wasn't so rigid after all.
And, in fact, what we're going to see is that
radiation or light exhibits both wave-like behavior and
particle-like behavior. And it depends on the kind of
experiment that you do, which one of those properties
you see, and that consequently matter also exhibits both
particle-like behavior and wave-like behavior.
And, again, it depends on the particular experiment that you
do as to which one of those properties you actually observe.
What we are going to do right now is we're actually going to
set aside the discussion of the structure of the atom,
how the electron and the nucleus are bound together,
how in the world do they remain stable?
We are going to set that discussion aside,
and we are not going to pick it up until next Monday.
And the reason for it is because in order to proceed with
that discussion, we actually have to go through
this other set of observations, this discussion about the wave
particle duality of light and matter in order to get the
clues, in order for us to make progress about what next big
leap we've got to make in order to talk about the structure of
the atom. We are going to change
subjects, or it's going to look like we're changing subjects,
but we're going to use the results of our discussions in
the next three days to come back to talk about the structure of
the atom. Then the first thing we've got
to talk about is we're going to talk about light and we're going
to talk about the wave-like nature to radiation.
And since that is really probably most intuitive,
let's do that.
Properties of waves. Well, what are waves?
Well, you are familiar with that.
Waves are a periodic variation of some quantity.
In the case of water waves, we have the periodic variation
of the height of water. At some points in space here,
the height of the water is high, at some point the height
of the water is low, and there is some average level
here. A sound wave.
Well, a sound wave is a periodic variation of the
density of air. At some points in space,
the air is very dense, at other points in space the
air is not so dense, and then there is an average
level here. Well, what we're going to do is
we're going to set this average level here to a zero.
And then, consequently, our waves are going to have a
positive amplitude that is measured from this zero level
up. That's a positive amplitude.
And measured from zero level down, that's our negative
amplitude. These waves are going to have
positive and negative amplitude. That is going to be very
important in how they actually behave.
Now, what we're going to do, oh, so what is light?
Well, light is a periodic variation of an electric field.
And that is what I plot here. Electric field versus distance.
And you can see, it is varying in intensity.
But what exactly is an electric field?
Well, let's just think about that for a moment.
Suppose I take two parallel plates, here is one and here is
another, and I am going to put a negative potential on that top
plate and a positive potential on the bottom plate.
And what is operating then between this negatively charged
plate and the positively charged plate is the Coulomb
interaction. And the electric field then is
the space through which that Coulomb force operates.
So, that electric field here is this space through which this
Coulomb force is operating. And, by convention,
we are going to say that it is operating here in this
orientation. In other words,
if I put a positive charge in between these two plates,
in this electric field, this space through which the
Coulomb force is operating, well, what's going to happen is
that this positive charge is going to drift to the negatively
charged plate. But if I turn the geometry
around, if I put the negatively charged plate on the bottom
using this coordinate system. And I now put that positive
charge in that electric field, it's going to drift to the
negatively charged plate again, but it's going to move in this
coordinate system down. So, this electric field has not
only magnitude because you can imagine, the larger the
potential difference between these two plates,
the larger the magnitude of the electric field.
But that electric field also has direction.
It is pointed in one way or the other.
And that is what represented here on my diagram of the
electric field versus distance. You can see that right here,
the electric field, the magnitude of it starts out
small, it increases, continues to increase until we
get a maximum, and then it starts getting
smaller. It gets smaller and goes all
the way to zero, at which point then it turns
around and the magnitude begins to increase in the opposite
direction. And, again, it becomes a
maximum in the opposite direction and then turns around
and goes to zero and continues. So, it's a periodic variation
here of an electric field. And so what that means is that
if we put some charge, say, at this point in space,
it will be pulled one way. And if we put it at this point
in space, it will be pulled yet another way.
And if we put it right here at zero, it's not going anywhere.
Now, what I've shown you here is a periodic variation of an
electric field. It is also the case that
whenever we have an electric field, we also have a magnetic
field that is actually perpendicular to this electric
field here. But since that magnetic field
is going to have all the same characteristics as the electric
field, from our point of discussion here,
I am not going to talk about it, but you are going to see
this a lot next semester in 8.02.
So, we are just going to characterize here the electric
field. And the way we're going to
characterize it is to put a functional form to it.
I am plotting the electric field here as a function of x.
And the functional form that we are going to use to characterize
it is A times cosine (2 pi x over lambda),
a cosine function. Now, what I want you to notice
is that this field oscillates between a positive amplitude A
This coefficient right out in front, right here,
that coefficient physically is the maximum amplitude.
So, if you were to be given a waveform that has a number out
here in front of a cosine x, that number,
immediately the physical significance of it is that it's
the maximum amplitude. The other parameter besides the
amplitude here, the maximum amplitude that we
use to characterize this cosine function, is the wavelength.
The wavelength is the distance between two successive maxima or
two successive minima. And that wavelength here comes
in the argument of that cosine function.
Again, if you were to be given a form that had a number times
cosine times a number times x,
well, that number is equal to 2 pi over lambda.
And so almost by inspection,
you can figure out what the wavelength is just from the
functional form of that cosine function.
And because the argument is 2 pi x over lambda,
well, you can easily see that you get
the maximum amplitude A here whenever x is an integral
multiple of lambda. When x is lambda,
2 lambda, 3 lambda or minus lambda, minus 2 lambda,
or zero. Most of that,
I think, you're familiar with. But not only is light,
or formally, electromagnetic radiation,
simply a periodic variation of the electric or magnetic field,
but light is also a period variation in time.
That is, light propagates, it moves.
For example, here is a cosine functional
form at some snapshot in time, t equals 0.
And I put a star here on the maximum of one of these maxima.
Now, what happens is that this wave moves.
It propagates. And the time it takes that wave
to move one wavelength, just like it did in that very
short clip, -- -- the time it takes to move
one wavelength so that this maxima that was here is now
there, is what we call a period. One period is the time it takes
the wave to move one wavelength. And we can also characterize
the period by the frequency of the wave.
That frequency is nu, here, and the period is 1 over.
And the frequency,
then, is the number of cycles per second.
In other words, if I were sitting right here at
x equals 0, and I am just watching the wave
come by me, the frequency is the number of times that a maxima
passes by me per second or the number of times a minima passes
by me per second. That is the frequency.
I will make that a little bit more obvious in just a moment.
Frequency, I said, is the number of cycles per
second. But the unit that we use to
convey frequency is something called the Hertz,
Hz. And formally it is just inverse
seconds. We drop the number of cycles,
but the number of cycles in that unit is implied.
Now suppose the following. I said that this electric field
has magnitude and it also has direction.
Suppose I take an electric field or an electromagnetic
radiation that has a frequency of 1 Hz, one cycle per second.
And right here at x equals zero, somehow I pin an
electron right there. It cannot move anywhere.
It's just stuck. What is going to happen to that
electron as this radiation of one Hertz passes by?
Well, what's going to happen is that that electron is going to
be now, since it's starting at the top, it is going to be
pulled down. And then it is going to be
pushed back up to its original position every one second
because this frequency here is one Hertz and because
this electric field has both magnitude and it has direction.
Now, we're going to let this electric field go by,
the electron gets pushed down and pulled back up once every
second. Hopefully that makes the idea
of frequency a little bit more real to you.
Not only do we have this periodic variation in time,
but we also have the wave propagate here in space.
That wave moves this way or this way or this way.
And so what we want to know is the speed of that propagation.
Now, speed is always distance traveled over the time elapsed.
And what we have already seen is that this wave will move by
one wavelength in the time period of a period.
That is, we already saw that this wave will move here one
wavelength, from here to here, in the time period of a period
or 1 over nu. That's the time elapsed.
So the speed of this propagation, the speed with
which it's moving, is the distance traveled,
which is lambda divided by the time elapsed,
which is 1 over nu.
And, therefore, the speed of that propagation
is lambda times nu. These are the important
concepts here. We have the speed of the
propagation, lambda times nu, we characterized our wave in
terms of amplitudes, in terms of wavelengths and in
terms of frequency. Those are the quantities that
are important that you really do have to understand very well.
Now, I think that you are all familiar with the fact that all
electromagnetic radiation has a constant speed.
That is, the wavelength of electromagnetic radiation times
its frequency is always equal to this number.
It's about 3x10^8 meters per second.
And we abbreviate it c, the speed of light.
No matter what frequency you have of radiation,
the corresponding wavelength is such that the wavelength times
the frequency is this constant, is this value of c.
And, of course, it is the different frequencies
or the different wavelengths of electromagnetic radiation that
gives it its different characters, --
-- even though it all has the same speed of propagation.
And those different kinds of properties, so to speak,
are this spectrum of electromagnetic radiation.
So electromagnetic radiation is an infinitely wide spectrum.
At very long wavelengths, we have what are known as radio
waves. At very short wavelengths,
we have what are known as gamma rays or cosmic rays.
The long wavelengths, the radio waves,
are going to have a relatively low frequency.
The short wavelengths are going to have a relatively high
frequency. Why?
Because the product of lambda times nu always has to be c.
Well, we're not going to go
very much in detail here, except maybe to tell you the
following. See this microwave range here?
Well, molecules, when they absorb microwave
radiation, they rotate. See the infrared radiation?
When molecules absorb infrared radiation they vibrate.
See the visible and ultraviolet range?
When molecules absorb visible or UV radiation,
they put an electron into a higher energy state.
And oftentimes, once you do that,
of course, the electron wants to relax to a lower energy
state, and so it relaxes by emitting a photon.
And that's how you see fluorescence or phosphorescence.
But it's usually a result of the absorption of visible or UV
radiation. Although, it is also possible
for a molecule to absorb some X-ray radiation and see that
kind of fluorescence, too.
And we will talk about it. But a little bit later on we're
going to talk about molecules absorbing X-ray radiation and
kicking out an electron. And we are going to use that to
figure out how strongly bound electrons are in atoms or how
strongly bound they are in molecules.
But that is all I will say about that right now.
What you are responsible for is, --
-- you are responsible in knowing that microwaves have
longer wavelengths than gamma rays or cosmic rays.
You are responsible for knowing the relative wavelengths here of
these different kinds of radiation.
Every MIT student should know that.
Of course, there is a very narrow range here of visible
radiation that our eye is sensitive to.
At the very long wavelengths is red.
At the very short wavelengths is blue and purple.
Again, you need to know that red has a longer wavelength than
blue or purple. What I have done right now is
really just to tell you that light is an electromagnetic
radiation characterized at the wave.
I put a functional form to it, cosine, characterized it by
frequency, wavelength. We talked about speed of
propagation. What I haven't done yet is I
haven't proved to you that light is, in fact, or light behaves
like a wave. And that is what we are going
to do now. And the experiment we are going
to do is the same experiment that a gentleman called Young
did in the late 1800s that seemed to really demonstrate,
without a question, that light is a wave,
or light behaves like a wave. We are going to do what's
called the two-slit experiment. And in this experiment,
we are going to take radiation of a single wavelength,
so we're going to make the radiation monochromatic.
And that wavelength that we are going to use is 6,330 angstroms
or 633 nanometers. And we are going to put in
front of that source, we are going to put a thin
piece of metal, or it could be any material,
but we are going to poke two holes in it.
We're going to make two little slits.
And naively, what you might expect is,
if you have a screen back here that kind of images this
radiation -- What you might expect is that
the spots on the screen right directly opposite these two
slits, well, these are the two spots that are going to light up
on the screen. That is where you're going to
see the image because, after all, light travels in
straight lines. That's where you should see
those two spots. If the slits here are 0.02
inches from each other, these two spots ought to be
0.02 inches separated from each other.
But, of course, when you do that experiment,
and Christine is actually going to do that experiment right now
for us, I think, there it is,
on the side board. You can sort of see it.
She is holding it. Well, what you can see here is
that we don't see just two bright spots.
But what we see is a bunch of spots here.
The brightest one happens to be right in the center of these two
slits. And then there is darkness and
then there is a bright spot and darkness, and a bright spot,
etc. We see lots of different bright
spots on the screen, not just two directly opposite
the openings of the slits. And this is just then a cartoon
of approximately what it is we see in real life.
Christine is up there in the projection booth,
holding a helium-neon laser behind these two slits separated
by 0.02 inches. Well, what's going on here?
Can we understand what the origin is of these lots of
different bright spots in this two-slit experiment?
Actually, if we were to measure that, the distance here should
be about 3 cm between those bright spots.
Well, what's going on? Well, what's going on is this.
Waves have the property called superposition.
That is, if I take one wave, but then I take a second wave
and I put that second wave in the same space,
the actual same space as the first wave, and I arrange the
two waves such that the maximum of both waves or the minimum of
both waves is at the same position, well,
the property of waves is that these two waves add up.
They superimpose. They superimpose to make a wave
that now has the amplitude of twice the amplitude of each of
those waves. And that is the condition that
we call constructive interference,
but we have to have the two waves at the same place in space
and the two maxima have to be in the exact same place in space.
But conversely, we can also take the wave and
bring in another wave, but we can move that other wave
over such that the maximum of one of the waves is at the same
place in space as the minimum of the other wave.
In this case, these two waves are out of
phase. And when we add them up,
there will be nothing. This is destructive
interference. Why?
Well, because we have positive amplitudes and negative
amplitudes. They cancel to give us nothing.
Destructive interference. And there is a website here at
the University of Colorado that lets you take two waves and
superimpose them. I need to play with it.
What I am going to do is I am going to make it so that these
two waves have the same frequency or the same wavelength
and the same amplitude. What I am doing here is I am
adding this wave, I am adding this wave right
here to this wave, and the result is shown here.
And what you cannot see is that the amplitude down here,
that amplitude is twice that of these two.
There is a two there. But, of course,
what I can do is I can move this wave over such that the two
waves are now at the same place in space, sort of,
where the maximum and the minima are at the same place in
space and exactly cancel them. And I am not having such a
great time doing this here, but I know you can do it
because I have done it before. And you can see that the
resulting amplitude is zero. And, of course,
then you can go home and you can play all you want with this,
make all kinds of different waveforms.
I am trying to change the amplitude here,
which I have done before, but I cannot right now.
You can do whatever waveform you want to.
You get the idea here with superposition.
So, that's a characteristic of waves.
They have positive and negative amplitudes.
They add and they cancel depending on the relative
positions in space. But now, let's talk a little
bit more about this pattern that we saw in the two-slit
experiment. This kind of row of bright
spots. And, to talk about it,
what we are going to do is we're going to use water waves
to try to explain it. Here is a picture of the beach.
Here is the water, here is the sand,
and here is where I wanted to be all weekend.
And the waves are coming in to the beach.
These straight blue lines are the crests of the waves rolling
into the beach. And suppose I get ambitious and
I erect a barrier right here at the shore, a sand barrier,
but for whatever perverse reason I poke two holes in that
sand barrier. Well, what is going to happen
as the waves roll up? Well, what is going to happen
is that a little bit of the wave here, this is kind of like a
pinhole, is going to leak through.
When it leaks through, it is going to spread out
equally in all directions. And so the wave front is going
to look like this semicircle here.
That is the wave crest. It is going to look like a
semicircle. And that is going to propagate
out. And when it propagates out to a
distance of 2 lambda, well, then another wave has
snuck in here through this hole and it propagates out.
Again, the distance between these two semicircles is a
lambda, is the wavelength, because these are the maxima of
the two waves. And, as time goes on,
that propagates further and another wave sneaks through that
hole and another wave sneaks through and the whole thing
propagates. Meanwhile, while that is
happening, there is also a part of the wave sneaking through
this hole and it emanates out in all directions and propagates
out in all directions. And let me, for the purpose of
talking about this, call this slit slit 1.
All the green waves are the waves that came through slit 1.
And I am going to call this slit 2.
And all the blue waves are the waves that came out slit 2.
And, again, the distance between any of these two crests
is the wavelength. And the wavelength is the same
for the waves coming from slit 1 and slit 2.
Now what I want you to do is I want you look at this point
right here. See that point?
Well, right at that point, what we have is we've got the
wave crest, the wave maximum from slit 2 at the same point in
space as the wave maxima from slit 1.
If we've got two maxima at the same point in space,
hey, we've got constructive interference right there,
because those amplitudes are going to add.
Look at this point. Well, again,
we've got the wave maxima from this blue wave coming through
slit 2 at the same point in space as the wave maxima from
the wave coming from slit 1. Again, this is a point of
constructive interference. And we can keep going.
Again, we have two maxima right here, a point of constructive
interference. And, you know what,
everywhere along this line we are going to have constructive
interference. We are going to have a
resultant wave, a water wave that is going to
be twice as high as each of the individual waves.
Look at this. What I want you to notice here,
for this point: for this point,
the wave that is constructively interfering, that has traveled
from the slit a distance r two, because this is slit 2.
But the distance from here to here is 4 lambda.
It is 1 lambda, 2 lambda, 3 lambda,
4 lambda, right? The radius r two of this
circle is 4 lambda. And it is interfering with this
wave crest. And the distance that that wave
traveled from slit 1, I am going to call r one,
and that is equal to 3 lambda, 1, 2,
3. This interference here is
between this wave that has traveled out a distance,
4 lambda, and this wave that has traveled out a distance,
3 lambda. And the difference in the
distance traveled by those two waves that are constructively
interfering is 4 lambda minus 3 lambda equals 1 lambda.
Is there a question over here? Yes?
I think I am coming to that. That was a snapshot of the next
slide. Sorry.
Yes? Because we are treating these
as regular waves, ordered waves.
Real waves on an ocean are not like that.
And then, you won't see this diffraction pattern that I am
erecting here. But, if you had pretty regular
waves coming in, in principle you could see it.
And I am going to show you, in a tank, some water waves
that do just this. Now I want you to look at this
point right here. Right here we have a point of
constructive interference because we have two maxima at
the same point in space. And at this point we've got
constructive interference. So everywhere along this line
here, we are going to have constructive interference.
And now I want you to look at the difference in the distance
of the waves that yield this constructive interference along
this line. I want you to look at this
green wave that is constructively interfering with
this blue wave. This green wave has traveled
out a distance r one or 2 lambda, and it is interfering
with this blue wave here that has traveled out a distance 4
lambda, r two equals 4 lambda.
So, the difference in the distance traveled for all of
these points that lead to constructive interference,
the difference in the distance traveled here is r two minus r
one or 4 lambda minus 2 lambda equals 2 lambda.
Does that agree with the snapshot in your picture?
Yes. I just didn't show you the
other one. Now I want you to look here.
Here is a point of constructive interference.
Here is another point of constructive interference.
Everywhere along this line we are going to have constructive
interference. And, if we check out what the
difference in the distance traveled is between the waves
that are constructively interfering there,
you are going to see that r two minus r one is equal to 0.
Now, what you would predict is
we would have along this angle some very bright spots,
along this angle some very bright spots.
And along this angle some very bright spots.
And you also can see that this is going to be symmetric about
the center, so you would have bright spots here,
bright spots there. Well, let's look at this in
real life, in a water tank, where we have plain waves
coming into a barrier and then we have two little slits right
there and right there. And what you can see is that
these bright semicircles are, in fact, the crest of the
waves. But what I want you to see is
right here -- What I want you to see,
is right here: These are this bright row,
here is a bright row, here is a bright row.
And in between there is kind of just a muddy diffuse pattern.
Well, this diffuse pattern is nothing.
This is a bright row. You get the idea.
You are going to have an angle which there is going to be
constructive interference, several angles of which there
is going to be constructive interference.
And in between those places where there is constructive
interference, there is destructive
interference, and you are not going to have
any light. How does that relate to what we
saw in the two-slit diffraction experiment?
Along this line we have waves that have constructively
interfered, and they constructively interfere so that
on the screen right there you have a bright spot.
And along this line, the difference in the distance
traveled by the waves that constructively interfere is 1
lambda. And then we've got another row
or another line, points of constructive
interference. That is going to image as a
bright spot. And if you analyze that,
the difference in the distance traveled by these two waves is 2
lambda. And then we're going to have
another bright spot here. For all the constructive
interference along this line, the difference in the distance
traveled is zero. And, of course,
I said before this was symmetric around the center.
And so this is what gives rise to the different bright spots on
the screen when we did the two-slit experiment.
Bright spot. Dark spot.
The bright spots are constructive interference.
The dark spots here are destructive interference.
Now, let me just tell you that the general condition for having
these bright spots -- The general condition that has
to obtain is that the difference in the distance traveled has to
be equal to N, an integer, times lambda.
This bright spot, the difference in the distance
traveled, N is 0. We call that the zero order
spot. This bright spot,
the difference in the distance traveled is lambda,
or N times lambda, where N is 1,
first order interference feature.
Here, the difference in the distance traveled is 2 lambda,
N times 2, called the second order interference features.
And the condition for that is this, r2 minus r1 equals N times
lambda. There is
also a condition for destructive interference,
which I ask you to think about, and we will talk just briefly
about its origin next time. All right.
See you on Wednesday.