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Where were we, last time?
Last time, we said we were setting aside the problem of the
structure of the atom. We were setting it aside
because we were stuck, and what we had to do was to
look at another area of discussion.
And that is this wave-particle duality of light and matter,
because it is that discussion that is going to give us the
clues about how to proceed. We are putting aside the
discussion of the structure of the atom all the way until next
Wednesday, because next Monday, I was reminded,
is a student holiday. [APPLAUSE]
And so we started. We talked about the wave-like
properties of light. We said that the property of
superposition, the fact that you can put waves
at the same point in space and their amplitudes add.
Since waves have both positive and negative amplitude,
that means that you have constructive and destructive
interference. And it is those interference
phenomena, then, that are evidence for wave-like
property. And we did the two-slit
experiment to try to give you an example of interference
phenomena. We were trying to understand
those interference phenomena, and did so in terms of this
diagram here. The interference phenomena that
we saw was an array, actually a row of bright spots,
dark spots, bright spots, dark spots.
And we drew these semicircles here around each one of the
slits. They represent a little bit of
the wave that emanated through those slits.
Because those slits are small, then, those waves emanated
equally in all directions. That is why these semicircles
represent the wave maxima. And what we discovered is that
if we looked along this line here, this line which led to
this bright spot, that all of the waves along
this line were constructively interfering.
That is, we had the maximum of the two waves at the same point
in space, or the minimum of the two waves at the same point in
space. And we noticed that everywhere
along this line, where we had that constructive
interference, the difference in the distance
of the waves traveled was one lambda.
We noticed, here, that everywhere along this line
that led to a very bright spot, we had constructive
interference. And the difference in distance
traveled by those two waves was two lambda.
And, likewise, up here, along this line,
the difference in the distance traveled was zero lambda.
And from just that, in a sense, qualitative
observation, we drew a conclusion.
And that conclusion was in order to get maximum
constructive interference, a condition that had obtain,
is that the difference in the distance traveled by the two
waves has to be an integral multiple of the wavelength.
N could be 0, 1, 2, 3, etc.
And so, the very bright spot here that always bisects the two
slits, since N is 0, we call that the zero-order
interference feature, or the zero-order fraction
feature. The bright spot that is either
to the left or to the right or up or down of that center bright
spot is the first-order diffraction feature or the
first-order interference feature.
And then the second bright spot up or down from the center is a
second-order diffraction feature.
And so on and so on. And in recitation section,
what you also should have done, is you should have reasoned
through what the condition was for destructive interference.
That is in between these bright spots, you have dark spots as a
result of destructive interference.
And that results because, say, right here you get a dark
spot -- Right here at that point you
would see the maximum of one wave at the same point in space
as the minimum of the other, and so they exactly cancel.
And so by just analyzing what waves were destructively
interfering, you should have come up with a general
expression for destructive interference,
of N plus one-half, that quanity,
times lambda. That is your general condition
for destructive interference, the difference in the distance
traveled by the two waves. This kind of diagram here we
are also going to see on Friday, because this interference
phenomenon is the property associated with waves.
And what we are going to do is see this diagram again when we
scatter electrons for particles. We are going to see that
particles, also, will destructively and
constructively interfere. They also have wave-like
properties. That is what we will do on
Friday. We have established,
now, the wave-like properties of radiation,
so I would like to move on and talk about the evidence for the
particle-like nature of radiation.
The evidence for the particle-like nature of
radiation comes from an effect called the photoelectric effect.
Shortly after Thompson discovered the electron,
scientists were noticing that if you took a metal and shined
radiation on that metal, that indeed electrons were
emitted. Electrons came off.
These were called photoelectrons.
However, the radiation that you shined on the metal had to have
a frequency nu that was greater than or equal to some threshold
frequency, nu nought. That is, if you took radiation
of some frequency here, nu, that was less than this
threshold frequency, well, you did not get any
electrons off. Well, another way to kind of
understand that data or that effect is to just plot the
number of the electrons that come off as a function of the
frequency of the radiation. And so at low frequency,
there are no electrons, but all of a sudden you get to
nu nought, and then electrons start coming off.
And no matter how high you increase the frequency here,
the number of electrons that come off remains the same,
remains constant. And it turned out that for the
metals that were looked at, at that time,
the threshold frequency, here, was in the UV range of
the electromagnetic spectrum. Well, in addition to just
measuring the number of electrons and generally
observing this effect, scientists did not understand
what was going on. So they just started measuring
everything they could think about measuring.
And one quantity that they measured was the kinetic energy
of these electrons that were being emitted.
And so you take kinetic energy here, KE, as a function of the
frequency. They found that at low
frequency, again, there is no kinetic energy,
because there are no electrons. And then at some frequency,
all of a sudden electrons started coming off.
And the kinetic energy of those electrons seemed to increase
with the frequency once past that threshold frequency,
nu nought. Well, this was really another
one of these conundrums, here, at that time,
because classical physics, classical electromagnetism,
classical physics had no way of explaining these data.
And, in fact, what classical physics
predicted is that the kinetic energy of these electrons,
that that kinetic energy should have nothing to do with the
frequency of the light. That is, the kinetic energy was
constant. As you increased the frequency
of the light, classical physics would tell
you that the kinetic energy should not be affected by the
frequency of the light. There was nothing in the
classical way of thinking, classical electromagnetism that
connected the frequency of the light to the kind of energy,
to the kinetic energy. There was no way for blue light
to make the electrons have a larger kinetic energy,
and for blue light to have an effect on the kinetic energy,
and red light to not have an effect on the kinetic energy.
In addition, what classical physics
predicted is that the kinetic energy of the electrons should
be dependent on the intensity of the light.
That is, the more and more intense the radiation on the
metal, the more and more kinetic energy those electrons should
have. Because, after all,
if you increase the intensity of the light going into the
metal, you are putting more and more energy into it.
That should be reflected in just how energetic those
electrons were picked out. The more energy in,
the electrons ought to come out with larger and larger kinetic
energy. Of course, the observation was
that the kinetic energy of the electrons had nothing to do with
the intensity of light. The kinetic energy of the
electrons did not increase as you made the light brighter and
brighter. As you put more and more energy
in, the kinetic energy remained the same.
It didn't have an effect. This was a real conundrum here.
The known classical physics was making predictions really just
contrary to what was being observed.
These data, here, of the kinetic energy versus
the frequency, were around for a few years
before Einstein took a look at them in 1905,
a hundred years ago. And he looked at these data for
many different metals. Here is some data for metal A,
for example, and here is some data for metal
B. And, in both cases,
it certainly looked like the kinetic energy was linearly
dependent on the frequency of the radiation.
But what was different for the two different metals was the
threshold frequency here, nu nought.
What Einstein did was, he went and fitted a straight
line to these data, y equals mx plus b.
And when he did that,
and he went to calculate the slope here, m,
of these lines, he thought "very interesting."
Because the slope of those lines was 6.626x10^-34 joule
seconds. The slope of those lines was
something called Planck's constant, h.
You say so what? Well, the reason he was so
interested in this is because just a few years earlier,
there was a scientist by the name of Max Planck who was
interested in understanding what was called the black-body
radiation data. What is a black-body?
Well, let's just, for simplicity purposes,
think of the black-body as the burner on an electric stove.
What you know is that if you turn up the voltage on that
burner, the burner gets hot. It increased in temperature.
And you increase the temperature, and sooner or
later, that burner is glowing bright red.
And you increase the temperature some more,
and the burner is glowing a brighter red.
And you increase it some more, and it is glowing orange.
And you increase it some more, which you shouldn't do,
and it's glowing yellow. And then if you could increase
it some more, it's glowing white.
What is happening, as you increase the
temperature, is that the radiation from that black-body,
this is the black-body radiation, is increasing in
intensity. But, more importantly,
the frequency is getting larger and larger.
Dark red, bright red, orange, yellow,
white, those are all frequencies.
The frequency is shifting to higher and higher values.
And what was actually done at that time is that the intensity
of that black-body radiation, oh, and this material is not in
your notes, because you are not responsible for it.
I am just trying to make this surprise that Einstein noticed
about the slope. I am just trying to put it in
some context, why he was so surprised and
amazed at it. This black-body intensity here,
people had dispersed that radiation and looked at the
frequencies that were coming off.
This is intensity versus frequency.
Here is a general shape of that intensity versus frequency.
That was observed for some temperature T one.
That was a low temperature. And then, when the temperature
was increased and the intensity versus frequency was observed,
well, the frequencies generally got higher.
This is higher temperature. Intensity goes up.
And you increase the temperature some more,
and you get even higher frequency.
T three is the highest temperature.
And that is what the data were. What Planck was trying to do
was to understand the origin of that black-body radiation.
What he said was that in these black-bodies,
in these materials, what there must be are
oscillators that are giving off this radiation.
But he had another little kick to these oscillators.
These oscillators were giving off radiation or energy in
chunks, in quanta, in particles.
And using that idea, plus some statistical
mechanics, he was able to calculate the shapes of these
curves. That is, he indeed got,
for the lowest temperature here, a curve that looked like
this. And for T two,
he got a curve that looked like this.
And for T three, he got a curve that looked like
this. He got the shape right,
but he wanted to get the intensity right,
too. What he realized he had to do
was he essentially needed to have a scaling factor,
actually, in front of his frequencies of his oscillators.
He wanted a constant. And so he said,
what constant do I have to have in order to make all of these
data fit the observation? That constant is Planck's
constant, his own 6.626x10^-34 joule seconds.
There it is. That is Planck's constant.
That is it. There is nothing deeper here.
It is a natural constant. It comes from our observations
of the world, of nature.
That is it. It is a fitting constant.
And so that is why Einstein was so amazed, here,
when he realized this number is the same as what comes out of
here. There must be something very
fundamental about this h, this Planck's constant.
That is the story. Isn't that amazing?
What Einstein then proceeded to do, of course,
is to write down the equation of the straight line that he
just put through these data. And so that equation is the
kinetic energy equal to nu, which is our x,
h, which is the slope. And then what he found was that
the intercept here, of course, is minus h times nu
nought. This is plus minus h times nu
nought. And that is the equation of a
straight line. Of course, he realized,
if this is energy on this side, boy, there better be energy on
this side. And so this h times nu
better be an energy. And since this nu was the
frequency of the incident radiation, therefore,
h times nu better be the energy of the incident radiation,
E sub i. That is where this expression,
energy equals h times nu comes from.
Nothing more. That is where it comes from,
the photoelectric effect. This was the first time that
there was any relationship between the frequency of the
radiation and the energy of the radiation.
In classical electromagnetism, there is no relationship
between the frequency and the energy.
This was the first time in which that relationship was
observed. And what this is saying is the
following. This is saying that you can
have any frequency of radiation you want, but the corresponding
energy comes in these chunks of h times nu.
h is a quantization constant. Radiation, nu,
is continuous, but the energy that corresponds
to any given frequency of radiation is h times nu.
This E equals h times nu, here, is thought of as a
quantum of energy. A particle of energy.
A chunk of energy. Later on, it became the photon,
the energy of a photon. If this h times nu here
was an energy, well, this h times nu nought
also better be an energy.
It is the threshold energy. Let's draw an energy level
diagram to try to understand that.
I am going to plot an energy here.
Let's draw an energy level for an electron.
This is an electron, here, bound to the metal.
And we know that it takes some energy to rip this electron off
of the metal. Up here, at higher energy,
is going to be our free electron.
I will just call it electron-free,
not bound to the metal anymore. And the energy that is required
to pull the electron off, from the bound state to its
free state, is this threshold energy, h times nu nought.
This threshold energy is like
an ionization potential. You know what the ionization
potential is. Just the energy required to
pull an electron off an atom or a molecule.
However, when we are pulling an electron off a chunk of a metal,
we actually have another name for it.
It is called the work function. That is just historical,
but it is the same thing as an ionization potential.
And we often give it the symbol phi.
That threshold energy, ionization energy for a metal,
is the work function, here, h times nu,
phi. The important point here is
this. In order to get an electron off
of the metal, what you have to have is
energy, E sub i. That energy E sub i has to be
equal to at least the threshold energy, h times nu nought.
If you come in with energy of
this radiation, of this wavelength and that
frequency, you pulled the electron off,
and then the electron is off the metal and it just kind of
stays there. It doesn't move away.
But you can also come in with energy here, E sub i,
that is greater than this work function, than the threshold
energy. And you can pull the electron
off. But then, the electron is
actually going to move away from the metal, and the energy with
which it moves away from the metal, its kinetic energy,
is just the incident energy minus this threshold energy.
It is the excess energy here. And I am going to write it as
the kinetic energy. From that energy level diagram,
this energy, the incident energy,
has to be equal to the work function, h nu nought,
plus the kinetic energy.
Or, if I turned this around, the kinetic energy is equal to
the incident energy minus h nu nought.
The actual expression for the energy that Einstein found,
I just turned that equation around.
This is an equation that you have to know.
I will not give this to you on an exam.
Now, you don't have to memorize it.
You just have to reason it. Draw yourself an energy
diagram. You know conservation of
energy. The sum of these two energies
has to equal the incident energy.
Then you will be all set. Now, what is very important
here is the following. If I come in with radiation
that is, say, a half of the threshold energy.
Suppose I come in with two photons, where each photon is
one half h nu --
The bottom line is that you are not going to get an electron
off. Even though you are coming in
with two photons, which together are going to
give you the threshold energy, you won't get a photon off.
You have to come in with at least the energy of the work
function. A photon has to have at least
this energy to get an electron off.
That is the particle-like nature of radiation.
Energy comes in chunks, in particles of energy,
in quanta of energy. Likewise, if I came in here
with a photon that had twice the energy of the work function or
the threshold energy, I would still only get one
electron off. I would not get two electrons
off, even though energetically you would be able,
in principle, to get two electrons off.
But you won't. You will only get one electron
off. Whenever you send a photon in,
if it has enough energy, that is if its energy is equal
to or greater than the work function, you will get an
electron off. There is one electron for every
photon. You never get two electrons for
every photon, or you can never get one
electron for two photons that are lower energy.
That is the particle quantum nature of radiation.
That is important. Yes?
What is the form of the photon? What do you mean by form?
Oh, you want a picture of the photon.
You're looking at them. You cannot draw a picture of a
photon because you want to relate it to something that is
within your classical experience.
And you cannot do that. It isn't a classical particle.
That is what you're working with right here.
You are trying to use your experiences, that are everyday
experiences, to explain something that isn't within your
everyday experiences. You don't have a frame or a
format to do that. Yeah?
Yes. If you have a constant flux of
photons onto the surface, you will have a constant flux
of electrons. Now, there is a probability.
It is not necessarily the case that every photon gets in,
will eject electrons, because there are other kinds
of competing processes. Whatever the rate with which
the photons come in. It depends on the flux of the
photons. And we will have some problems
like that, where we are going to assume that the probability of
the electron coming off is going to be one, so one electron for
every photon. But, in reality,
there are competing processes. Are all electrons being
ejected? Actually, some electrons.
Again, this goes to the probability.
Some are actually kind of going in, too, into deeper the metal.
Not the probability, right.
The rate at which the electrons come out with is dependent on
the intensity. The more photons you send in,
the larger the number of photons per second coming in,
the larger the number of electrons per second coming out.
This is a plot, here, of the energy of the
electrons. That's all right.
Other questions? Yeah?
Eventually. Yes.
Absolutely. There are other problems that
will come in. Usually, your light source
isn't so energetic that you could possibly do that.
Now, also usually what happens is that you've got your metal
grounded, so that as you lose electrons, new electrons come in
and fill it up to the fermi level.
And so you don't charge up your sample.
In an experiment, if you had your metal just not
grounded and you did shine some radiation, what would happen is
the metal would start to charge up.
Then, that would make it difficult to get electrons off.
Yes?
Yes. How strongly those electrons
are bound to the metal depends on the electronic structure of
the metal. And we are going to talk a
little bit about what determines the strength of the interaction
for electrons on atoms and molecules, but it is similar to
what it is for metals. That is coming in a few days.
Yes?
Yes, there is. Right.
You don't actually have to have a metal.
You could do it. There are usually higher
frequencies on insulators and semiconductors,
right. It is harder to see the effect,
but it can be done and has been done.
Well, what I want to do right now is to show you an experiment
we are going to do. We are going to do a
photoelectron experiment.
And the experiment is this. We have a device up here.
What we have is an aluminum plate.
And that aluminum plate is mounted on this blue metal rod.
And in the middle of this rod is a needle on a pivot.
And this is a fairly frictionless pivot.
This black ring, here, is just a support
structure, an insulating support structure.
What we are going to do is put some excess charge on this
aluminum plate. And that excess charge is going
to run down this metal rod and then onto this needle.
And because that excess charge, the electrons on the needle and
the electrons on the metal rod are repulsive,
since this is rather frictionless,
that needle is going to move because of the repulsive
interactions between these electrons.
What we're then going to do is try to do the photoelectron
experiment. We are going to take some UV
radiation and shine it on this metal and drive the electrons
off. And we should see,
then, the needle swing back to its original position.
I need a couple of volunteers in order to do this experiment
here. Come on up.
All right. Fantastic.
I think that is all right. Good.
Okay. One of you needs to be the
charger, and the other needs to be the discharger.
Which one? You want to discharge?
Discharge, okay. You come over here.
And what you are going to do is discharge the aluminum plate
after we get some excess charge on it.
You are going to do it just by holding it up to here.
You have to get it kind of close because it is not a very
intense UV source. Could you get the video cam on
the side or on the center to get where we are putting it?
I guess we are putting it on the side, right?
Okay. There we go.
There is the device. You are the charger.
What we are going to do, to get the excess charge,
we are taking a piece of natural fur.
We are going to rub it on this Lucite rod.
You have to keep your fingers on the yellow tape here.
And we are going to transfer some of the natural oils here
onto this rod. And there are plenty of
negative ions around here and free electrons.
That oil likes those negative charges, and so there are going
to be excess negative charges on this Lucite rod.
And then you are going to come over here and just touch the
edge of this and let the electrons flow onto there.
Are you right-handed? You have to rub that really,
really hard. [LAUGHTER] Great.
Go over there. Touch the end.
Cool. Why don't you give it another
jolt here and we will really move that needle over.
Okay. Discharger.
We have to give it another jolt.
That is okay. You may have touched it with
something else and discharged it a little bit.
No, that's okay. You have to get the hang of
this, here. You are doing fantastic.
Take it off. That's what it is.
You are holding it on too long. Okay, that is pretty good.
Put it in front there. Get it a little bit closer.
Here comes the UV radiation. We did it.
Try it again really hard. Just touch it.
Take it off. Do it again.
You need to do it again. You've got to get it right
here. Not too much,
not too little. Okay.
It is doing it. Discharge in front.
Electrons off. Now we have to do a control
experiment. That is, you have got to charge
it up again, but now, when you put the light there,
what I am going to do is hold up a Pyrex plate in between the
light and the metal. It is going to block the
radiation, and it should not discharge.
You need to get on there. Fantastic.
There it goes. Here is the plate.
Get it to a little bit lower. Do a good discharge.
[APPLAUSE] Fantastic. Thank you very much.
Thank you for being a good sport.
That is the photoelectron experiment.
Hey, it works. Well, what I want to do now is
just spend a few minutes working on a few problems.
I think these are pretty straightforward,
but I just want to make sure that everybody is on the same
page here in terms of being able to do the homework.
Here is the first problem. The first problem says,
how many photons? And remember what we said a
photon was? E equals h nu.
This is the number of joules. Implied is the number of joules
per photon, although we don't usually write this,
but that is what that is, joules per photon.
You may want to write it as you do these problems.
How many photons associated with radiation of a wavelength
lambda equals one picometer, which is 1.0x10^-12 meters,
how many of these do you need in order to create,
say, a laser pulse of energy that is one joule?
Lasers are pulsed, so I am talking about a pulse
of energy, one joule. You want to draw yourself a
picture, here. We are drawing a picture of one
pulse of energy, one joule.
Now, we have not been given the frequency, here,
of this radiation, but we know the wavelength,
and we know the relationship between frequency and
wavelength. It is just c over lambda.
I know what lambda is. I can calculate nu.
And, when I do that, I find that the energy of the
photon, hc over lambda, that energy of the
photon is 1.99x10^-13 joules per photon.
And I am using one more figure than is significant since this
an intermediate step in the calculation.
If I want a pulse of one joule of energy and I am asked how
many photons do I need to get that, and each photon is
1.99x10^-13 joules, well, that means that I am
going to need 5.0x10^12 photons. There was a question here?
All right. Let's work another one.
Here, we want to define what we mean by power.
This says the power of radiation from the continuous
laser is three milliwatts.
We have some laser, and the power of that radiation
coming out is three milliwatts. Well, what is power?
Power is energy per unit time. It is the energy delivered or
the energy expended per unit time.
The unit of power that we are going to use is a watt.
A watt is a joule per second. We are told that we have
radiation of three milliwatts. That is 3.0x10^-3 joules per
second. And the question asks,
how long will it take for a total energy of one joule to be
supplied? Well, one joule.
And we have the rate of energy supply as 3x10^-3 joules per
second. That gives us 330 seconds.
That is straightforward. Finally, we have one more.
It says, how many photons per second of, again,
the same radiation, of the wavelength of one
picometer. That means, again,
we are dealing with photons that have an energy of
1.99x10^-13 joules per photon. How many photons per second,
the rate of photons at the wavelength do you have to have,
or do you have, if the power of the radiation
is three milliwatts? Well, the power of the
radiation is 3.0x10^-3 joules per second, --
-- and we have 1.99x10^-13 joules per photon.
In order to have this kind of power, what we have to have
being emitted is 1.5x10^10 photons per second.
All right. The photoelectric effect is one
of the experiments that demonstrated the particle-like
nature of light, of radiation.
Particle-like nature because you have to have these chunks of
energy to make some process occur.
Next time, we are going to look at and will just talk briefly
about the other experiments that demonstrated the particle-like
nature of radiation. And that other experiment is
the demonstration that a photon has momentum,
even though it doesn't have any mass.
See you Wednesday. See you Friday.