PROFESSOR: Last time,
we spoke about photons in the context of
an interferometer. The Mach-Zehnder interferometer. And we saw the very
unusual properties of photons and interference,
and how relatively simple interference a effect
can be used to produce a very surprising measurement. Today we're going to backtrack
and go from the beginning, and think about photons as
physicists did 100 years ago, and how, by thinking
about photons, they pretty much came up
with quantum mechanics. So we want to trace this back. And the best place
to start, probably, is with a photoelectric effect. The photoelectric
effect is an experiment done by Hertz in 1887, in
which he irradiated plates. That means shine light,
high energy beams of light, on metal plate, and he found
that electrons were released. Those were called
photo electrons. And therefore, you would
get a photoelectric current from those electrons. So this is the effect
we want to discuss now, is the photoelectric effect. And it's Hertz, 1887. So first a description. So polished metal plates
irradiated may emit electrons. And these are called
photo electrons sometimes. Photo electrons is
just an electron that was released due to a photon. And therefore, we get a
photoelectric current. OK so so far, so good. But what was special
about this experiment? The first step that was
special was that there was a critical frequency. If you would take a sample
and you would irradiate it with light, and you
would begin with light with very low frequency,
nothing would happen. And all of a sudden after
a certain frequency, boom. You would get a current. So there is a threshold
frequency, nu0, such that only for Nu greater
than nu0 there is a current. So no current for
lower frequencies. Now as it turned out,
nu0 depends on the metal you're irradiating. And even more, it's a
complicated thing to calculate. It depends on the
surface of the metal, so that's why Hertz apparently
had to polish the metal. And this frequency, if
the metal is irregular, may depend on where you shine. So you'd better prepare
the metal very nicely. And it may even depend
on the crystalline nature of the metal, because
it's a many body effect. You see, anticipating
the resolution, there is this piece of metal and
there are a few free electrons running around. And they run around among
the crystalline structure of the metal. And removing them, it's
going to take some energy, and that energy depends on
the metal and the arrangement and all kinds of things. So this nu0 depends on the
metal and the configuration of atoms at the surface. Third property was
kind of interesting. The magnitude of the
current was proportional to the intensity of the light. Magnitude current
is proportional to the light intensity. And perhaps the last one and
fourth, a rather important one, a very crucial
property, is that you could observe the energy
of the photo electrons, and it seemed to be independent
of the light intensity. So energy of the
photo electrons is independent of the
intensity of light The number of photo
electrons would depend on the
intensity of light, but not the energy of
the photo electrons. Now there is more to that,
but I think it was not quite exactly noticed by Hertz. So Hertz probably didn't
notice all these things. But the last one, that maybe
we can put in brackets here, is that the energy of the
photo electrons E gamma-- oh, no-- E of the electrons
increases linearly with the frequency of the light. So this photoelectric
effect was not easy to understand if you
thought of light as a wave. And Einstein came
up with an answer that he almost said
what was going on, but didn't quite use the word. He said that light comes
in bundles of energy. And in a beam, you have
bundles of energy quanta. Didn't quite say
light is a particle. He himself was a little
non-committal, I think, about this concept. But Einstein, in 1905, gives
the natural explanation and says that light
is composed of quanta. He would have to
wait until 1920s until the name photons came
up, given by a chemist, Lewis. So he called them quanta. These are later photons, and
I will use the name photons from the beginning. With energy, E equal h nu,
where nu is the frequency and h was Planck's constant. Planck had already
introduced the constant in trying to fit the
black body spectrum. The black body spectrum,
the intensity of light is a function of frequency
in black body radiation, had a particular curve. Planck tried to fit it and he
realized he needed one constant and he called it h. That's Planck's constant. And the same constant
that Planck introduced, reappeared in
Einstein's proposition. This is Planck's constant. So the picture that
Einstein and others had was that you would have
kind of a potential here and plot energy
over here, and maybe this is some distance. And you have a metal and there
is the electrons captured here. And here is zero energy. So they have negative
energy, they're captured. And you need some
amount of energy, w, which is called the work
function, that depends on the type of metal you have. And if you could
supply that energy, w, to any one of these electrons
that are bound in this metal, they would come out and
not be attracted any more and would be able to fly free. So it is like an
escape velocity, you're bound by the
gravity of earth. You need something velocity,
some energy to shoot you out. Same thing here. So this w, or work function,
is defined as the energy needed to release an electron. And that work
function is that thing that depends on the metal
you have and the structure and how well you've
polished the surface. So if this is true,
then Einstein, if he was right
with this property, there would be the following
statement that you could make. The energy of the electron,
which is, roughly speaking, one half mv squared, would
be equal to the energy that the photon [INAUDIBLE],
minus the work function. So you supply a photon. Some of the energy goes
into the work function, but the rest of the energy
goes into giving free energy to these electrons. So you have the
energy of the photon minus w, which is what you
need to just take it out with 0 velocity. And then the rest of
the energy of the photon would be transmitted as
kinetic energy of the electron. So if this is true, this
would be h nu minus omega. And this was considered
a prediction, because that statement that
the energy of the electron increases linearly
with the frequency, was not quite obvious to people. Experiments were
not fine enough. Measuring the energy of
the emitted particles was not all that easy either. So this was
Einstein's prediction. And the experimental
confirmation took 10 years to come. It was verified by
Millikan in 1915. So Millikan, in 1915, measures
the energy of the photo electrons, verifies
Einstein's conjecture, and actually produces,
by measuring so carefully the energy of the photo
electrons, produces a measurement of h, which
is the best to that point. And h is measured
to better than 1%, so a very accurate
measurement of h. And perhaps you would say,
OK, so this is all wonderful, now everybody
believes in photons. But that's quite
far from the truth. They didn't believe
in photons too much, because Maxwell had
been too successful. And Einstein himself knew that
once you started believing in particles like
photons, you had this subject with this case of
loss of determinism and waves that we have sometimes
as particles and things he didn't like much. So people were quite reluctant
to believe in these things. It's quite amazing. So it took a while still. So let's do a simple exercise
to introduce some numbers here, and show you how to do some
very simple computation. So let me do an example. You shine UV light with lambda
290 nanometers on a metal with work function 4.05 EV. What is the energy, E
of the photo electrons, and what is their speed? Now it is a goal of mine, and of
the instructors in this course, that a calculation
like that, you should be able to do without
turning on your iPhone and checking what the h bar is,
and getting a few constants, and what is an EV
or all these things. Well nowadays, you can
just check Wolfram Alpha and they will give
you the answer for this, in beautiful,
beautiful calculations. Just copy the
question like that, pretty much, I think it
will answer it for you. But you should be
able to do back of the envelope calculations,
in which you estimate things quickly. And with one significant
digit, you don't even need a calculator to do this. So let's see how
one does this thing. So the first thing to
do is to figure out what is the energy of this photon. That's the first problem. So the energy of
a photon is h nu. But nu it's not lambda. So nu time lambda is c,
so this is hc over lambda, where c is the speed of light. OK, hc lambda, we could
do it if we knew h. I must say, I never remember
what h is in normal units. Joule seconds, six
point something, I don't quite remember it. So what do I do? I use h bar, which
is h over 2 pi. So h is 2 pi h
bar c over lambda. And here is where you-- here is the first
thing that maybe you want to remember by heart. h bar c is a pretty
nice number, it's about 200mev times a fermi,
If you want it more precise, it's 197.33, if you
want to get five digits, but it's pretty close
to 200 mev fermi. And what is a fermi? It's 10 to the minus 15 meters. OK, so with this number,
I claim you can do pretty much all you want to do. So here you have 2pi
times 200 mev times 10 to the minus 15
meters divided-- I'll put 197 here-- divided by lambda, which
is 290 nanometers, which is 10 to the minus 9 meters. So 10 to the minus 9
and 10 to the minus 15 is 10 to the minus 6 up. And this is a million
ev, which is 10 to 6 ev. So all these meters cancel
and there's just an ev left. So this is 2 pi times
197 over 290 ev. And you certainly could estimate
this like 2 over 3 times 2 pi, which is 6. And that's about four. And if you want to
do it more carefully, it comes out to 4.28 ev. And the nice thing is that
the answer comes in ev's. And the work
functions, everybody gives them in ev's, so
it's a convenient thing. So at this moment, you
have this electron energy of the photon being this. So energy of the
electron is energy of the photon minus the
work function, which is 428 minus 405 ev, and it's 0.23 ev. That's a kinetic
energy and that should be a non-relativistic
electron because the rest mass of an electron is
about half a million ev. It is 511,000 ev. So this is fairly
non-relativistic, but how slow is it? Is it moving a
centimeter per second? No, it's moving pretty fast. You can write this as
one half mv squared. And then what do you do? You put one half m
of the electron c squared v squared
over c squared. And this is one half of 511,000
ev times v over c squared. So do the arithmetic, it's
20.46 over that, square root and multiply it by
the speed of light. You can do this
roughly in your head. And the velocity comes out
to 284 kilometers per second, so it's pretty fast.
