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PROFESSOR: All right. So today's task is going
to be to outline some of the basic experimental
facts that we will both have to deal with
and that our aim should be to understand and
model through the rest of the course.
Physics doesn't tell you some abstract truth
about why the universe is the way it is. Physics
gives you models to understand how things
work and predict what will happen next. And
what we will be aiming to do is develop models
that give us an intuition for the phenomena
and allow us to make predictions. And these
are going to be the experimental facts I would
like to both explain, develop an intuition
for, and be able to predict consequences of.
So we'll start off with-- so let me just outline
them. So, first fact, atoms exist. I'll go
over some of the arguments for that. Randomness,
definitely present in the world. Atomic spectre
are discrete and structured. We have a photoelectric
effect, which I'll describe in some detail.
Electrons do some funny things. In particular
electron diffraction. And sixth and finally,
Bell's Inequality. Something that we will
come back to at the very end of the class,
which I like to think of as a sort of a frame
for the entirety of 8.04.
So... we'll stick with this for the moment.
So everyone in here knows that atoms are made
of electrons and nuclei. In particular, you
know that electrons exist because you've seen
a cathode ray tube. I used to be able to say
you've seen a TV, but you all have flat panel
TVs, so this is useless. So a cathode ray
tube is a gun that shoots electrons at a phosphorescent
screen. And every time the electron hits the
screen it induces a little phosphorescence,
a little glow. And that's how you see on a
CRT.
And so as was pithily stated long ago by a
very famous physicist, if you can spray them,
they exist. Pretty good argument. There's
a better argument for the existence of electrons,
which is that we can actually see them individually.
And this is one of the most famous images
in high-energy physics. It's from an experiment
called Gargamelle, which was a 30-cubic meter
tank of liquid freon pulsing just at its vapor
pressure 60 times a second. And what this
image is is, apart from all the schmut, you're
watching a trail of bubbles in this de-pressurizing
freon that wants to create bubbles but you
have to nucleate bubbles. What you're seeing
there in that central line that goes up and
then curls around is a single electron that
was nailed by a neutrino incident from a beam
at CERN where currently the LHC is running.
And this experiment revealed two things. First,
to us it will reveal that you can see individual
electrons and by studying the images of them
moving through fluids and leaving a disturbing
wake of bubbles behind them. We can study
their properties in some considerable detail.
The second thing it taught us is something
new-- we're not going to talk about it in
detail-- is that it's possible for a neutrino
to hit an electron. And that process is called
a weak neutral current for sort of stupid
historical reasons. It's actually a really
good name. And that was awesome and surprising
and so this picture is both a monument to
the technology of the experiment, but also
to the physics of weak neutral currents and
electrons. They exist if you can discover
neutrinos by watching them. OK.
Secondly, nuclei. We know that nuclei exist
because you can shoot alpha particles, which
come from radioactive decay, at atoms. And
you have your atom which is some sort of vague
thing, and I'm gonna make the-- I'm gonna
find the atom by making a sheet of atoms.
Maybe a foil. A very thin foil of stuff. And
then I'm gonna shoot very high-energy alpha
particles incident of this.
Probably everyone has heard of this experiment,
it was done by Rutherford and Geiger and Marsden,
in particular his students at the time or
post-docs. I don't recall-- and you shoot
these alpha particles in. And if you think
of these guys as some sort of jelly-ish lump
then maybe they'll deflect a little bit, but
if you shoot a bullet through Jello it just
sort of maybe gets deflected a little bit.
But Jello, I mean, come on.
And I think what was shocking is that you
should these alpha particles in and every
once in a while, they bounce back at, you
know, 180, 160 degrees. Rutherford likened
this to rolling a bowling ball against a piece
of paper and having it bounce back. Kind of
surprising. And the explanation here that
people eventually came up upon is that atoms
are mostly zero density. Except they have
very, very high density cores, which are many
times smaller than the size of the atom but
where most of the mass is concentrated. And
as a consequence, most of the inertia.
And so we know that atoms have substructure,
and the picture we have is that well if you
scrape this pile of metal, you can pull off
the electrons, leaving behind nuclei which
have positive charge because you've scraped
off the electrons that have negative charge.
So we have a picture from these experiments
that there are electrons and there are nuclei--
which, I'll just write N and plus-- which
are the constituents of atoms.
Now this leads to a very natural picture of
what an atom is. If you're a 19th-century
physicist, or even an early 20th-century physicist,
it's very natural to say, aha, well if I know
if I have a positive charge and I have a negative
charge, then they attract each other with
a 1 over r minus q1 q2-- sorry, q1 q2 over
r potential. This is just like gravity, right.
The earth and the sun are attracted with an
inverse-r potential. This leads to Keplerian
orbits. And so maybe an atom is just some
sort of orbiting classical combination of
an electron and a nucleus, positively charged
nucleus.
The problem with this picture, as you explore
in detail in your first problem on the problem
set, is that it doesn't work. What happens
when you accelerate a charge? It radiates.
Exactly. So if it's radiating, it's gotta
lose energy. It's dumping energy into this--
out of the system. So it's gotta fall lower
into the potential. Well it falls lower, it
speeds up. It radiates more. Because it's
accelerating more to stay in a circular orbit.
All right, it radiates more, it has to fall
further down.
So on the problem set you're going to calculate
how long that takes. And it's not very long.
And so the fact that we persist for more than
a few picoseconds tells you that it's not
that-- this is not a correct picture of an
atom. OK.
So in classical mechanics, atoms could not
exist. And yet, atoms exist. So we have to
explain that. That's gonna be our first challenge.
Now interestingly Geiger who is this collaborator
of Rutherford, a young junior collaborator
of Rutherford, went on to develop a really
neat instrument. So suppose you want to see
radiation. We do this all the time. I'm looking
at you and I'm seeing radiation, seeing light.
But I'm not seeing ultra high energy radiation,
I'm seeing energy radiation in the electromagnetic
waves in the optical spectrum.
Meanwhile I'm also not seeing alpha particles.
So what Geiger wanted was a way to detect
without using your eyes radiation that's hard
to see. So the way he did this is he took
a capacitor and he filled the-- surrounded
the capacitor with some noble gas. It doesn't
interact. There's no-- it's very hard to ionize.
And if you crank up the potential across this
capacitor plate high enough, what do you get?
A spark. You all know this, if you crank up
a capacitor it eventually breaks down because
the dielectric in between breaks down, you
get a spontaneous sparking.
So what do you figure it would look if I take
a capacitor plate and I charge it up, but
not quite to breakdown. Just a good potential.
And another charged particle comes flying
through, like an alpha particle, which carries
a charge of plus 2, that positive charge will
disturb things and will add extra field effectively.
And lead to the nucleation of a spark.
So the presence of a spark when this potential
is not strong enough to induce a spark spontaneously
indicates the passage of a charged particle.
Geiger worked later with-- Marsden? Muller.
Heck. I don't even remember. And developed
this into a device now known as the Geiger
counter. And so you've probably all seen or
heard Geiger counters going off in movies,
right. They go ping ping ping ping ping ping
ping ping ping, right. They bounce off randomly.
This is an extremely important lesson, which
is tantamount to the lesson of our second
experiment yesterday. The 50-50, when we didn't
expect it. The white electrons into the harness
box then into a color box again, would come
out 50-50, not 100 percent. And they come
out in a way that's unpredictable. We have
no ability to our knowledge-- and more than
our knowledge, we'll come back to that with
Bell's Inequality-- but we have no ability
to predict which electron will come out of
that third box, white or black, right.
Similarly with a Geiger counter you hear that
atoms decay, but they decay randomly. The
radiation comes out of a pile of radioactive
material totally at random. We know the probabilistic
description of that. We're going to develop
that, but we don't know exactly when. And
that's a really powerful example-- both of
those experiments are powerful examples of
randomness.
And so we're going to have to incorporate
that into our laws of physics into our model
of quantum phenomena as well. Questions? I
usually have a Geiger counter at this point,
which is totally awesome, so I'll try to produce
the Geiger counter demo later. But the person
with the Geiger counter turns out to have
left the continent, so made it a little challenging.
OK. Just sort of since we're at MIT, an interesting
side note. This strategy of so-called hard
scattering, of taking some object and sending
it at very high velocity at some other object
and looking for the rare events when they
bounce off at some large angle, so-called
hard scattering. Which is used to detect dense
cores of objects. It didn't stop with Rutherford.
People didn't just give up at that point.
Similar experience in the '60s and '70s which
are conducted at Slack, were involved not
alpha particles incident on atoms but individual
electrons incident on protons. So not shooting
into the nucleus, but shooting and looking
for the effect of hitting individual protons
inside the nucleus. And through this process
it was discovered that in fact-- so this was
done in the '60s and '70s, that in fact the
proton itself is also not a fundamental particle.
The proton is itself composite.
And in particular, it's made out of-- eventually
people understood that it's made out of, morally
speaking, and I'm gonna put this in quotation
marks-- ask me about it in office hours--
three quarks, which are some particles. And
the reason we-- all this tells you is that
it's some object and we've given it the name
quark. But indeed there are three point-like
particles that in some sense make up a proton.
It's actually much more complicated than that,
but these quarks, among other things, have
very strange properties. Like they have fractional
charge.
And this was discovered by a large group of
people, in particular led by Kendall and Friedman
and also Richard Taylor. Kendall and Friedman
were at MIT, Richard Taylor was at Stanford.
And in 1990 they shared the Nobel Prize for
the discovery of the partonic structure out
of the nucleons. So these sorts of techniques
that people have been using for a very long
time continue to be useful and awesome.
And in particular the experiment, the experimental
version of this that's currently going on,
that I particularly love is something called
the relativistic heavy ion collider, which
is going on at Brookhaven. So here what you're
doing is you take two protons and you blow
them into each other at ultra high energy.
Two protons, collide them and see what happens.
And that's what happens. You get massive shrapnel
coming flying out. So instead of having a
simple thing where one of the protons just
bounces because there's some hard quark, instead
what happens is just shrapnel everywhere,
right. So you might think, well, how do we
interpret that at all. How do you make sense
out of 14,000 particles coming out of two
protons bouncing into each other. How does
that make any sense? And the answer turns
out to be kind of awesome.
And so this touches on my research. So I want
to make a quick comment on it just for color.
The answer turns out to be really interesting.
First off, the interior constituents of protons
interact very strongly with each other. But
at the brief moment when protons collide with
each other, what you actually form is not
a point-like quirk and another point-like
quark. In fact, protons aren't made out of
point-like quarks at all. Protons are big
bags with quarks and gluons and all sorts
of particles fluctuating in and out of existence
in a complicated fashion.
And what you actually get is, amazingly, a
liquid. For a brief, brief moment of time
the parts of those protons that overlap--
think of them as two spheres and they overlap
in some sort of almond-shaped region. The
parts of those protons that overlap form a
liquid at ultra high temperature and at ultra
high density. It's called the RHIC fireball
or the quark-gluon plasma, although it's not
actually a plasma. But it's a liquid like
water. And what I mean by saying it's a liquid
like water, if you push it, it spreads in
waves. And like water, it's dissipative. Those
waves dissipate.
But it's a really funny bit of liquid.
Imagine you take your cup of coffee. You drink
it, you're drinking your coffee as I am wont
to do, and it cools down over time. This is
very frustrating. So you pour in a little
bit of hot coffee and when you pour in that
hot coffee, the system is out of equilibrium.
It hasn't thermalized. So what you want is
you want to wait for all of the system to
wait until it's come to equilibrium so you
don't get a swig of hot or swig of cold. You
want some sort of Goldilocks-ean in between.
So you can ask how long does it take for this
coffee to come to thermal equilibrium. Well
it takes a while. You know, a few seconds,
a few minutes, depending on exactly how you
mess with it. But let me ask you a quick question.
How does that time scale compare to the time
it takes for light to cross your mug? Much,
much, much slower, right? By orders of magnitude.
For this liquid that's formed in the ultra
high energy collision of two protons, the
time it takes for the system-- which starts
out crazy out of equilibrium with all sorts
of quarks here and gluons there and stuff
flying about-- the time it takes for it to
come to thermal equilibrium is of order the
time it takes for light to cross the little
puddle of liquid. This is a crazy liquid,
it's called a quantum liquid. And it has all
sorts of wonderful properties. And the best
thing about it to my mind is that it's very
well modeled by black holes. Which is totally
separate issue, but it's a fun example. So
from these sorts of collisions, we know a
great deal about the existence of atoms and
randomness, as you can see. That's a fairly
random sorting.
OK so moving on to more 8.04 things. Back to
atoms. So let's look at specifics of that.
I'm not kidding, they really are related to
black holes. I get paid for this. So here's
a nice fact, so let's get to atomic spectra.
So to study atomic spectra, here's the experiment
I want to run. The experiment I want to run
starts out with some sort of power plant.
And out of the power plant come two wires.
And I'm going to run these wires across a
spark gap, you know, a piece of metal here,
a piece of metal here, and put them inside
a container, which has some gas. Like H2 or
neon or whatever you want. But some simple
gas inside here.
So we've got an electric potential established
across it. Again, we don't want so much potential
that it sparks, but we do want to excite the
H2. So we can even make it spark, it doesn't
really matter too much. The important thing
is that we're going to excite the hydrogen,
and in exciting the hydrogen the excited hydrogen
is going to send out light. And then I'm going
to take this light-- we take the light, and
I'm gonna shine this on a prism, something
I was taught to do by Newton. And-- metaphorically
speaking-- and look at the image of this light
having passed through the prism.
And what you find is you find a very distinct
set of patterns. You do not get a continuous
band. In fact what you get-- I'm going to
have a hard time drawing this so let me draw
down here. I'm now going to draw the intensity
of the light incident on the screen on this
piece of paper-- people really used to use
pieces of paper for this, which is kind of
awesome-- as a function of the wavelength,
and I'll measure it in angstroms.
And what you discover is-- here's around 1,000
angstroms-- you get a bunch of lines. Get
these spikes. And they start to spread out,
and then there aren't so many. And then at
around 3,000, you get another set. And then
at around 10,000, you get another set.
This is around 10,000.
And here's the interesting thing about these.
So the discovery of these lines-- these are
named after a guy named Lyman, these are--
these are named after a guy named-- Ballmer.
Thank you. Steve Ballmer. And these are passion,
like passion fruit. So. Everyone needs a mnemonic,
OK. And so these people identified these lines
and explained various things about them.
But here's an interesting fact. If you replace
this nuclear power plant with a coal plant,
it makes no difference. If you replace this
prism by a different prism, it makes no difference
to where the lines are. If you change this
mechanism of exciting the hydrogen, it makes
no difference. As long as it's hydrogen--
as long as it's hydrogen in here you get the
same lines, mainly with different intensities
depending upon how exactly you do the experiment.
But you get the same position of the lines.
And that's a really striking thing.
Now if you use a different chemical, a different
gas in here, like neon, you get a very different
set of lines. And a very different effective
color now when you eyeball this thing. So
Ballmer, incidentally-- and I think this is
actually why he got blamed for that particular
series, although I don't know the history--
Ballmer noticed by being-- depending on which
biography you read-- very clever or very obsessed
that these guys, this particular set, could
be-- they're wavelengths. If you wrote their
wavelengths and labeled them by an integer
n, where n ran from 3 to any positive integer
above 3, could be written as 36. So this is
pure numerology. 36, 46 angstroms times the
function n squared over n squared minus 4,
where N is equal to 3, 4, dot dot dot-- an
integer.
And it turns out if you just plug in these
integers, you get a pretty good approximation
to this series of lines. This is a hallowed
tradition, a phenomenological fit to some
data. Where did it come from? It came from
his creative or obsessed mind. So this was
Ballmer. And this is specifically for hydrogen
gas, H2.
So Rydberg and Ritz, R and R, said, well actually
we can do one better. Now that they realized
that this is true, they looked at the whole
sequence. And they found a really neat little
expression, which is that 1 over the wavelength
is equal to a single constant parameter. Not
just for all these, but for all of them. One
single numerical coefficient times 1 over
m squared minus 1 over n squared-- n is an
integer greater than zero and greater in particular
than m. And if you plug in any value of n
and any value of m, for sufficiently reasonable--
I mean, if you put in 10 million integers
you're not going to see it because it's way
out there, but if you put in or-- rather,
in here-- if you put any value of n and m,
you will get one of these lines. So again,
why? You know, as it's said, who ordered that.
So this is experimental result three that
we're going to have to deal with. When you
look at atoms and you look at the specter
of light coming off of them, their spectra
are discrete. But they're not just stupidly
discrete, they're discrete with real structure.
Something that begs for an explanation. This
is obviously more than numerology, because
it explains with one tunable coefficient a
tremendous number of spectral lines. And there's
a difference-- and crucially, these both work
specifically for hydrogen. For different atoms
you need a totally different formula.
But again, there's always some formula that
nails those spectral lines. Why? Questions?
OK. So speaking of atomic spectra-- whoops,
I went one too far-- here's a different experiment.
So people notice the following thing. People
notice that if you take a piece of metal and
you shine a light at it, by taking the sun
or better yet, you know, these days we'd use
a laser, but you shine light on this piece
of metal. Something that is done all the time
in condensed matter labs, it's a very useful
technique. We really do use lasers not the
sun, but still it continues to be useful in
fact to this day.
You shine light on a piece of metal and every
once in a while what happens is electrons
come flying off. And the more light and the
stronger the light you shine, you see changes
in the way that electrons bounce off. So we'd
like to measure that. I'd like to make that
precise. And this was done in a really lovely
experiment. Here's the experiment. The basic
idea of the experiment is I want to check
to see, as I change the features of the light,
the intensity, the frequency, whatever, I
want to see how that changes the properties
of the electrons that bounce off.
Now one obvious way-- one obvious feature
of an electron that flew off a piece of metal
is how fast is it going, how much energy does
it have. What's its kinetic energy. So I'd
like to build an experiment that measures
the kinetic energy of an electron that's been
excited through this photoelectric effect.
Through emission after shining light on a
piece of metal. Cool?
So I want to build that experiment. So here's
how that experiment goes. Well if this electron
comes flying off with some kinetic energy
and I want to measure that kinetic energy,
imagine the following circuit. OK first off
imagine I just take a second piece of metal
over here, and I'm going to put a little current
meter here, an ammeter. And here's what this
circuit does. When you shine light on this
piece of metal-- we'll put a screen to protect
the other piece of metal-- the electrons come
flying off, they get over here. And now I've
got a bunch of extra electrons over here and
I'm missing electrons over here. So this is
negative, this is positive. And the electrons
will not flow along this wire back here to
neutralize the system.
The more light I shine, the more electrons
will go through this circuit. And as a consequence,
there will be a current running through this
current meter. That cool with everyone? OK.
So we haven't yet measured the kinetic energy,
though. How do we measure the kinetic energy?
I want to know how much energy, with how much
energy, were these electrons ejected.
Well I can do that by the following clever
trick. I'm going to put now a voltage source
here, which I can tune the voltage of, with
the voltage V. And what that's going to do
is set up a potential difference across these
and the energy in that is the charge times
the potential difference. So I know that the
potential difference it takes, so the amount
of energy it takes to overcome this potential
difference, is q times V. That cool?
So now imagine I send in an electron-- I send
in light and it leads an electron to jump
across, and it has kinetic energy, kE. Well
if the kinetic energy is less than this, will
it get across? Not so much. It'll just fall
back. But if the kinetic energy is greater
than the energy it takes to cross, it'll cross
and induce a current.
So the upshot is that, as a function of the
voltage, what I should see is that there is
some critical minimum voltage. And depending
on how you set up the sign, the sign could
be the other way, but there's some critical
minimal voltage where, for less voltage, the
electron doesn't get across. And for any greater
voltage-- or, sorry, for any closer to zero
voltage, the electron has enough kinetic energy
to get across. And so the current should increase.
So there's a critical voltage, V-critical,
where the current running through the system
runs to zero. You make it harder for the electrons
by making the voltage in magnitude even larger.
You make it harder for the electrons to get
across. None will get across. Make it a little
easier, more and more will get across. And
the current will go up. So what you want to
do to measure this kinetic energy is you want
to measure the critical voltage at which the
current goes to zero.
So now the question is what do we expect to
see. And remember that things we can tune
in this experiment are the intensity of the
light, which is like e squared plus b squared.
And we can tune the frequency of the light.
We can vary that. Now does the total energy,
does that frequency show up in the total energy
of a classical electromagnetic wave? No. If
it's an electromagnetic wave, it cancels out.
You just get the total intensity, which is
a square of the fields. So this is just like
a harmonic oscillator. The energy is in the
amplitude. The frequency of the oscillator
doesn't matter. You push the swing harder,
it gets more kinetic energy. It's got more
energy. OK.
So what do we expect to see as we vary, for
example, the intensity? So here's a natural
gas. If you take-- so you can think about
the light here as getting a person literally,
like get the person next to you to take a
bat and hit a piece of metal. If they hit
it really lightly they're probably not going
to excite electrons with a lot of energy.
If they just whack the heck out of it, then
it wouldn't be too surprising if you get much
more energy in the particles that come flying
off. Hit it hard enough, things are just gonna
shrapnel and disintegrate.
The expectation here is the following. That
if you have a more intense beam, then you
should get more-- the electrons coming off
should be more energetic. Because you're hitting
them harder. And remember that the potential,
which I will call V0, the stopping voltage.
So therefore V0 should be greater in magnitude.
So this anticipates that the way this curve
should look as we vary the current as a function
of v, if we have a low voltage-- sorry, if
we have a low-intensity beam-- it shouldn't
take too much potential just to impede the
motion.
But if we have a-- so this is a low intensity.
But if we have a high-intensity beam, it should
take a really large voltage to impede the
electric flow, the electric current, because
high-intensity beam you're just whacking those
electrons really hard and they're coming off
with a lot of kinetic energy. So this is high
intensity. Everyone down with that intuition?
This is what you get from Maxwell's electrodynamics.
This is what you'd expect.
And in particular, as we vary-- so this is
our predictions-- in particular as we vary--
so this is 1, 2, with greater intensity. And
the second prediction is that V-naught should
be independent of frequency. Because the energy
density and electromagnetic wave is independent
of the frequency. It just depends on the amplitude.
And I will use nu to denote the frequency.
So those are the predictions that come from
8.02 and 8.03. But this is 8.04. And here's what
the experimental results actually look like.
So here's the intensity, here's the potential.
And if we look at high potential, it turns
out that-- if we look, sorry, if we look at
intermediate potentials, it's true that the
high intensity leads to a larger current and
the low intensity leads to a lower current.
But here's the funny thing that happens. As
you go down to the critical voltage, their
critical voltages are the same. What that
tells you is that the kinetic energy kicked
out-- or the kinetic energy of an electron
kicked out of this piece of metal by the light
is independent of how intense that beam is.
No matter how intense that beam is, no matter
how strong the light you shine on the material,
the electrons all come out with the same energy.
This would be like taking a baseball and hitting
it with a really powerful swing or a really
weak swing and seeing that the electron dribbles
away with the same amount of energy. This
is very counter-intuitive.
But more surprisingly, V-naught is actually
independent of intensity. But here's the real
shocker. V-naught varies linearly in the frequency.
What does change V-naught is changing the
frequency of the light in this incident. That
means that if you take an incredibly diffuse
light-- incredibly diffuse light, you can
barely see it-- of a very high frequency,
then it takes a lot of energy to impede the
electrons that come popping off.
The electrons that come popping off have a
large energy. But if you take a low-frequency
light with extremely high intensity, then
those electrons are really easy to stop. Powerful
beam but low frequency, it's easy to stop
those electrons. Weak little tiny beam at
high frequency, very hard to stop the electrons
that do come off. So this is very counter-intuitive
and it doesn't fit at all with the Maxwellian
picture. Questions about that?
So this led Einstein to make a prediction.
This was his 1905 result. One of his many
totally breathtaking papers of that year.
And he didn't really propose a model or a
detailed theoretical understanding of this,
but he proposed a very simple idea. And he
said, look, if you want to fit this-- if you
want to fit this experiment with some simple
equations, here's the way to explain it. I
claim-- I here means Einstein, not me-- I
claim that light comes in packets or chunks
with definite energy. And the energy is linearly
proportional to the frequency. And our energy
is equal to something times nu, and we'll
call the coefficient h.
The intensity of light, or the amplitude squared,
the intensity is like the number of packets.
So if you have a more intense beam at the
same frequency, the energy of each individual
chunk of light is the same. There are just
a lot more chunks flying around. And so to
explain the photoelectric effect, Einstein
observed the following. Look, he said, the
electrons are stuck under the metal. And it
takes some work to pull them off. So now what's
the kinetic energy of an electron that comes
flying off-- whoops, k3. Bart might have a
laugh about that one. Kinetic, kE, not 3.
So the kinetic energy of electron that comes
flying off, well, it's the energy deposited
by the photon, the chunk of light, h-nu well
we have to subtract off the work it took.
Minus the work to extract the electron from
the material. And you can think of this as
how much energy does it take to suck it off
the surface. And the consequence of this is
that the kinetic energy of an electron should
be-- look, if h-nu is too small, if the frequency
is too low, then the kinetic energy would
be negative.
But that doesn't make any sense. You can't
have negative kinetic energy. It's a strictly
positive quantity. So it just doesn't work
until you have a critical value where the
frequency times h-- this coefficient-- is
equal to the work it takes to extract. And
after that, the kinetic energy rises with
the frequency with a slope equal to h. And
that fits the data like a champ.
So no matter-- let's think about what this
is saying again. No matter what you do, if
your light is very low-frequency and you pick
some definite piece of metal that has a very
definite work function, very definite amount
of energy it takes to extract electrons from
the surface. No matter how intense your beam,
if the frequency is insufficiently high, no
electrons come off. None.
So it turns out none is maybe a little overstatement
because what you can have is two photon processes,
where two chunks hit one electron at the right,
just at the same time. Roughly speaking the
same time. And they have twice the energy,
but you can imagine that the probability of
two photon hitting one electron at the same
time of pretty low. So the intensity has to
be preposterously high. And you see those
sorts of multi-photon effects. But as long
as we're not talking about insanely high intensities,
this is an absolutely fantastic probe of the
physics.
Now there's a whole long subsequent story
in the development of quantum mechanics about
this particular effect. And it turns out that
the photoelectric effect is a little more
complicated than this. But the story line
is a very useful one for organizing your understanding
of the photoelectric effect. And in particular,
this relation that Einstein proposed out of
the blue, with no other basis. No one else
had ever seen this sort of statement that
the electrons, or that the energy of a beam
of light should be made up of some number
of chunks, each of which has a definite minimum
amount of energy.
So you can take what you've learned from 8.02
and 8.03 and extract a little bit more information
out of this. So here's something you learned
from 8.02. In 8.02 you learned that the energy
of an electromagnetic wave is equal to c times
the momentum carried by that wave-- whoops,
over two. And in 8.03 you should have learned
that the wavelength of an electromagnetic
wave times the frequency is equal to the speed
of light, C.
And we just had Einstein tell us-- or declare,
without further evidence, just saying, look
this fits-- that the energy of a chunk of
light should be h times the frequency. So
if you combine these together, you get another
nice relation that's similar to this one,
which says that the momentum of a chunk of
light is equal to h over lambda. So these
are two enormously influential expressions
which come out of this argument from the photoelectric
effect from Einstein. And they're going to
be-- their legacy will be with us throughout
the rest of the semester.
Now this coefficient has a name, and it was
named after Planck. It's called Planck's Constant.
And the reason that it's called Planck's Constant
has nothing to do with the photoelectric effect.
It was first this idea that an electromagnetic
wave, that light, has an energy which is linearly
proportional not to its intensity squared,
none of that, but just linearly proportional
to the frequency. First came up an analysis
of black body radiation by Planck. And you'll
understand, you'll go through this in some
detail in 8.044 later in the semester. So I'm
not going to dwell on it now, but I do want
to give you a little bit of perspective on it.
So Planck ran across this idea that E is equal
to h/nu. Through the process of trying to
fit an experimental curve. There was a theory
of how much energy should be emitted by an
object that's hot and glowing as a function
of frequency. And that theory turned out to
be in total disagreement with experiment.
Spectacular disagreement. The curve for the
theory went up, the curve for the experiment
went down. They were totally different.
So Planck set about writing down a function
that described the data. Literally curve-fitting,
that's all he was doing. And this is the depths
of desperation to which he was led, was curve-fitting.
He's an adult. He shouldn't be doing this,
but he was curve-fitting. And so he fits the
curve, and in order to get it to fit the only
thing that he can get to work even vaguely
well is if he puts in this calculation of
h/nu. He says, well, maybe when I sum over
all the possible energies I should restrict
the energies which were proportional to the
frequency.
And it was forced on him because it fit from
the function. Just functional analysis. Hated
it. Hated it, he completely hated it. He was
really frustrated by this. It fit perfectly,
he became very famous. He was already famous,
but he became ridiculously famous. Just totally
loathed this idea. OK. So it's now become
a cornerstone of quantum mechanics. But he
wasn't so happy about it.
And to give you a sense for how bold and punchy
this paper by Einstein was that said, look,
seriously. Seriously guys. e equals h/nu.
Here's what Planck had to say when he wrote
a letter of recommendation to get Einstein
into the Prussian Academy of Sciences in 1917,
or 1913. So he said, there is hardly one among
the great problems in physics to which Einstein
has not made an important contribution. That
he may sometimes have missed the target in
his speculations as in his hypothesis of photons
cannot really be held too much against him.
It's not possible to introduce new ideas without
occasionally taking a risk.
Einstein who subsequently went on to develop
special relativity and general relativity
and prove the existence of atoms and the best
measurement of Avogadro's Constant, subsequently
got the Nobel Prize. Not for Avogadro's Constant,
not for proving the existence of atoms, not
for relativity, but for photons. Because of
guys like Planck, right. This is crazy.
So this was a pretty bold idea. And here,
to get a sense for why-- we're gonna leave
that up because it's just sort of fun to see
these guys scowling and smiling-- there is,
incidentally there's a great book about Einstein's
years in Berlin by Tom Levenson, who's a professor
here. A great writer and a sort of historian
of science. You should take a class from him,
which is really great. But I encourage you
to read this book. It talks about why Planck
is not looking so pleased right there, among
many other things. It's a great story.
So let's step back for a second. Why was Planck
so upset by this, and why was in fact everyone
so flustered by this idea that it led to the
best prize you can give a physicist. Apart
from a happy home and, you know. I've got
that one. That's the one that matters to me.
So why is this so surprising? And the answer
is really simple. We know that it's false.
We know empirically, we've known for two hundred
and some years that light is a wave. Empirically.
This isn't like people are like, oh I think
it'd be nice if it was a wave. It's a wave.
So how do we know that? So this goes back
to the double-slit experiment from Young.
Young's performance of this was in 1803. Intimations
of it come much earlier. But this is really
where it hits nails to the wall. And here's
the experiment.
So how many people in here have not seen a
double-slit experiment described? Yeah, exactly.
OK. So I'm just going to quickly remind you
of how this goes.
So we have a source for waves. We let the
waves get big until they're basically plane
waves. And then we take a barrier. And we
poke two slits in it. And these plane waves
induce-- they act like sources at the slits
and we get nu. And you get crests and troughs.
And you look at some distant screen and you
look at the pattern, and the pattern you get
has a maximum. But then it falls off, and
it has these wiggles, these interference fringes.
These interference fringes are, of course,
extremely important. And what's going on here
is that the waves sometimes add in-- so the
amplitude of the wave, the height of the wave,
sometimes adds constructively and sometimes
destructively. So that sometimes you get twice
the height and sometimes you get nothing.
So just because it's fun to see this, here's
Young's actual diagram from his original note
on the double-slit experiment. So a and b
are the slits, and c, d and f are the [INAUDIBLE]
on the screen, the distant screen. He drew
it by hand. It's pretty good.
So we've known for a very long time that light,
because of the double-slit experiment, light
is clearly wavy, it behaves like a wave. And
what are the senses in which it behaves like
a wave? There are two important senses here.
The first is answered by the question, where
did the wave hit the screen? So when we send
in a wave, you know, I drop a stone, one big
pulsive wave comes out. It splits into-- it
leads to new waves being instigated here and
over here. Where did that wave hit the screen?
Anyone?
AUDIENCE: Everywhere.
PROFESSOR: Yeah, exactly. It didn't hit this
wave-- the screen in any one spot. But some
amplitude shows up everywhere. The wave is
a distributed object, it does not exist at
one spot, and it's by virtue of the fact that
it is not a localized object-- it is not a
point-like object-- that it can interfere
with itself. The wave is a big large phenomena
in a liquid, in some thing.
So it's sort of essential that it's not a
localized object. So not localized. The answer
is not localized. And let's contrast this
with what happens if you take this double-slit
experiment and you do it with, you know, I
don't know, take-- who. Hmm. Tim Wakefield.
Let's give some love to that guy.
So, baseball player. And have him throw baseballs
at a screen with two slits in it. OK? Now
he's got pretty good-- well, he's got terrible
accuracy, actually. So every once in a while
he'll make it through the slits. So let's
imagine first blocking off-- what, he's a
knuckle-baller, right-- so every once in a
while it goes, the baseball will go through
the slit.
And let's think about what happens, so let's
cover one slit. And what we expect to happen
is, well, it'll go through more or less straight,
but sometimes it'll scrape the edge, it'll
go off to the side, and sometimes it'll come
over here. But if you take a whole bunch of
baseballs, and-- so any one baseball, where
does it hit? Some spot. Right? One spot. Not
distributed. One spot.
And as a consequence, you know, one goes here,
one goes there, one goes there. And now, there's
nothing like interference effects, but what
happens is as it sort of doesn't-- you get
some distribution if you look at where they
all hit. Yeah? Everyone cool with that? And
if we had covered over this slot, or slit,
and let the baseballs go through this one,
same thing would have happened.
Now if we leave them both open, what happens
is sometimes it goes here, sometimes it goes
here. So now it's pretty useful that we've
got a knuckle-baller. And what you actually
get is the total distribution looks like this.
It's the sum of the two. But at any given
time, any one baseball, you say, aha, the
baseball either went through the top slit,
and more or less goes up here. Or it went
through the bottom slit and more or less goes
down here. So for chunks-- so this is for
waves-- for chunks or localized particles,
they are localized. And as a consequence,
we get no interference.
So for waves, they are not localized, and
we do get interference. Yes, interference.
OK. So on your problem set, you're going to
deal with some calculations involving these
interference effects. And I'm going to brush
over them.
Anyway the point of the double-slit experiment
is that whatever else you want to say about
baseballs or anything else, light, as we've
learned since 1803 in Young's double-slit
experiment, light behaves like a wave. It
is not localized, it hits the screen over
its entire extent. And as a consequence, we
get interference. The amplitudes add. The
intensity is the square of the amplitude.
If the intensities add-- so sorry, if the
amplitudes add-- amplitude total is equal
to a1 plus a2, the intensity, which is the
square of a1 plus a2 squared, has interference
terms, the cross terms, from this square.
So light, from this point of view, is an electromagnetic
wave. It interferes with itself. It's made
of chunks. And I can't help but think about
it this way, this is literally the metaphor
I use in my head-- light is creamy and smooth
like a wave. Chunks are very different. But
here's the funny thing. Light is both smooth
like a wave, it is also chunky. It is super
chunky, as we have learned from the photoelectric
effect. So light is both at once. So it's
the best of both worlds. Everyone will be
satisfied, unless you're not from the US,
in which case this is deeply disturbing. So
of course the original Superchunk is a band.
So we've learned now from Young that light
is a wave. We've learned from the photoelectric
effect that light is a bunch of chunks. OK.
Most experimental results are true. So how
does that work? Well, we're gonna have to
deal with that.
But enough about light. If this is true of
light, if light, depending on what experiment
you do and how you do the experiment, sometimes
it seems like it's a wave, sometimes it seems
like it's a chunk or particle, which is true?
Which is the better description?
So it's actually worthwhile to not think about
light all the time. Let's think about something
more general. Let's stick to electrons. So
as we saw from yesterday's lecture, you probably
want to be a little bit wary when thinking
about individual electrons. Things could be
a little bit different than your classical
intuition. But here's a crucial thing. Before
doing anything else, we can just think, which
one of these two is more likely to describe
electrons well.
Well electrons are localized. When you throw
an electron at a CRT, it does not hit the
whole CRT with a wavy distribution. When you
take a single electron and you throw it at
a CRT, it goes ping and there's a little glowing
spot. Electrons are localized. And we know
that localized things don't lead to interference.
Some guys at Hitachi, really good scientists
and engineers, developed some really awesome
technology a couple of decades ago. They were
trying to figure out a good way to demonstrate
their technology. And they decided that you
know what would be really awesome, this thought
experiment that people have always talked
about that's never been done really well,
of sending an electron through a two-slitted
experiment. In this case it was like ten slits
effectively, it was a grading. Send an electron,
a bunch of electrons, one at a time, throw
the electron, wait. Throw the electron, wait.
Like our French guy with the boat.
So do this experiment with our technology
and let's see what happens. And this really
is one of my favorite-- let's see, how we
close these screens-- aha. OK. This is going
to take a little bit of-- and it's broken.
No, no. Oh that's so sad.
AUDIENCE: [LAUGHTER]
PROFESSOR: Come on. I'm just gonna let-- let's
see if we can, we'll get part of the way.
I don't want to destroy it. So what they actually
did is they said, look, let's-- we want to
see what happens. We want to actually do this
experiment because we're so awesome at Hitachi
Research Labs, so let's do it. So here's what
they did. And I'm going to turn off the light.
And I set this to some music because I like
it.
OK here's what's happening. One at a time,
individual photons.
[MUSIC PLAYING]
PROFESSOR: So they look pretty localized.
There's not a whole lot of structure. Now
they're going to start speeding it up. It's
100 times the actual speed.
[MUSIC PLAYING]
PROFESSOR: Eh? Yeah.
AUDIENCE: [APPLAUSE]
PROFESSOR: So those guys know what they're
doing. Let's-- there were go. So I think I
don't know of a more vivid example of electron
interference than that one. It's totally obvious.
You see individual electrons. They run through
the apparatus. You wait, they run through
the apparatus. You wait. One at a time, single
electron, like a baseball being pitched through
two slits, and what you see is an interference
effect. But you don't see the interference
effect like you do from light, from waves
on the sea.
You see the interference effect by looking
at the cumulative stacking up of all the electrons
as they hit. Look at where all the electrons
hit one at a time. So is an electron behaving
like a wave in a pond? No. Does a wave in
a pond at a spot? No. It's a distributed beast.
OK yes, it interferes, but it's not localized.
Well is it behaving like a baseball? Well
it's localized.
But on-- when I look at a whole bunch of electrons,
they do that. They seem to interfere, but
there's only one electron going through at
a time. So in some sense it's interfering
with itself. How does that work? Is an electron
a wave?
AUDIENCE: Yes.
PROFESSOR: Does an electron hit at many spots
at once?
AUDIENCE: No.
PROFESSOR: No. So is an electron a wave. No.
Is an electron a baseball? No. It's an electron.
So this is something you're going to have
to deal with, that every once in awhile we
have these wonderful moments where it's useful
to think about an electron as behaving in
a wave-like sense. Sometimes it's useful to
think about it as behaving in a particle-like
sense. But it is not a particle like you normally
conceive of a baseball. And it is not a wave
like you normally conceive of a wave on the
surface of a pond. It's an electron.
I like to think about this like an elephant.
If you're closing your eyes and you walk up
to an elephant, you might think like I've
got a snake and I've got a tree trunk and,
you know, there's a fan over here. And you
wouldn't know, like, maybe it's a wave, maybe
it's a particle, I can't really tell. But
if you could just see the thing the way it
is, not through the preconceived sort of notions
you have, you'd see it's an elephant. Yes,
that is the Stata Center. So-- look, everything
has to happen sometime, right?
AUDIENCE: [LAUGHTER]
PROFESSOR: So Heisenberg-- it's often, people
often give the false impression in popular
books on physics, so I want to subvert this,
that in the early days of quantum mechanics,
the early people like Born and Oppenheimer
and Heisenberg who invented quantum mechanics,
they were really tortured about, you know,
is it an electron, is it a wave. It's a wave-particle
duality. It's both. And this is one of the
best subversions of that sort of silliness
that I know of.
And so what Heisenberg says, the two mental
pictures which experiments lead us to form,
the one of particles the other waves, are
both incomplete and have the validity of analogies,
which are accurate only in limited cases.
The apparent duality rises in the limitation
of our language. And then he goes on to say,
look, you developed your intuition by throwing
rocks and, you know, swimming. And, duh, that's
not going to be very good for atoms.
So this will be posted, it's really wonderful.
His whole lecture is really-- the lectures
are really quite lovely. And by the way, that's
him in the middle there, Pauley all the way
on the right. I guess they were pleased. OK
so that's the Hitachi thing.
So now let's pick up on this, though. Let's
pick up on this and think about what happens.
I want to think in a little more detail about
this Hitachi experiment. And I want to think
about it in the context of a simple two-slit
experiment. So here's our source of electrons.
It's literally a gun, an electron gun. And
it's firing off electrons. And here's our
barrier, and it has two slits in it.
And we know that any individual electron hits
its own spot. But when we take many of them,
we get an interference effect. We get interference
fringes. And so the number that hit a given
spot fill up, construct this distribution.
So then here's the question I want to ask.
When I take a single electron, I shoot one
electron at a time through this experiment,
one electron. It could go through the top
slit, it could go through the bottom slit.
While it's inside the apparatus, which path
does it take?
AUDIENCE: Superposition.
PROFESSOR: Good. So did it take the top path?
AUDIENCE: No.
PROFESSOR: How do you know?
[INTERPOSING VOICES]
PROFESSOR: Good, let's block the bottom, OK,
to force it to go through the top slit. So
we'll block the bottom slit. Now the only
electrons that make it through go through
the top slit. Half of them don't make it through.
But those that do make it through give you
this distribution. No interference. But I
didn't tell you these are hundreds of thousands
of kilometers apart, the person who threw
in the electron didn't know whether there
was a barrier here. The electron, how could
it possibly know whether there was a barrier
here if you went through the top.
This is exactly like our boxes. It's exactly
like our box. Did it go through-- an electron,
when the slits are both open and we know that
ensemble average it will give us an interference
effect, did the electron inside the apparatus
go through the top path? No. Did it go through
the bottom path? Did it go through both? Because
we only see one electron. Did it go through
neither? It is in a--
AUDIENCE: Superposition.
PROFESSOR: --of having gone through the top
and the bottom. Of being along the top half
and being along the bottom path. This is a
classic example of the two-box experiment.
OK. So you want to tie that together.
So let's nuance this just a little bit, though,
because it's going to have an interesting
implication for gravity. So here's the nuance
I want to pull on this one. Let's cheat. OK.
Suppose I want to measure which slit the electron
actually did go through. How might I do that?
Well I could do the course thing I've been
doing which is I could block it and just catch
the-- catch electrons that go through in that
spot. But that's a little heavy-handed. Probably
I can do something a little more delicate.
And so here's the more delicate thing I'm
going to do. I want to build a detector that
uses very, very, very weak light, extremely
weak light, to detect whether the particle
went through here or here. And the way I can
do that is I can sort of shine light through
and-- I'm gonna, you know, bounce-- so here's
my source of light. And I'll be able to tell
whether the electron went through this slit
or it went through this slit. Cool?
So imagine I did that. So obviously I don't
want to use some giant, huge, ultra high-energy
laser because it would just blast the thing
out of the way. It would destroy the experiment.
So I wanna something very diffuse, very low
energy, very low intensity electromagnetic
wave. And the idea here is that, OK, it's
true that when I bounce this light off an
electron, let's say it bounces off an electron
here, it's true it's going impart some momentum
and the electron's gonna change its course.
But if it's really, really weak, low energy
light, then it's-- it's gonna deflect only
a little tiny bit.
So it will change the pattern I get over here.
But it will change it in some relatively minor
way because I've just thrown in very, very
low energy light. Yeah? That make sense? So
this is the experiment I want to do. This
experiment doesn't work. Why.
AUDIENCE: You know which slit it went through.
PROFESSOR: No. It's true that it turns out
that those are correlated facts, but here's
the problem. I can run this experiment without
anyone actually knowing what happens until
long afterwards. So knowing doesn't seem to
play any role in it. It's very tempting often
to say, no, but it turns out that it's really
not about what you know. It's really just
about the experiment you're doing.
So what principle that we've already run into
today makes it impossible to make this work?
If I want to shine really low-energy, really
diffuse light through, and have it scatter
weakly. Yeah.
AUDIENCE: Um, light is chunky.
PROFESSOR: Yeah, exactly. That's exactly right.
So when I say really low-energy light, I don't--
I really can't mean, because we've already
done this experiment, I cannot possibly mean
low intensity. Because intensity doesn't control
the energy imparted by the light. The thing
that controls the energy imparted by a collision
of the light with the electron is the frequency.
The energy in a chunk of light is proportional
to the frequency.
So now if I want to make the effect the energy
or the momentum, similarly-- the momentum,
where did it go-- remember the momentum goes
like h over lambda. If I want to make the
energy really low, I need to make the frequency
really low. Or if I want to make the momentum
really low, I need to make the wavelength
what? Really big. Right? So in order to make
the momentum imparted by this photon really
low, I need to make the wavelength really
long.
But now here's the problem. If I make the
wavelength really long, so if I use a really
long-wavelengthed wave, like this long of
a wavelength, are you ever going to be able
to tell which slit it went through? No, because
the particle could have been anywhere. It
could have scattered this light if it was
here, if it was here, if it was here, right?
In order to measure where the electron is
to some reasonable precision-- so, for example,
to this sort of wavelength, I need to be able
to send in light with a wavelength that's
comparable to the scale that I want to measure.
And it turns out that if you run through and
just do the calculation, suppose I send in--
and this is done in the books, in I think
all four, but this is done in the books on
the reading list-- if you send in a wave with
a short enough wavelength to be able to distinguish
between these two slits, which slit did it
go through, the momentum that it imparts precisely
watches-- washes out is just enough to wash
out the interference effect, and break up
these fringes so you don't see interference
effects.
It's not about what you know. It's about the
particulate nature of light and the fact that
the momentum of a chunk of light goes like
h over lambda. OK? But this tells you something
really interesting. Did I have to use light
to do this measurement? I could have sent
in anything, right? I didn't have to bounce
light off these things.
I could have bounced off gravitational waves.
So if I had a gravitational wave detector,
so-- Matt works on gravitational wave detectors,
and so, I didn't tell you this but Matt gave
me a pretty killer gravitational wave detector.
It's, you know, here it is. There's my awesome
gravitational wave detector. And I'm now going
to build supernova. OK.
And they are creeping under this black hole,
and it's going to create giant gravitational
waves. And we're gonna use those gravitational
waves and detect them with the super advanced
LIGO. And I'm gonna detect which slit it went
through. But gravitational waves, those aren't
photons. So I really can make a low-intensity
gravitational wave, and then I can tell which
slit it went through without destroying the
interference effect. That would be awesome.
What does that tell you about gravitational
waves? They must come in chunks. In order
for this all to fit together logically, you
need all the interactions that you could scatter
off this to satisfy these quantization properties.
But the energy is proportional to the frequency.
The line I just gave you is a heuristic. And
making it precise is one of the great challenges
of modern contemporary high-energy physics,
of dealing with the quantum mechanics and
gravity together.
But this gives you a strong picture of why
we need to treat all forces in all interactions
quantum-mechanically in order for the world
to be consistent. OK. Good. OK, questions
at this point? OK. So-- oh, I forgot about
this one-- so there are actually two more.
So I want to just quickly show you-- well,
OK.
So, this is a gorgeous experiment. So remember
I told you the story of the guy with the boat
and the opaque wall and it turns out that's
a cheat. It turns out that this opaque screen
doesn't actually give you quantum mechanically
isolated photons. They're still, in a very
important way, classical. So this experiment
was done truly with a source that gives you
quantum mechanically isolated single photons,
one at a time.
So this is the analogue of the Hitachi experiment.
And it was done by this pretty awesome Japanese
group some number of years ago. And I just
want to emphasize that it gives you exactly
the same effects. We see that photons-- this
should look essentially identical to what
we saw at the end of the Hitachi video. And
that's because it's exactly the same physics.
It's a grating with something like 10 slits
and individual particles going through one
at a time and hitting the screen and going,
bing.
So what you see is the light going, bing,
on a CCD. It's a pretty spectacular experience.
So let's get back to electrons. I want another
probe of whether electrons are really waves
or not. So this other experiment-- again,
you're going to study this on your problem
set-- this other experiment was done by a
couple of characters named Davisson and Germer.
And in this experiment, what they did is they
took a crystal, and a crystal is just a lattice
of regularly-located ions, like diamond or
something. Yeah?
AUDIENCE: Before you go on I guess,
I wanted to ask if the probability of a photon or an electron going through the 10 slits is about the same?
PROFESSOR: Is what, sorry?
AUDIENCE: Is exactly the same.
PROFESSOR: You mean for different electrons?
AUDIENCE: Yeah.
PROFESSOR: Well they can be different if the
initial conditions are different. But they
could be-- if the initial conditions are the
same, then the probabilities are identical.
So every electron behaves identically to every
other electron in that sense. Is that what
you were asking?
AUDIENCE: It is actually like through any [INAUDIBLE] the probability of it going like [INAUDIBLE]?
PROFESSOR: Sure, absolutely. So the issue
there is just a technological one of trying
to build a beam that's perfectly columnated.
And that's just not doable. So there's always
some dispersion in your beam. So in practice
it's very hard to make them identical, but
in principle they could be if you were infinitely
powerful as an experimentalist, which-- again,
I was banned from the lab, so not me.
So here's our crystal. You could think of
this as diamond or nickel or whatever. I think
they actually use nickel but I don't remember
exactly. And they sent in a beam of electrons.
So they send in a beam of electrons, and what
they discover is that if you send in these
electrons and watch how they scatter at various
different angles-- I'm going to call the angle
here of scattering theta-- what they discover
is that the intensity of the reflected beam,
as a function of theta, shows interference
effects.
And in particular they gave a whole calculation
for this, which I'm not going to go through
right now because it's not terribly germane
for us-- you're going to go through it on
your problem set, so that'll be good and it's
a perfect thing for your recitation instructors
to go through. But the important thing is
the upshot. So if the distance between these
crystal planes is L-- or, sorry, d-- let me
call it d. If the distance between the crystal
planes is d, what they discover is that the
interference effects that they observed, these
maxima and minima, are consistent with the
wavelength of light. Or, sorry, with the electrons
behaving as if they were waves with a definite
wavelength, with a wavelength lambda being
equal to some integer, n, over 2d sine theta.
So this is the data-- these are the data they
actually saw, data are plural. And these are
the data they actually saw. And they infer
from this that the electrons are behaving
as if they were wave-like with this wavelength.
And what they actually see are individual
electrons hitting one by one. Although in
their experiment, they couldn't resolve individual
electrons. But that is what they see.
And so in particular, plugging all of this
back into the experiment, you send in the
electrons with some energy, which corresponds
to some definite momentum. This leads us back
to the same expression as before, that the
momentum is equal to h over lambda, with this
lambda associated. So it turns out that this
is correct.
So the electrons diffract off the crystal
as if they have a momentum which comes with
a definite wavelength corresponding to its
momentum. So that's experimental result--
oh, I forgot to check off four-- that's experimental
result five, that electrons diffract. We already
saw the electron diffraction.
So something to emphasize is that-- so these
experiments as we've described them were done
with photons and with electrons, but you can
imagine doing the experiments with soccer
balls. This is of course hard. Quantum effects
for macroscopic objects are usually insignificantly
small. However, this experiment was done with
Buckyballs, which are the same shape as soccer
balls in some sense. But they're huge, they're
gigantic objects. So here's the experiment
in which this was actually done. So these
guys are just totally amazing. So this is
Zellinger's lab. And it doesn't look like
all-- I mean it looks kind of, you know. It's
hideous, right? I mean to a theorist it's
like, come on, you've got to be kidding that
that's--
But here's what a theorist is happy about.
You know, because it looks simple. We really
love lying to ourselves about that. So here's
an over. We're going to cook up some Buckyballs
and emit them with some definite known thermal
energy. Known to some accuracy. We're going
to columnate them by sending them through
a single slit, and then we're going to send
them through a diffraction grating which,
again, is just a whole bunch of slits.
And then we're going to image them using photo
ionization and see where they pop through.
So here is the horizontal position of this
wave along the grating, and this is the number
that come through. This is literally one by
one counts because they're going bing, bing,
bing, as a c60 molecule goes through. So without
the grating, you just get a peek. But with
the grating, you get the side bands. You get
interference fringes.
So these guys, again, they're going through
one by one. A single Buckyball, 60 carbons,
going through one by one is interfering with
itself. This is a gigantic object by any sort
of comparison to single electrons. And we're
seeing these interference fringes.
So this is a pretty tour de force experiment,
but I just want to emphasize that if you could
do this with your neighbor, it would work.
You'd just have to isolate the system well
enough. And that's a technological challenge
but not an in-principle one.
OK. So we have one last experimental facts
to deal with. And this is Bell's Inequality,
and this is my favorite one. So Bell's Inequality
for many years languished in obscurity until
someone realized that it could actually be
done beautifully in an experiment that led
to a very concrete experiment that they could
actually do and that they wanted to do.
And we now think of it as an enormously influential
idea which nails the coffin closed for classical
mechanics. And it starts with a very simple
question. I claim that the following inequality
is true: the number of undergraduate-- of
the number of people in the room who are undergraduates,
which I'll denote as U-- and not blonde, which
I will denote as bar B-- so undergraduates
who are not blonde-- actually let me write
this out in English. It's gonna be easier.
Number who are undergrads and not blonde plus
the number of people in the room who are blonde
but not from Massachusetts is strictly greater
than or equal to the number of people in the
room who are undergraduates and not from Massachusetts.
I claim that this is true. I haven't checked
in this room. But I claim that this is true.
So let's check. How many people are undergraduates
who are not blonde? OK this is going to--
jeez. OK that's-- so, lots. OK. How many people
are blonde but not from Massachusetts? OK.
A smattering. Oh God, this is actually going
to be terrible.
AUDIENCE: [LAUGHTER]
PROFESSOR: Shoot. This is a really large class.
OK. Small. And how many people are undergraduates
who are not from Massachusetts? Yeah, this--
oh God. This counting is going to be-- so
let's-- I'm going to do this just so I can
do the counting with the first two rows here.
OK. My life is going to be easier this way.
So how many people in the first two rows,
in the center section, are undergraduates
but not blonde? One, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen. We could dispute some
of those, but we'll take it for the moment.
So, fourteen. You're probably all undergraduates.
So blonde and not from Massachusetts. One.
Awesome. Undergraduates not from Massachusetts.
One, two, three, four, five, six, seven, eight,
nine, ten, eleven, twelve, thirteen, fourteen,
fifteen. Equality.
AUDIENCE: [LAUGHTER]
PROFESSOR: OK. So that-- you might say well,
look, you should have been nervous there.
You know, and admittedly sometimes there's
experimental error. But I want to convince
you that I should never, ever ever be nervous
about this moment in 8.04. And the reason is
the following. I want to prove this for you.
And the way I'm gonna prove it is slightly
more general, in more generality. And I want
to prove to you that the number-- if I have
a set, or, sorry, if the number of people
who are undergraduates and not blonde which,
all right, is b bar plus the number who are
blonde but not from Massachusetts is greater
than or equal to the number that are undergraduates
and not from Massachusetts.
So how do I prove this? Well if you're an
undergraduate and not blonde, you may or you
may not be from Massachusetts. So this is
equal to the number of undergraduates who
are not blonde and are from Massachusetts
plus the number of undergraduates who are
not blonde and are not from Massachusetts.
It could hardly be otherwise. You either are
or you are not from Massachusetts. Not the
sort of thing that you normally see in physics.
So this is the number of people who are blonde
and not from Massachusetts, number of people
who are blonde, who are-- so if you're blonde
and not from Massachusetts, you may or may
not be an undergraduate. So this is the number
of people who are undergraduates, blonde,
and not from Massachusetts plus the number
of people who are not undergraduates, are
blonde and are not from Massachusetts.
And on the right hand side-- so, adding these
two together gives us plus and plus. On the
right hand side, the number of people that
are undergraduates and not from Massachusetts,
well each one could be either blonde or not
blonde. So this is equal to the number that
are undergraduates, blonde, and not from Massachusetts,
plus-- remember that our undergraduates not
blonde and not from Massachusetts. Agreed?
I am now going to use the awesome power of--
and so this is what we want to prove, and
I'm going to use the awesome power of subtraction.
And note that U, B, M bar, these guys cancel.
And U, B bar, M bar, these guys cancel. And
we're left with the following proposition:
the number of undergraduates who are not blonde
but are from Massachusetts plus the number
of undergrad-- of non-undergraduates who are
blonde but not from Massachusetts must be
greater than or equal to zero.
Can you have a number of people in a room
satisfying some condition be less than zero?
Can minus 3 of you be blonde undergraduates
not from Massachusetts? Not so much. This
is a strictly positive number, because it's
a numerative. It's a counting problem. How
many are undergraduates not blonde and from
Massachusetts. Yeah? Everyone cool with that?
So it could hardly have been otherwise. It
had to work out like this.
And here's the more general statement. The
more general statement is that the number
of people, or the number of elements of any
set where each element in that set has binary
properties a b and c-- a or not a, b or not
b, c or not c. Satisfies the following inequality.
The number who are a but not b plus the number
who are b but not c is greater than or equal
to the number who are a but not c. And this
is exactly the same argument.
And this inequality which is a tautology,
really, is called Bell's Inequality. And it's
obviously true. What did I use to derive this?
Logic and integers, right? I mean, that's
bedrock stuff.
Here's the problem. I didn't mention this
last time, but in fact electrons have a third
property in addition to-- electrons have a
third property in addition to hardness and
color. The third property is called whimsy,
and you can either be whimsical or not whimsical.
And every electron, when measured, is either
whimsical or not whimsical. You never have
a boring electron. You never have an ambiguous
electron. Always whimsical or not whimsical.
So we have hardness, we have color, we have
whimsy. OK. And I can perform the following
experiment. From a set of electrons, I can
measure the number that are hard and not black,
plus the number that are black but not whimsical.
And I can measure the number that are hard
and not whimsical. OK?
And I want to just open up the case a little
bit and tell you that the hardness here really
is the angular momentum of the electron along
the x-axis. Color is the angular momentum
of the electron along the y-axis. And whimsy
is the angular momentum of the electron along
the z-axis. These are things I can measure
because I can measure angular momentum.
So I can perform this experiment with electrons
and it needn't be satisfied. In particular,
we will show that the number of electrons,
just to be very precise, the number of electrons
in a given set, which have positive angular
momentum along the x-axis and down along the
y-axis, plus up along the y-axis and down
along the z-axis, is less than the number
that are up. Actually let me do this in a
very particular way. Up... zero down at theta.
Up at theta, down at-- two theta is greater
than the number that are up at zero and down at theta.
Now here's the thing-- two theta. You can't
at the moment understand what this equation
means. But if I just tell you that these are
three binary properties of the electron, OK,
and that it violates this inequality, there
is something deeply troubling about this result.
Bell's Inequality, which we proved-- trivially,
using integers, using logic-- is false in
quantum mechanics.
And it's not just false in quantum mechanics.
We will at the end of the course derive the
quantum mechanical prediction for this result
and show that at least to a predicted violation
of Bell's Inequality. This experiment has
been done, and the real world violates Bell's
Inequality. Logic and integers and adding
probabilities, as we have done, is misguided.
And our job, which we will begin with the
next lecture, is to find a better way to add
probabilities than classically. And that will
be quantum mechanics See you on Tuesday.
AUDIENCE: [APPLAUSE]
