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visit MIT OpenCourseWare at ocw.mit.edu. 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 804. 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 804 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 802 and 803. But this is 804. 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 chain. 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 802 and 803 and extract a little bit
more information out of this. So here's something
you learned from 802. In 802 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 803 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 804
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: [INAUDIBLE]. 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: [INAUDIBLE] position. 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: [INAUDIBLE]. 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: [INAUDIBLE]. 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. 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: [INAUDIBLE]? PROFESSOR: Is what, sorry? AUDIENCE: [INAUDIBLE]? 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: [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 804. 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 at zero, down at theta. Up at theta, down
at-- to theta is greater than the number that are
up at zero and down at theta. Now here's the thing-- to 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]
