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ROBERT FIELD: Last
time, we talked
about the photoelectric effect.
What was that?
And what were the
important points?
Yes?
AUDIENCE: It's
quantized and has energy
associated with its frequency.
ROBERT FIELD: Yes.
OK, so, the idea of
quantization of electromagnetic
radiation and photons.
And the photon has
an energy h times nu.
Nu is the frequency.
And the evidence was mostly
from a plot of what versus what.
Somebody else?
Yes?
AUDIENCE: Frequency of the
incoming photon versus--
or, that's the x-axis.
So, kinetic energy
of the ejected
electron versus frequency.
ROBERT FIELD: Exactly.
And the slope of that plot,
which was h, is universal.
It doesn't matter where
the electron came from.
And this was really
an amazing thing.
And then, the other
thing we talked about
was Compton scattering.
And what did Compton
scattering tell us?
Yes.
AUDIENCE: Photon
has a momentum--
ROBERT FIELD: Yes.
AUDIENCE: --transfer.
ROBERT FIELD: We're interested
in the particle-like character
of what we think of as waves.
And we saw that the
waves were particles.
And particles-- or
at least packets.
And these packets had
definite momentum.
And that was a
wonderful observation.
So, today, this is the menu of
what I'm going to talk about.
And at the end, there's
a magic word, "spectra."
And I like that
because what we're
going to be discovering today
is that the internal structure
of atoms and molecules--
we are not allowed to observe it
directly, but it's surprising.
And it's encoded in something
which we can observe,
which is a spectrum.
And the spectrum that I
will show you at the end
is one that is--
contains essentially
no information,
but acts as a template
for what we really
want to know about how things
are different from hydrogen
atom.
And that's the beginning of our
exploration of the structure
of atoms and molecules.
It's through the spectrum and
it's how it's different from--
in subtle ways-- the spectrum
of the hydrogen atom.
OK, so, we're going to turn our
focus today to the electron,
as opposed to light.
And we're going to
play the same game.
We know the electron
is a particle.
Does anyone want to tell
me why we know that?
Is there any-- anybody
who's got a clue?
You can-- oh, good.
Yes.
AUDIENCE: There's an experiment
with the little oil drops,
where they suspended
them and found that--
ROBERT FIELD: Yes.
AUDIENCE: --charge is quantized.
ROBERT FIELD: I love
that experiment.
That's the Millikan experiment.
One of the reasons I love it is
because Millikan and Mulliken
are two different people.
So I find that it's really easy
to come up with one of them.
And I remember Mulliken
is a spectroscopist
and Millikan was a
different kind of physicist.
But they're both famous and
they both have connections
to important universities--
University of Chicago for
Millikan and Mulliken, MIT
and Caltech.
So we're going to be
looking at something
that we know is a particle.
And we're going to show that
it has wave characteristics.
We want to be able to show that
the electron has a wavelength
and it follows the
same equation that we
use to describe the behavior
of electromagnetic radiation.
So how would we control the
momentum of an electron?
Yes.
AUDIENCE: So you can
control the kinetic energy
that it has by putting it
through a certain potential.
ROBERT FIELD: OK, so, we
know we can easily measure
the momentum of a particle.
That's not a big challenge.
But then how would we
measure its wavelength?
Yes.
AUDIENCE: Some type
of diffraction?
ROBERT FIELD: Yes.
We basically use some
material, which acts a ruler.
We have a thin metal foil and
the distances between atoms
in the foil are constant.
So that's the ruler
against which we measure
the wavelength of light.
And we'll talk about the
Davisson-Germer experiment,
where we measure the wavelength.
And then we'll talk about the
Geiger-Marsden experiment,
where we say, well, atoms
have electrons in them.
And what is the
structure of an atom?
Remember, we can't look inside.
So we have to use some
kind of an experiment
to be able to look
inside the atom.
Now, physicists have one
trick they often use.
To find out the internal
structure of something,
they shoot a
particle, or a wave,
at it that has a wavelength
comparable to the distances
you're hoping to measure.
So if you have a very high
energy probe particle,
it will have a very
short wavelength
and it will look sort
of like a bullet.
And we know how bullets scatter
off of targets or hit targets.
We also, if we
choose the wavelength
to be comparable to
the distances we're
expecting to measure, then
we're going to see diffraction.
This is a kind of subject that
lends itself to exam questions.
So let's start out by talking
about the Geiger-Marsden
experiment--
I mean, the
Davisson-Germer experiment.
So we have a beam of X-rays
or a beam of electrons--
either one.
And we have an aluminum foil.
And we have-- it's
just an intense beam.
We want to stop most of it
before it hits a detector.
And, so, this is some
kind of a screen or--
So what we're
looking for is, when
the X-rays or the electrons
scatter off of this ruler,
we get something that appears on
the screen as pairs of circles.
This is the powder
pattern because this
is a multi-crystalline object.
But in each object, we have a
bunch of equally-spaced atoms,
where this is the
lattice constant
and this is the square root of
2 times the lattice constant.
So each atom has
nearest neighbors
and second-nearest neighbors.
So we have two rulers going on.
And one ruler will
give one set of rings
and the other ruler will give
a different set of rings.
And because these particles
are randomly oriented,
instead of having
spots, you have circles.
And there's all sorts of
information in these powder
patterns.
But basically,
they're saying, well,
we're seeing a
structure which is
related to something we know.
How would we know the distance
between atoms in a foil?
Using macroscopic measurements?
Yes.
You're hot today.
AUDIENCE: You have
access to density
and you have access to
the non-atomic weight.
ROBERT FIELD: That's it, yes.
So it's a simple matter
to know at least what is
the magnitude of the distance.
There is an issue of what
is the crystal structure.
And there are
different structures
and that will give rise
to different features
in the powder pattern.
But the important
thing is we do this.
We look at the pattern that
emerges when we shoot X-rays
at this screen.
And we already know that
X-rays have wavelengths.
And we know that
they have momentum.
We know about the
scattering of photons.
And as a result,
we know that we can
predict exactly what the pattern
associated with the X-ray
scattering is going to be.
And then we do the same
thing with electrons.
And now for the electrons,
we can control the momentum.
That's easy.
And what we want to know
is what is the wavelength.
And we have the
wavelength of the X-rays.
And so what we do is we vary the
momentum of the electrons until
the powder pattern
for the electron--
the diffraction pattern
for the electron--
is exactly the same
as the diffraction
pattern for the X-ray.
And we discover that,
for the electron,
we have the same result.
OK, so, we have, now, photons.
They have wavelengths.
And the momentum was the
surprise for the photons.
And we have particles, which
have wavelengths and momentum.
And the wavelength was a
surprise for the particle.
So it doesn't matter.
Everything follows
this equation.
And this equation was
anticipated by de Broglie who,
in his PhD thesis--
you know, he's a
person about your age--
and he wrote his thesis in 1924.
And, among other
brilliant things,
he said that everything should
follow this simple equation.
And that was a brave statement.
And it predicted
that de Broglie was
going to make a lot of brilliant
statements in his career.
And this was just the first of--
and one of the nicest--
but we'll hear a little
bit more about de Broglie
by the time I'm finished
with this lecture.
So, we're now
worried about atoms.
And we already know that atoms
have a diameter, roughly.
We know that from the
density, the typical size
we like to have--
quantities that describe
macroscopic objects.
Which are not like 10 to the
minus 20, but like 1 to 100.
And so, the angstrom unit, which
is 10 to the minus 10 meters,
is a very useful thing
for talking about sizes
of atoms and molecules.
So if we have an atom of
a size about one angstrom,
we can use this
equation to say, well,
what would it take for an
electron to fit inside an atom?
So we specify this--
we know this, we know that--
and that determines what the
momentum would have to be.
And these are all
simple calculations.
And since I don't like doing
calculations on the board,
and I don't really
need to do this now--
you need to be able
to do them, quickly,
if I ask you on the exam.
But basically, what
we end up finding out
is that the velocity
of the electron
would have to be 7.25 times
10 to the 6 meters per second.
Which is OK, it's pretty fast.
It's a few percent of
the speed of light.
But that would correspond
to a kinetic energy,
which is 2.4 times 10
to the minus 7 joules.
Remember, I don't like
these kinds of units.
But it also corresponds to--
doing a unit conversion--
149 electron volts.
Electron volts are a good
unit for energy for atoms
because the ionization energy--
the energy it takes to pull
an electron off of an atom--
is always somewhere between
five and 15 electron volts.
So you always want to calibrate
yourself, your insight,
in terms of numbers which
are in the small scale,
rather than having to
remember the exponent.
All right, 149 electron volts.
Should that bother you?
Well, it can't bother
you yet because you
don't know about what
the ionization energy is.
But I just told you.
It's between five and 15.
This is a factor
of 10-- too big.
So this is going
to be a problem.
How is it possible
for things so small
to have an electron fitting in
that small size without it just
leaving because it's
just way too high energy?
And so that leads us to
ask questions about, well,
what is the internal
structure of an atom?
How can an atom somehow
accommodate this electron which
needs to somehow fit inside?
So that was the basis for the
Geiger-Marsden experiment.
Now, the Geiger-Marsden
experiment
looks very similar to
the previous experiment.
And here we have--
whoops-- alpha particles.
Alpha particles are
helium-2 plus ions.
And they're produced
by radioactive decay.
And they have a tremendous
amount of energy.
More energy than was
possible in the days
these experiments we're doing
to create for a particle.
In fact, one of the
earliest experiments,
or apparatuses, capable
of producing very high
energy electrons was built
by Robert Van de Graaff, here
at MIT.
And this was in the form
of cylindrical towers,
right near the parking garage.
And it was there for the
first 10 years I was at MIT.
I'm very old, but that's
still fairly recent.
But anyway, Van de Graaff could
make high energy particles
and it was really neat.
And what he could do
was dwarfed by what
can be done in electron
accelerators, now.
But in the days when the
Geiger-Marsden experiment was
done, which was 1911,
there was no way of making
and controlling the
energy of an electron--
or of any particle.
And here we have
some particles which
are produced by
radioactive decay, which
have tremendous energy.
So they're heavy and
they have high energy.
And so that means the
wavelength is very small.
We want to use
these helium ions.
Yes?
AUDIENCE: Could you
not also control
the energy of each particle as
they're passing through, right?
ROBERT FIELD: Yes.
But you would need
a very high voltage.
And though that is
something we could imagine
doing now, but in
1911, the ability
to do that sort of thing
with control was not there.
There needed to be advances
in vacuum technology.
There needed to be
advances in power supplies.
I mean, we're talking
about very high voltages.
And you wouldn't want to
do that in your laboratory,
even now with the capability.
I remember when I was
a graduate student,
we had these things called
Spellman power supplies.
And they could
produce 40 kilovolts.
They were really scary.
But that's nothing
compared to what you need.
OK, so, we want bullets.
We want to have these
helium particles interacting
with a thin metal foil.
We have a whole
array of atoms here.
And we have a little hole here.
And what's going to
happen is this radiation
is going to hit these atoms.
And there's going
to be backscatter
and forward scattering.
And the crucial
experiment was to measure
the ratio of backscattering
to forward scattering.
Now, if we have a target that
looks sort of like a smear,
then the forward and backward
scattering would be similar.
If we have a target that looked
like a bunch of tiny points,
there would very rarely
be backscattering.
All of the scattering
would be forward
because most of the
particles don't hit anything.
And so what was found,
and what was the surprise,
is that there was very
little backscattering.
And that implied that
the ratio of the size
of the target to the size of
the particle was enormous.
The particles that scattered--
the alpha particles-- were tiny.
They had a size--
something like 10 to the minus
4 times the typical dimension
of an atom.
So this is jellium.
And this is a perfectly
reasonable approach.
That the positive and negative
charges that make up an atom
are distributed uniformly.
This was the surprise.
How do we explain,
now, if atoms that
are scattering the
alpha particles
are really small, even
compared to the one angstrom?
Well, how do they
stick together?
Why is matter not compressable?
So what is going on here?
Now, I'm not exactly sure
of the genealogy here,
but Geiger and Marsden
were workers, or students,
in the Rutherford lab.
And the old man wanted to save
face or to say, oh, here's
an experiment.
We learned something from this.
You know, this is what
we do-- this is my job.
But anyway, Rutherford said,
well, maybe it's like this.
We have a nucleus and
we have the electrons.
So we have a nucleus where
all the positive charge
of the atom, and most
of the mass, resides.
And we have electrons
in circular orbits.
So maybe these
circular orbits explain
why you can't compress
atoms to something
that would be commensurate
with the size of the nuclei.
That the electrons cause a
repulsion and the structure
is stable.
So this is a pretty
reasonable hypothesis
until one analyzes it.
So what we have is a
positive charge here,
negative charge here.
And so, there is
Coulomb attraction
and there's centrifugal force,
or centripetal acceleration.
And we have to have
these two things match.
So the inward force
is minus the charge
on the electron squared
over 4 by epsilon 0
and 1 over r squared.
And the centrifugal, it's--
OK, this is-- you know all this.
You know to do this.
Have known it since
high school, probably.
And so, you can combine all
these things, say the inward
and outward-- the
inward force is exactly
canceling the outward force.
And you can solve
for the velocity.
And the velocity is q over
the electron squared over 4
by epsilon 0 mass
of the electron
and the radius of the
orbit, square root.
This is a trivial derivation.
I won't insult you by
attempting to do it and try
to increase your
understanding of the equation,
because you already
understand it.
So this is the
requirement for the radius
of the circular orbit.
And it has the mass--
I mean, this is the
requirement for the velocity.
And this is the radius, here.
So we know all that.
There is nothing about
quantization, yet.
We know that for any
radius, the electron
will have a certain velocity.
And we can choose
whatever radius,
we want whatever
velocity we want.
And that corresponds to
whatever energy we get.
What we're interested in is
the frequency of the orbit.
And so that will
be 1 over the time
it takes for the
electron to go around.
And 1 over the time it takes
for electron to go around
is 2 pi r, the circumference,
divided by the velocity.
So we have the
velocity is equal to--
I mean, the frequency is
equal to 1 over 2 pi r.
And we can write an
equation for this.
And that's in the notes.
In fact, there was
a typo in the notes.
Which has been
corrected and read.
But I don't need to
tell you what it is.
We can calculate this frequency.
The reason we
calculate the frequency
is because we know if we have
electrons moving back and forth
at some frequency,
they're going to radiate
electromagnetic radiation
at that frequency.
Well, where did that
energy come from?
It came from the
motion of the electron.
So it has to give
up kinetic energy.
Now, the energy is
the kinetic energy
plus the potential energy.
So if it gives up energy,
some of these two things
has to decrease.
And what happens
is this decreases
faster that this increases.
And what ends up happening
is that the electron
has a death spiral.
What happens will be
that the electron will
go in a spiral, going
faster and faster as it
goes to smaller and smaller
radius, and annihilate itself.
So this is garbage.
This can't be true.
It violates laws that
everybody knows are right.
So one needs to find a
way to live with this.
Now, one really doesn't
need to find a way
to live with it until
you realize what happens.
Because this picture, subject
to a couple of hypotheses,
predicts an infinite
number of 10-digit numbers.
It's not an accident.
Maybe one prediction
would be fine.
But all of the lines in the
spectra of hydrogen atom,
helium ion, lithium
doubly-charged ion, all
of those are predicted with
no adjustable parameters
to measurement accuracy.
Now, at the time
this was being done,
the measurement accuracy might
have been only a part in 1,000,
or maybe a part in a million.
But a part in 10 to the
10th, beyond that we're
starting to get into
fundamental physics.
But this is an
astonishing thing.
And so I have to explain what
the additional assumptions were
because we've got something
that predicts things
we have no business knowing.
And there was no
explanation for the spectrum
before these experiments, or
this picture, was developed.
So we have to first find a way,
whether it's believable or not,
of getting rid of the
radiative collapse.
So Bohr proposed that
angular momentum, l vector r
cross p is conserved.
Well, we know that angular
momentum is conserved.
But for a microscopic system,
what it means to be conserved
may be a little bit more subtle.
He proposed that angular
momentum is conserved
and that the angular momentum,
the magnitude of the angular
momentum, had to have
a particular value.
And that value was
integer times h bar.
h bar is the Planck's
constant divided by 2 pi.
Now, this is complete nonsense.
Why should it be conserved?
Why should it be restricted
to this set of values?
Well, the reason we accept
that it's restricted
is because it gives the energy
levels that are observed.
Now, before I get to
energy level-- well, I do.
OK, we have energy levels.
We find that the energy is equal
to minus some constant over n
squared.
Same n as in here.
This is the Rydberg constant.
And it's something
that you can measure.
It's basically a whole bunch of
fundamental constants combined.
And so, it has a value.
This is one of the numbers
in my permanent memory.
And that's the value
of the Rydberg constant
in reciprocal centimeter units.
And to get it into energy units
you multiply by h times c,
or to get it into frequency,
you just multiply it
by the speed of light.
So, anyway, this
is a number that is
known to many decimal places.
And it is generated by this
idea that the angular momentum
has to be certain values.
Conservation is
good, that's easy.
This is weird.
And it's also wrong
because we find out
later that the possible
values of n include 0.
Which would completely
mess up the Bohr model.
But, anyway, this,
then, in combination
with this amazing
statement that--
OK, we have the nth energy
and the n prime-th energy.
And the spectrum corresponds
to the frequency, corresponds
to e n minus e n prime, over h.
So everything we see in
the emission spectrum
of the hydrogen
atom, or in a gas--
which is mostly H2--
there are transitions
associated with the free atoms.
And they're always around
this simple equation,
based on the Rydberg equation.
This says the spectra are
telling us about the energy
level differences.
And it's a simple equation.
And it's true.
It's true at incredibly
high accuracy.
And it tells you nothing except
the mass of the particle.
Because the mass
of the particle--
for this Rydberg
equation-- we have
it expressed in terms of
the mass of the electron.
But it really should be the
reduced mass of the mass
of the nucleus times the
mass of the electron,
or the mass of the nucleus
plus the mass of the electron.
And this is something we know
for all two-body interactions.
That was known well before
the time of these experiments.
If you have two
things interacting,
the reduced mass is
what you want rather
than the individual masses.
And so, the only information
in these one-electron spectra
is the mass of the nucleus.
And there's not much difference
in this reduced mass effect.
But it's enough to say this
is a spectrum of hydrogen,
as opposed to lithium 2 plus.
OK, but we still
have a problem--
a very serious problem.
Why is the angular
momentum conserved?
Why is the angular momentum
forced to have a certain value?
Well, I've really finished
this lecture pretty fast.
Let me just get to the
end and I will go back.
All right, so, the
problem is this electron
is assumed to be a particle
and it's assumed to be moving.
So the equation--
Maxwell's equations--
all of the equations
about motion of
charged particles
say if it's moving
it's going to radiate,
whereas if it's oscillating,
it's going to radiate.
So maybe the problem is
that it's not moving.
Remember, the particles are
both particles and waves.
So we could imagine that,
around this circular orbit,
we have standing waves--
no motion.
And this led to this
Schrodinger equation,
which talks about the states of
the electrons that are allowed.
It's basically the
classical wave equation,
with a couple little twists.
And the thing about
waves, you remember,
we can have constructive and
destructive interference.
We can have standing waves.
So there doesn't need to be
a motion of our particle.
There could be some
static description
of the probability of finding
the electron everywhere
around this orbit.
And that's the
Schrodinger equation.
In the next lecture,
I'm going to talk
about the classical
wave equation, which
will be the warm-up for
the Schrodinger equation.
And the Schrodinger equation
explains everything.
People have made
some really fantastic
philosophical statements
about the Schrodinger.
It contains everything
that we need to know.
The problem is we can't
solve the equation exactly.
But it's true.
And it contains
everything that we'd ever
want to know about the
microscopic structure of atoms
and molecules.
So we've been led by these
very simple experiments, which,
now, you could do--
really trivially.
You wouldn't have to be a smart
student of a smart advisor,
or stupid advisor.
You would be able to
do these experiments.
And you could say, yeah,
this is all very weird,
but know now we know that
spectra are everything.
And I'm a spectroscopist.
I'm very proud of
this because the idea
that you can make
a few measurements
and say something about the
internal structure of an atom
or molecule--
that's a fantastic thing.
And what we've seen--
the spectrum of
one-electron atoms--
is something which
is really simple.
It's the template for
understanding all complexity.
Because everything is
different from hydrogen.
Hydrogen has a point
charge at the center.
It's not quite a point charge,
and that's actually a subject
of even modern physics.
And other atoms are
not a point charge
because they have electrons.
And so there is a
concentration of charge--
the thing to which the electron
is attached is space filling.
So that results in a shift
at the energy levels.
And the shift at the
energy levels, and how
that shift depends on the
orbital angular momentum,
tells you something about
the shape of this charge
distribution--
the radial shape.
So everything we
do in spectroscopy
is somehow referenced
to something
we understand perfectly, but
which is not of much interest.
But it's a template for
building up our understanding
of everything.
And this is a kind of
a radical statement.
And I get to say this
because I'm up here
and I do this for a living.
I mean-- not teaching,
but research.
And I really believe
that the things
that we are enabled to observe
about the microscopic structure
of things are encoded
in something completely
unlike looking at it.
And our job is to figure
out how to break that code.
And that's what I've done for
the last 50 years and it's fun.
OK, so, what more could
I say to amuse you
for five-- six minutes?
Not much, because I've skipped
a lot of really great stuff.
But go back to de Broglie.
De Broglie had the
hypothesis that there
is an integer number
of wavelengths
around the circular orbit.
And that was telling
you that it's stable
because if it weren't an
integer number of wavelengths,
the electron would
self annihilate.
But that takes a valid
point and moves it
into something which
is a little bit wrong.
Because we're not trying
to have a particle moving,
we're having a distribution
of probabilities.
And there are still wavelengths
and nodal structures.
And the stable solutions
do involve the particle
not self annihilating.
And so, de Broglie
scores another triple.
I mean, he didn't come up
with the Schrodinger equation.
So that's the home run that
says, OK, we have the material,
now to explain everything.
The next lecture will
be just introducing
the mathematics of the wave
equation and the crucial ideas.
And that will lead into the
following lecture, where we all
talk about the
Schrodinger equation.
So I can stop now.
Thanks.
