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PROFESSOR: So, a couple
of announcements.
Tomorrow we've got Quiz Two.
Based on Homework Two
And then, Thursday,
we'll have the
periodic table quiz.
I give you the numbers,
you give me the letters.
And Friday is the last
day for the contest.
I'll have office hours.
I know when others
have office hours.
I'll be available 4:30 to 5:30
down the hall in my office.
Where is Grant's, who's
buried in Grant's Tomb?
Grant.
Where are my office hours held?
In my office.
Good.
Last day.
Last day.
It was a rough weekend.
I can't get a smile,
can't get a laugh.
So, last day we looked at the
validation of the Bohr model.
And we had two pieces
of experimental data.
First were the hydrogen
spectrum lines,
measured in 1853 by Angstrom
and fit to an equation by J.J.
Balmer in 1885.
And, secondly, we saw the
Franck-Hertz experiment,
in which we were able to
get the sense that energy
levels within a multi-electron
atom, like, mercury,
those energy levels
are also quantized,
which gave credence to
the quantum condition.
Which said that the movement
of an electron, the movement
of an electron, is quantized.
And then we started looking
at the limitations of the Bohr
model, problems
with the Bohr model.
And I said, in two
words, fine structure.
And we started looking
at fine structure.
We see that the 656
nanometer line in hydrogen,
in point of fact, is a doublet.
It's actually two lines
very closely spaced.
Bohr model is incapable
of explaining this.
Zeeman looked at
gas discharge tube,
measured hydrogen spectrum
in a magnetic field,
and found line
splitting proportional
to the intensity of the field.
Stark did similar experiments
only in an electric field,
and also found line
splitting, the degree of which
was proportional to the
intensity of the field.
And the Bohr model is
incapable of explaining this.
Sommerfeld, in 1916, put
a patch on the Bohr model
and proposed that
the electron moves
not only in a circular orbit
but also in elliptical orbits.
And there's a plurality
of these orbits.
But overall, the distance from
the nucleus to the electron
orbits is more or less given by
the principal quantum number n,
and the degree of
eccentricity is tiny.
But, more or less,
described by n.
And then to further
depict what's going on,
he introduced the
orbital quantum number
or, as your book calls it,
the azimuthal quantum number.
And the magnetic quantum number.
And further said that the
energy of the electron
is given by the specifications
according to all three quantum
numbers.
And then, lastly, in
order to get completeness,
we jumped ahead in
time to 1922 with
the Stern-Gerlach experiment,
which was the beam of silver.
Atoms through a divergent
magnetic field, the beam
split in two symmetrically
about the point
at which the beam would land
on the slide in the absence
of a magnetic field.
And that can led to
some deeper thinking
by two graduate students in
Leiden, Goudsmit and Uhlenbeck,
who proposed the notion
of electron spin.
And from that came the
fourth quantum number,
s, and we're going to
throw that into the mix
even though Sommerfeld didn't
give it to us back in 1916.
But I just want to move
forward with all of them.
And we recognize
that s could take
on two values,
plus or minus 1/2,
or we could call it
spin up or spin down.
And, as you're going
to learn in 802,
the convention in
electromagnetism
is the right-hand rule.
That is to say, the thumb
indicates the vector
and the curl of the fingers
indicate the rotation.
So we would argue that
this is an electron
spin from looking top-down
anticlockwise and then
vice versa.
So that's as far as we got.
And so now what
I'd like to do is
to take a look at
the periodic table
and get a sense of
electron filling.
And whether that explains the
trends in the periodic table.
And for that I'm going to go
to table 6-3 in your reading.
And all we're going
to do, basically,
is say, well can we can we
use this idea of n, l and m,
and explain the order of
filling in a periodic table.
So when n equals 1,
l must be equal to 0.
And the m must be equal to 0.
So that means there's
only one orbital.
And, in that orbital, we
can have two electrons.
So we've got the possibility of
two different electrons going
into the 1s orbital.
Then we go, n equals 2.
l can take a value of 0.
Which is the same as
what we had before.
Just one orbital
in that subshell.
Or we can have l equals
1, in which case m can
take three different values;
minus 1, 0, and plus 1.
So there's three orbitals there.
3 plus 1 is 4.
And that gives us
the possibility
of putting 8 electrons in.
Because this is n, l, and m.
And then we've got the choice
of s plus or minus 1/2.
And we'll do one more.
We'll go to n equals 3.
So at l equals 0,
we just have 1.
When l equals 1, we
have the same thing
as we had with the 2p.
And then when l equals 2, we
have what's known as the 3d.
Remember the
spectroscopists there.
They don't like numbers.
So they use s, p, d, f.
But we're smart.
We can go 0, 1, 2.
It didn't hurt us.
And we go minus 2,
minus 1, 0, 1, 2.
So there's 5.
So 5 plus 3 is 8, and 9.
9 times 2 is 18.
Now let's go to
the periodic table
and see if this reconciles.
So, when we have 1s,
there's hydrogen, helium.
And then the next is n equals 2.
So that should give
us 4 times 2 is 8.
So, 3, 4, 5, 6, 7, 8, 9, 10.
So there's the 8 different
electrons that get us
all the way over to neon.
And now let's go to n equals 3.
So, I have 1, 2,
3, 4, 5, 6, 7, 8.
And then I'm over to 4.
4s.
But look, this is
saying I should have 18.
18 electrons in n equals 3.
So what I'm pointing out here
is that there's a disconnect,
there's a disconnect between
the populating of electrons
just in ascending
quantum number,
and the way the elements are
arranged in the periodic table.
There's some other
factor at work here.
So we want to take a look at
what that could possibly be.
And, so what we need is to
go to a modified energy level
diagram.
A modified energy
level diagram, that
can explain what the
filling sequence is
in the periodic table.
And the modified
energy level diagram
is drawn on the basis
of the Aufbau principle.
The Aufbau principle.
Aufbau, German,
meaning construction.
I think it actually
means out-build.
But, anyways, it generates
the filling sequence.
So, there are three parts
to the Aufbau principle.
And the Aufbau principle
is going to govern,
it's going to govern or direct
the electron filling sequence.
Directs the electron
filling sequence.
So, the first component
of the Aufbau principle
is the Pauli
exclusion principle.
Pauli exclusion principle.
Wolfgang Pauli was an Austrian.
He did his Ph.D under
Sommerfeld in Munich,
and then he post-docced
with Born in Gottingen
and on to Niels Bohr Copenhagen.
These people worked together.
They traveled from
lab to lab, and it
was a very vibrant
conversation going on.
Eventually, he became a
professor physics in Hamburg.
And the Pauli exclusion
principle, simply stated,
is that in any electron
system, each electron
has a unique set of
four quantum numbers.
A unique set of four
quantum numbers.
Any atom set, four
quantum numbers
are unique for each electron.
For each electron.
So you can think of this
as the set of n, l, m and s
as sort of the social
security number,
if you like, for each of
the electrons in the set.
And he eventually gets the
Nobel Prize for this, in 1945.
Virtually everybody
I'm going to talk
about today, with the
exception of Sommerfeld,
gets the Nobel Prize.
Sommerfeld is a very
interesting character, though.
While he himself never won
the Nobel Prize, many, many
of his students and proteges
won Nobel Prizes, to which you
must include that there
was something very, very
special about the quality of
the mentoring that he gave,
that so many of his proteges
went on to win the Nobel Prize.
So this is the first part
of the Aufbau principle.
The second part is
that the electrons fill
from lowest to highest energy.
So electrons fill orbitals.
You can think of the
orbitals as placeholders.
They're really energy concepts.
But we populate orbitals from
lowest to highest energy.
From lowest to highest energy.
Or if I wanted to be a
wise guy, I would say,
I'm going to define
energy in such a way
as I get that as the
filling sequence.
One's got to fit the other.
And the energy is
itself a function
of the four quantum numbers.
So energy, once I
specify n, l, m, and s,
you can give me the
energy and away we go.
And the thing is that
you need to realize
that the energy levels are
actually a function of electron
occupancy.
So you can look
carefully, if you get down
into the d orbitals
and the f orbitals,
as you add one more
electron moving one element
to the right on
the periodic table,
sometimes you see the
population not adding by one,
because the addition
of an electron
changes the relative energies
of the various orbitals.
So let's remember that,
that energy is a function
of electron occupancy.
Energy is a function
of electron occupancy.
And then the third part
of the Aufbau principle
is called Hund's rule.
Hund's rule.
Now, Hund is German for dog, but
this isn't named after a dog.
This is named after Friedrich
Hund, a professor of physics
at Frankfurt.
He taught for many years.
He died at the
tender age of 101.
And he spoke about degeneracy.
Not societal degeneracy,
but degeneracy in an atom.
And this degeneracy
is the condition
where you have a plurality of
orbitals at the same energy
level.
So, in orbitals of
equivalent energy,
in orbitals of
equivalent energy,
we strive for
unpaired electrons.
This is filling, now.
This is how to direct
the filling sequence.
Strive for unpaired electrons.
Let's say unpaired
electron spins.
Let's make it a little clearer.
Unpaired electron spins.
So I'm going to
give you an example,
work through an example.
So, what have we got here?
Here's carbon.
And if you look at any
element on the periodic table,
you'll see this green.
And this is the
electronic configuration.
This tells you what
the sequence looks
like for that particular element
in the neutral form, et cetera,
et cetera, subject to
many, many considerations.
So let's look at carbon.
So if we look at carbon, it
tells us 1 s2, 2 s2, 2 p2.
So what's all this mean?
Well, it's all written
in code for us.
So the first number
here is the n number.
So this means n equals 1.
s is the spectrocopist's
notation,
so this means that l equals 0.
And then the 2 here means
2 electron occupancy.
So the 1s orbital is filled
to the tune of two electrons.
And the 2s orbital
has two electrons,
and the 2p orbital
likewise has two electrons.
So now we want to do is
put these electronics
into their orbitals.
And we can use, there's
a variety of notations.
This is one I'll use today.
This is a box notation.
So this is 1s.
This is 2s.
And then this is 2p.
If you recall, 2p there's
going to be three of them.
If you go back here,
2p has three orbitals
in that sub-shell.
And we said that
this is the one time
that m makes some
sense with respect
to Cartesian coordinates.
m is -1, 0, +1 so we can call
this 2p x, 2p y, and 2p z.
And I don't know which is which.
I don't know what the arbitrary
standards of x, y, and z,
I don't know where the
origin of the universe is,
so I can't tell you all this.
But, arbitrarily, if
I choose one as x,
I know where the other two are
according to right-hand rule.
So let's start filling.
This says 2, so I'll put in
one spin up, and one spin down.
Now it says 2s 2.
One spin up, one spin down.
And now here's where
the Hund rule comes in.
I've got to put two electrons
into these three boxes.
Now, I could be a librarian
and start left to right,
and make them nice and neat.
Or, I could just put
them in wherever I want.
What the Hund rule says is,
strive for unpaired electron
spin.
So for the Hund rule, you'd
put them in both same spin
and in different orbitals.
And then when you
get to nitrogen,
nitrogen will have
a third one here.
And when you get
to oxygen you've
got three plus the fourth one's
going to be one of these three
and I don't care.
I mean, some people are
really anal about it,
and they want you to
go from left to right.
As long as you've got two of
them unpaired and one of them
paired for oxygen I'll be happy.
So now we can use
this concept and look
at the energy level diagram
for multi-electron atoms.
And this is taken from the book.
So this is different
from the energy level
diagram for
hydrogen, because you
can see some compression here.
And I want you to know first
of all, all of these values
are negative.
It's true that energy
increases vertically,
but the zero is way up here.
So these are all
negative values.
And I want to zoom in here.
Because the energy difference
between 1s and 2s is large.
And we know in hydrogen it's 3/4
of the total energy difference.
Because there's some compression
in a multi-electron atom.
But nevertheless,
n equals 1 to n
equals 2 is a huge
energy difference.
So let's zoom in on here.
And what do we see?
Well, 2s, 2p, 3s, 3p and look.
4s lies below 3d.
4s lies below 3d.
So that tells us, there's my 3s.
3s 1, 3s 2.
There's 3p x1, 3p
y1, 3p z1, 2, 2, 2.
And now what's next?
It says go to 4s.
So potassium is 4s 1.
Calcium's 4s 2.
And scandium is 3d 1.
So this energy level
diagram is the map.
It tells you how to put
electronics in sequence.
So, this now gives us
the rational basis for e
equals n, l, m and s.
All right.
Well this is nice.
It's been graphical, and so on.
Now I want to go to
the same position.
I want to get back to this, I
want to get back to this point.
But I want to go
by different route.
I want to go by different
route, and for that we're
going to go by wave mechanics.
So, same destination.
Same destination,
via wave mechanics.
Now, you don't have partial
differential equations
as a prerequisite
for 3.091, so I'm not
going to go through the math.
I'm going to give you the
features of the wave mechanics.
So that later on you're going
to spiral around and study this
again.
You'll have seen it before.
And again, there's going
to be people involved,
and they're all giants
in modern physics.
The first one is de Broglie.
The first one is
Louis-Victor de Broglie.
Let's get his name on the board.
Louis-Victor de Broglie, he was
an aristocrat from Normandy,
who had gone to the Sorbonne.
He was studying humanities.
Political science,
literature, and around
about the time of his
senior year he decided
to switch horses
for graduate school
and forget about a career
in the diplomatic corps,
do a Ph.D in physics.
So, he did a Ph.D in physics.
And in 1924 he published this
Ph.D thesis, beautiful piece
of elegant writings.
Less than 30 pages long.
And I'll give you sort of the
summary, the one-liner that
summarizes his Ph.D thesis.
Most of you are going to
have to write a thesis.
The word thesis comes from the
Greek, and it means, sort of,
my position.
My statement.
Before I have a thesis, I have
something that is not a thesis.
It's my trial balloon.
That's a hypo-thesis,
a hypothesis.
Now, the key to
writing a good thesis
is to ask a really
good question.
If you ask a
pedestrian question,
you're probably going to get
some pedestrian answers and ho
hum.
If you ask a really
interesting question,
you give rise to the possibility
of interesting answers.
And what de Broglie did is,
he asked a really interesting
question.
So here's his question: He
says, if a photon, which
has no mass, photon's
just an energy packet.
If a photon which has no mass
can behave as a particle,
and we've seen.
We model ray optics
as particle beams.
And a photon, and Max Planck
said equals h nu got no mass,
but we can think of it in our
little anthropomorphic brains
as, photon photon photon.
So if a photon, which has no
mass, can behave as a particle,
does it follow that an
electron, which has mass,
can behave as a wave?
It's beautiful.
Let's do it one more time.
If a photon, which has no
mass, can behave as a particle,
does it follow that an
electron which has mass,
can behave as a wave?
So he asked the question.
And he answers it.
In less than 30 pages.
So, let's get this on the board.
Because this is beautiful.
It's like, ask not what you
can do for your country.
If a photon, which has no
mass, can behave as a particle,
or can be modeled as a
particle, behave really means,
so that the theoreticians
can model it as a particle,
does it follow that an
electron, which has mass,
can behave as a wave?
See, if you understand
the question then
the impact of the answer, and
the answer is, if it does,
this is what its
wavelength is going to be.
So de Broglie said, the
wavelength of an electron,
if it were to behave
as a wave, would
be given by the ratio
of the Planck constant
to the Newtonian momentum,
which you know from 8.01
is simply h over the product of
electron mass and its velocity.
So that's de Broglie's thesis.
So let's take a look
what we can do with this.
Now, you remember in the
Bohr model, recall Bohr.
Well, Bohr taught us that mvr,
that's the quantum, condition,
mvr equals the ratio
of h over 2 pi times n,
where n takes on
the discrete values.
1, 2, 3, et cetera.
That's the quantum condition.
Now let's take this
idea of de Broglie.
And first of all we have to
put the electron in its orbit.
So we'll put the
electron in its orbit.
And now I'm going to
have it behave as a wave.
So if it behaves as a wave,
I'm going to draw it as a wave.
Now, why did I draw it this way?
There's two kinds of
waves in this world.
There's standing waves and
there's traveling waves.
Now, this orbit's station,
it's in this orbit.
So it better be a standing wave.
I think there's a
cartoon in the book.
There you go, standing wave.
In order for this to
be a standing wave,
there's a geometric
constraint on this.
Listen carefully
geometric constraint.
I'm not saying anything
about quantum mechanics.
Geometric constraint,
the geometric constraint
for a standing wave is what?
You know what this
distance is, right?
I mean, this is not to scale.
This should be
10,000:1, in which case
these ripples are
barely visible.
But here it looks
kind of exaggerated.
This is a hyper wave.
This is emphasis added in proof.
So that means that the
circumference here, 2 pi
r, the circumference must be an
integral number of wavelengths
for a standing wave.
But we know that
from de Broglie,
I can write n lambda
as n h over m v. I've
just put in de Broglie's
definition of the wavelength
of an electron.
And now I can
cross-multiply and I get mvr
equals h over 2 pi times n.
Which is Bohr's
quantum condition.
So we've got validation
of the Bohr model,
so that's a pretty
compelling case
that maybe the electron
really does behave as a wave,
and that explains why we
have the quantum condition
that we do.
So de Broglie, that's
his Ph.D in 1924.
Einstein read the
thesis, loved the thesis.
But we don't care
what Einstein says,
because he's a theoretician.
So one theoretician praising
another theoretician.
That's not how science works.
How does science work?
Data.
We need data.
And the data come in 1927.
1927, at Bell Labs
in New Jersey.
At Bell Labs in New Jersey
come the critical data.
And they were taken by
Davisson and Germer.
Davisson and Germer.
Davisson and Germer
were studying crystals.
They were studying crystals
of various elements,
and in particular
metal crystals.
Metal crystals,
by X-ray analysis.
And in order for
you to appreciate
what I'm going to show you of
Davisson and Germer's work,
I'm going to take you
back to high school
to those thrilling days
with the wave tanks.
Remember the wave tank?
This is the top view, this is
the side view of the wave tank.
And you might have some kind
of a mechanical device here
that has a paddle.
And it starts
vibrating up and down,
and it starts sending
waves into the tank.
So the waves come like
this, from the edge
it looks like this.
Remember that?
Sure you do.
You're toying with me.
Oh, I don't remember
anything, we never did that.
Sure you did.
OK, so you can send waves down.
Now, what we can do is,
we can put a dam here.
I'm going to put a dam.
And depending on, if
this is the wavelength,
this is the wavelength,
it's the distance
between two successive crests.
And, if this spacing
here, the gap
between the wall and
the edge of the dam, d,
if d is greater than
lambda, the waves
just propagate but
for the place where
they're blocked by the dam.
So you get, you
essentially cast a shadow.
You've seen that.
And so I could model
this system as a beam.
This is a beam.
This is a water beam.
This is a water beam shadow.
And this is equivalent
to ray optics.
Straight lines, if something
gets in the way it's opaque.
Blocks transmission.
End of story.
So you've seen all that.
But you also did this other
experiment, I'm willing to bet.
So let's do the
other experiment.
We're going to do
the same thing.
I'm going to send
waves down here.
Only, this time we're going
to make the dam a little bit
different.
This time we're going to bring
the dam in from the wall.
And we're going to
put a tiny opening.
And I'm going to go
some more into the tank.
And then another tiny opening.
It could be the same.
It's probably best to keep
it the same dimension.
Now, in this case, the spacing,
d, is much less than lambda.
d is much less than lambda.
And what happens in this case?
When d is much less than lambda,
you don't get the shadow.
You don't get something
like this instead, remember.
You've got the rings.
This is called diffraction.
Diffraction.
And there is no way to
explain diffraction modeling
water as a beam.
You must implore the
wave-like behavior of water
in order to explain diffraction.
Explain only by invoking
wave-like properties.
So with wave-like properties, we
get something that makes sense
in terms of the data.
Now, let's do this
same experiment,
let's do this same experiment
on a metal crystal.
So if you go back to
the gas discharge tube.
Remember the gas discharge
tube that we were looking
at for lecture after lecture?
If you take a look and go
through the energetics of it.
If you put one volt
across the plates,
you know the energy is going
to be a product of charge times
voltage.
And that's equal
to 1/2 mv squared.
Which you know from 801
is p squared over 2m.
And p is equal to h over lambda.
Pretty soon you can come
up with the wavelength.
And that will give you the
wavelength of the electron.
This is a ballistic
electron now.
See what I'm doing?
See, once I said
that this indicates
that the electronic
in stationary orbit
can be modeled as a wave
of a certain wavelength, so
now the free electron.
It's got m, it's got v,
there's Planck constant.
I can go ahead and
compute its wavelength.
I can compute the
wavelength of a baseball.
So, you go through it.
And you get a value
of about 12 Angstroms.
12 Angstroms.
Now, if I want to see whether
there's wave-like properties,
I need to have a condition
that gives me diffraction.
So I'm going to have
to find something
that gives me a dam
with an opening that's
less than 12 Angstroms.
If I'm going to use 1 volt.
So what can I do?
Well, turns out you're going
to learn this in greater detail
later, but if this is
a crystal of nickel.
Crystal of nickel, the atoms
are arranged in regular arrays.
And this is what the face of
nickel looks like, 4 atoms each
at the corner.
And one in the
center of that face.
This distance is 3.53 Angstroms.
Perfect.
Perfect.
So what Davisson and Germer
did is, they irradiated this.
They irradiated this first with
X-rays on the order of lambda,
on the order of,
say, 10 Angstroms.
And what did they get?
This is the output.
This is the output.
You get a diffraction pattern.
It's a set of rings,
concentric rings.
So this is the
X-ray diffraction.
This is the X-ray
diffractogram, if you like.
And then, what did they do next?
They irradiate the same
crystal with an electron beam.
lambda of the electron
beam, 10 Angstroms.
And what did they get?
Are you ready?
Drum roll.
Now, there is no way
that you can get a ring
pattern from a beam of electrons
acting as a particle beam.
The only explanation for
this is that the electrons
were behaving as
waves of this value
to give us the same spacing
as we got with X-rays.
And you're comfortable
if I say that X-rays
are a type of light.
So, therefore, it's got
wave-like properties.
And it's got
particle-like properties.
Well, now I've
just made the point
that this is the
electron diffractogram.
So this is evidence of
electron diffraction.
And this shook the world.
Because now it's real.
There's no way you can
get this otherwise.
This is electron diffraction.
This was 1927.
1929, de Broglie
gets the Nobel Prize.
1937, Davisson gets
the Nobel Prize.
So this means the wave-particle
duality is complete.
It applies not only to light,
but it applies to matter.
So, wave-particle
duality, they call it.
Wave-particle
duality is complete.
Matter can act as waves.
Electromagnetic radiation
can act as particles.
So, sometimes people refer to
de Broglie's accomplishment
as matter waves.
Matter waves.
And what do you call the
behavior of billiard balls
banging around and so on?
We call that mechanics.
So now we're going
to use what might
seem as an oxymoron,
contradiction, wave mechanics.
Wave mechanics.
That means matter
behaving as a wave.
But still behaving as a matter.
So this is the dynamic.
So that's pretty
good for de Broglie.
So let's go to number two.
I said there were going
to be three people here.
Number two is Werner Heisenberg.
Werner Heisenberg.
Heisenberg studied with Pauli.
Sommerfeld.
Did his Ph.D in Munich in 1923.
Got his Ph.D with
Sommerfeld at the age of 22.
And then he decided to
take a postdoc with Bohr.
And he was working with
Bohr for a couple of years,
he was feeling a
little bit burned-out.
And decided to take
three weeks off.
Went up to a deserted island
off the coast of Norway.
Came back three weeks later with
the mathematical formulation
of quantum mechanics.
I'm not kidding you, that's
what he did in his time
off, to kind of unwind.
And so one of the things that
he used as a critical piece
of this derivation is that
the position and velocity
of an electron cannot
be fully specified.
They cannot be fully specified
below certain limits.
There's a threshold
below which we can't go.
This is sort of
like, if I asked you
to time a 100-meter
sprint, which typically
takes less than 10 seconds.
But I give you a clock, and
the nearest unit on the clock
is the minute.
So you wouldn't be
able to distinguish.
So, he says-- and
the reason for this
is, it's a consequence
of quantization.
Light itself is quantized.
So, at some point you're asking
for a continuous splitting
and splitting and splitting into
finer and finer time segments.
And you can't get there.
So we already knew
this from Planck.
And so one of the ways that
he expressed the inability
to go below a certain threshold
is the uncertainty principle.
The uncertainty principle.
And that's
unfortunate that, see,
it was originally
published in German.
And the idea really is the
indeterminacy principle.
But, English says uncertainty.
So it's a limit to
determination, but there it is.
And so one expression
of this is,
the product of the velocity.
Only, he wrote it in
terms of momentum.
So the mass isn't
going to change.
So, think of this as the
uncertainty and the velocity.
And the uncertainty
in the position.
So this is the
x-coordinate of particle.
You can break its
trajectory into three
orthogonal components.
So the uncertainty
in the x direction
of the momentum times the
uncertainty in the position
is greater than or equal to
the Planck constant divided
by 2 pi.
The Planck constant
divided by 2 pi.
And so what this
means is that we're
going to see a transition in
models from individual atoms.
If we want to describe what
happens with individual atoms,
we need what is known as
a deterministic model.
Sort of, Newtonian mechanics.
You tell me the initial
position and velocity.
You tell me the forces, and I
can predict where it's going
and where it's going to be.
So that's deterministic models.
So, deterministic models
describing individual atoms
are going to give rise
to probabilistic models.
Probabilistic models.
And probabilistic
models obviously
can't be talking about
individual atoms.
Must be talking about
ensembles of atoms.
So I can't say where any
individual atom will be,
because I don't have
the ability to do so.
But I can tell you, if you
give me a large number of them,
I'll tell you roughly
what the expected outcome
could be in terms of energy.
And ultimately predict
the spectrum, and so on.
So instead of chicken
and egg we have
now chickenality and egg-ness.
Everything is just sort of
getting a little bit murky.
Little bit murky.
You can do a
calculation on this.
You can do a calculation on
this, a very simple take.
Take the Bohr model and take
the ground state electron
in hydrogen. n equals
1 in atomic hydrogen.
And you know this is
about half an Angstrom.
So the distance across
here is 1 Angstrom.
So make 1 Angstrom
your uncertainty,
and you'll find that the
uncertainty in the momentum
is on the order of 15%.
But what this is saying
is, when you get down
to atomic dimensions, you
can't just shine light on it
and reveal what's going on.
Because you're going to
disturb the very thing
you're trying to measure.
Some people say
that, every time you
try to work at the
atomic level, it's
as though you're trying
to take a picture
with the sun at your back and
your shadow is in the picture.
So you can't get there without
disturbing the very thing.
Another way to
think about it is,
the photons that are
capable of this resolution
are going to have such
high energies they'll
knock the very thing
you're trying to measure.
All right.
He gets Nobel Prize in 1932.
And then the third one
is Erwin Schroedinger.
Let's get him up here.
Erwin Schroedinger.
Also an Austrian,
University of Zurich.
He too was burned out.
They get burned out, these guys.
So at Christmastime,
1925, took a vacation.
At Villa Herwig, in Aurora.
And comes back two weeks
later with the wave mechanics
formulation, of
quantum mechanics.
See, sometimes going
away on a vacation.
So he took de Broglie's notion
of the electron as a wave
and wrote equations to
model wave-like behavior.
So, let's look at
how to get there.
And here's what he did.
So, you know, for
example, that we could
start with a violin string.
And it has a
geometric constraint.
It must be fixed at both ends.
And if I pluck
that string, it can
vibrate as long as it conforms
to the geometric constraint
of a standing wave.
So here's one possibility.
Wherein we would
call this n equals 1.
I have simply the
entire string vibrating
in the matter that's shown.
But here's a second possibility.
I could have it vibrating
as is shown here,
n equals 2 with a
node in the middle
where that node
doesn't move at all.
The string is stationary
at its mid-point.
And what's the
characteristic here?
This is operating at
a certain frequency.
Let's say it's middle C.
And this is the overtone.
This is the first harmonic.
And it's going to
be an octave higher,
because it's as though we
have two strings, each fixed.
See, from a physics
standpoint I could literally
cut this string in half and
fix it there, and this is now n
equals 1 for the half-length.
So it's going to have the
same pitch as the half-length.
Which means that this
is an octave higher,
and this is going to be two
octaves higher, and so on.
And all of these conform.
So you get a plurality
of solutions.
You get a plurality
of solutions.
And the solutions look
something like this.
They'll eventually teach you
the string as a simple harmonic
oscillator.
Simple harmonic oscillator.
And it has equations
that look like this.
If you want to
plot its position,
this is x going from 0 to l.
And this is the y-coordinate.
So you can, for example,
write something like this.
So, the function
will look like this.
Some pre-multiplier
times cosine of kx
plus another pre-multiplier
b times the sine of kx.
And the geometry will
dictate that the value of k
is n pi over l.
So, pi over l is the geometry.
And n takes multiple values.
Just as you see here, there's
not a unique solution.
So listen carefully.
Wave equation,
plurality of solutions.
But subject to some constraints.
Subject to some constraints.
So what Schroedinger did
is, he wrote a wave equation
to describe the motion
of electron in its orbit.
And guess what he gets?
He gets a plurality
of solutions.
And when you look at the
plurality of solutions,
the plurality of
solutions ultimately
map into what we know as the
distinct values of n, l, m
and s.
See, this is the
one-dimensional.
So this is giving us
n numbers. n equals 1,
this is now n equals 2.
So I'm getting
quantum numbers here.
Now, if I did this
in three dimensions,
I'd have a plurality
of quantum numbers.
And Schroedinger gets us all
the way to n, l, m and s.
And so here's what
it looks like.
This is the equation,
it's a wave equation,
so there's a double
derivative in space.
There's a forcing function.
And this is i, square root of
minus 1 in a time base here.
So it's a harmonic
kind of equation.
Psi is the wave function,
it's an abstract concept
but we'll show you how
to make sense of it.
And these are the
various solutions,
the plurality of solutions.
And we can now map those into
what we know as 1s, 2s, 2px,
2py, 2pz, et cetera, et cetera.
And you see this number a sub 0.
That's our Bohr radius.
Comes right out
of the equations.
0.529 Angstroms.
So, this is quite good.
But, as I said, the psi
is the wave function.
Or psi, however you
want to call this.
This is called
the wave function.
Wave function.
And we have plurality
of solutions.
We call these Eigenfunctions.
Eigenfunctions.
And the closest we can
get to something physical
is the product of psi and
its complex conjugate.
And that is related
to the probability
of finding the electron.
Probability of
finding the electron.
Which, in essence, gives the
boundaries of the orbitals.
So now I'm going to put, we're
going to get a Cartesian shape.
I told you, 1s, 2s.
What do they look like?
Here's what they look like.
So these are the square of
the wave function plotted.
So this is in a radial
distribution function.
It's only in one direction,
out from the radius.
Now, if you whip
this around in 3-D,
you'll generate the surface.
But already you
can see, here's 1s,
and it peaks at about
1/2 an Angstrom.
And here's 2s, and it peaks
at 4 times the Bohr radius.
And psi squared
3s about 9 times.
This is from your book.
So it's a maximum.
But there's some uncertainty.
See, it's not a simple line
fixed at 0.529 Angstroms.
This is another way of plotting.
So these are spherical.
What we were calling circular
now becomes namely spherical.
And there's this node here.
This is the p orbitals.
They're dumbbell-shaped.
With two lobes.
And if you have a
single electron,
it doesn't reside in one lobe.
It can jump from one
side to the other.
You might say, well, how does it
get from one lobe to the other
when halfway between,
there's a nodal plane.
It has zero probability.
Well, it's behaving as a wave.
Behaves as a particle, you
can't get through a wall
that says zero permission.
That's how, you can transfer
energy from here to here
and have that node's
perfectly stationary.
Anybody skip rope?
You know how this works.
Now, this is where I
quarrel with the book.
This is another drawing.
But I'm uncomfortable
with the fact
that they chose
different colors.
Because I think to the
first time learner,
you might be tempted to think,
well, one electron lives here
and the other
electron lives here.
No, the electron,
if there's only one,
it can go from one to the other.
If there are two, they can
go from one to the other.
And, see, they do this
all the way through.
So please don't start
rationalizing in your mind that
one electron goes in the yellow
and the other electron goes
in the grey.
There's the d.
Aren't they pretty?
If you find the f
orbitals, that's wild.
So I think this is probably
a good place to stop.
We've got a few minutes here.
If you want to read
more about uncertainty,
this is a very nice
book by David Peat that
goes into the meanings,
including this indeterminacy,
and so on.
Good book here on hydrogen.
Please, I don't want noise.
We've got, still,
a few more minutes.
Still got a few more minutes.
This book here talks
about hydrogen.
Goes right back to Democritus.
One chapter is a beautiful
thing on the use of hydrogen
as a potential fuel.
All the Bohr and whatnot.
This is a play that won the
Tony Award in the year 2000.
Written by Michael Frayn.
And it's about the fact that
Niels Bohr was the mentor
to Werner Heisenberg.
And now it's 1941, the
Nazis have invaded Denmark.
And Bohr is essentially
waiting to get out
of Denmark before the war
overtakes the rest of Europe.
Meanwhile, Heisenberg is now
the head of the Nazi equivalent
of the Manhattan Project.
And he goes to Copenhagen
to visit his old mentor.
That's a fact.
They have dinner.
That's a fact.
They go for a walk.
That's a fact.
They never speak to each
other after that night.
That's a fact.
So the question is,
what went on that night.
And that's what Michael Frayn
uses as the dramatic point
of departure.
So, did Heisenberg go to
Bohr to get Bohr's opinion
about nuclear weaponry?
Did he try to find out
whether the Allies were
working on a bomb?
Did he go to say, look, we
should on both sides not
develop nuclear weaponry?
What went on in
that conversation?
And so you see
Bohr at the center.
There's Heisenberg
who's the one electron.
And there's Margaret,
who is Bohr's wife, who
was the observer.
And so there's the play
between the uncertainty
in quantum mechanics and the
uncertainty in human relations.
It's a really nice play.
Here's a rendition of it.
This is Stephen Rea playing
Bohr, Francesca Annis playing
Margaret Bohr, and playing
Werner Heisenberg is, do you
recognize him,
it's Daniel Craig.
This is a book that came out not
too long ago about Heisenberg.
A lot of controversy about him.
Some people accused him
of being a collaborator.
Other people say that he
was absolutely brilliant
in the right amount
of foot-dragging.
He did not want to give
Hitler nuclear weapons.
If he'd been a total
disaster he would
have been replaced
by somebody who
might have been more zealous.
And if he went too fast,
he might have figured out
how to make nuclear weapons.
So, very interesting
book about him.
And here's a nice photo.
This is Bohr.
This is Werner Heisenberg.
And this is Wolfgang Pauli.
Undoubtedly talking
about what goes
on when you pluck that string.
So we'll see you on Wednesday.
