Let's get started.
The weekend is over.
Time to get back to work,
back to learning. A few
announcements.
Tomorrow Quiz 2 based on Homework 2.
Thursday will be the Periodic Table
quiz. Remember,
all you have to do is fill in the
one or two-letter abbreviation for
the chemical symbol.
And Friday is the last day of the
contest, 5:00 PM. Send
it to me by email.
You can enter more than once,
if you want. You can have multiple
entries. You can do the lanthanides
or the actinides or both.
Test 1 will be a week from
Wednesday, and I will say something
about that later in the week.
And I will be available 3:30 to
4:30 today for office hours if
someone wishes to drop in.
Last day we talked about the
validation of the Bohr model.
We saw that the Bohr model was able
to correlate the observations of
Angstrom, which had been formulated
by Balmer. And,
furthermore, the concept of energy
level quantization was observed in
mercury during the course of the
Franck-Hertz experiment giving
credence to the notion that
quantization of energy levels is not
the property of one-electron atoms
alone but is something that is more
broadly applicable.
So Bohr was validated in very,
very strong measure, but there were
also some contrary data.
And we saw those. First there was
the observation by Michelson who
back in the late 1880s had done very
precise interferal metric
measurements of the hydrogen lines
and had observed that the 656
nanometer line associated with the
transition of n equals 3 to n equals
2 was, in fact, a doublet.
If you look really carefully,
you could find it was not a single
line but two lines,
and Bohr is silent on such matters.
And then we had Zeeman who
conducted the experiment of hydrogen
emission spectrum in a magnetic
field. And he found line splitting
in the magnetic field.
And, furthermore, that the
intensity of the splitting was
proportional to the intensity of the
applied magnetic field.
Again, the Bohr model silent about
these splitting into
multiple levels.
And subsequently Starck did the
analogous experiment but with an
electric field.
Again, in an electric field line
splitting observed,
the intensity of the line splitting
varying with the intensity of the
applied electric field.
So there are some problems here.
Sommerfeld came to the rescue in
1916. He preserved the broad
concept of energy levels but put in
some fine structure.
And the fine structure involved
taking those levels that Bohr
quantified with the number n and
assigning to those n levels a little
bit of energy width so that there is
a plurality of orbits,
some circular, as in the Bohr case,
some a little bit off circular. And
that bandwidth he called the shell.
This is the shell model. And in
order to label the various orbitals,
as he called them, he introduced two
more quantum numbers, l and m.
And we saw l and m last day.
And then, at the very end of the
lecture, we saw the last quantum
number introduced,
sort of a back to the future where
we brought in the experiment by
Stern and Gerlach where they sent
the beam of silver atoms through an
asymmetrical magnetic field and
observed deflection,
but deflection in two directions.
Deflection up and deflection down
indicating that there was something
strange that had yet to be accounted
for. And several years later these
two graduate students at Leiden by
the name of Goudsmit and Uhlenbeck
proposed that the electron in fact
spins. It doesn't simply orbit the
nucleus, but as it orbits it spins.
And since we don't know the
absolute up and down in the universe,
it is possible that some of the
electrons may be spinning up.
And, according to the right-hand
rule, they would be spinning,
by our convention, anticlockwise.
And some of them might be spinning
clockwise, in which case,
according to the right-hand rule we
would consider them spin-down.
And, in any arbitrary crucible full
of molten silver,
half of them are going to be spin-up
and half of them are going to be
spin-down. And,
sure enough, we see the splitting.
And so that is where we left it on
Friday. I thought,
well, maybe we can now go to the
Periodic Table and figure out what
is going on there.
If we look at the kinds of electron
configurations we would expect,
when we are at n equals one, which
is k shell, we know that the
selection rules allow for only one
value of l, and that must be l
equals zero, which the
spectroscopists would call lowercase
s. And, under these circumstances,
m must be uniquely zero. And s can
take on values of plus or minus
one-half. So there are really
two choices here.
So there are two electron
configurations in the n equals one
shell, if we follow according to the
selection rules that we spelled out
last day. And,
if we go to n equals two,
this would be the l shell.
And when n equals two, well,
l could take on value of zero or one.
And when it takes on value of zero
then m must be zero uniquely.
S can equal plus or minus one-half.
We have two electron configurations
when l equals zero.
When l equals one, m can vary over
one, zero and minus one.
S will always be allowed to take
plus or minus one-half.
So I have three times two is six.
All in all, in the l shell, I have
the possibility of eight different
configurations.
And we will just do one more.
And so when we get to n equals
three that would be m shell by the
spectroscopists' notation.
By the way, l equals one would be
the p-orbitals.
And so, in this case,
l could equal zero, l could equal
one or l could equal two.
When l equals zero, again,
m is zero and s is plus or minus
one-half, so I can pick up two
electrons here.
In other words,
I have occupancy states for two
electrons.
The orbitals you consider as bins in
which you can throw electrons,
one per bin. Here I have m is one,
zero, minus one. S again is plus or
minus one-half.
There are six here.
And when l equals two,
m can vary from two down to minus
two. S, again,
is plus or minus one-half.
So five times two is ten. This
gives me, in total, 18.
So this seems to be making sense.
And we talked about periodicity and
properties, so let's go and take a
look at what we have and see if we
can reconcile this.
If we look at n equals one,
here is n equals one, hydrogen and
helium. And, indeed,
we have two electron configurations.
And then we are over to lithium.
Well, lithium is n equals two.
Here is 2s1, 2s2 and here are the
six associated with the 2p.
And then here is 3s,
3s2. And then something is going
wrong. What I have here is there is
the 3p, there are six of them,
and now it is telling me I should be
able to put in ten more.
But before I can put in those ten
more, I am already over to 4s.
Potassium is 4s, and I've only
finished with 3p.
Clearly, there is something else at
work here.
In other words,
these energy states are not filling
just in ascending n number.
We need to add something else in
order to explain what is going on in
the Periodic Table.
And that something else is
essentially the filling sequence.
What is the filling sequence of
electrons in orbitals?
Just by n, l, m,
s is inadequate. There is something
else at work here.
It is close but not complete.
And the answer to this question is
given by the Aufbau Principle.
The Aufbau Principle comes from a
German word which means it is
essentially construction or building,
build-up principle.
And there are three components to
the Aufbau Principle.
The first one is the Pauli Exclusion
Principle which was enunciated by
Wolfgang Pauli.
Pauli was an Austrian who did his
PhD under Sommerfeld in Munich and
then post-doced with Max Born in
Gottingen and also with Niels Bohr
in Copenhagen.
So you can see these people went to
comparable schools.
They worked with the same people in
this group and kept the vigorous
discussion going.
He eventually got a position at the
University of Hamburg where he was a
Professor of Physics.
And what he said first of all is
that in any system of electrons that
the set of quantum numbers n,
l, m and s, unique to each electron
in a system.
That is to say in a given atom.
So each electron has a distinct set
of quantum numbers,
the first important idea.
You can think of each electron as
having its own social security
number or IP address or something
like this. That is important.
Once we have chosen a certain mix
of n, l, m and s,
it is used once for that particular
atom.
So that is the first point.
The second point of the Aufbau
Principle is just energetics.
Electrons will occupy orbitals in
order of ascending energy,
occupying the lowest energy first
and up. Clearly,
this n, l, m, s sequence is
different from the energy sequence
once we get beyond 3p.
So there is something else going on.
Let's talk about electrons.
Electrons fill orbitals from lowest
energy to highest energy.
But there is a wrinkle here.
And that is that the energy levels
themselves change with
electron occupancy.
And as you go to higher and higher
levels, those energy levels,
as you saw in the case of hydrogen,
are more closely spaced. The
differences between those energy
levels are becoming smaller and
smaller as the n number rises.
And what the second point of Aufbau
says is that as electrons begin to
fill those levels the differences in
energy may shift so that in an
unoccupied state certain levels may
be in the inverse order from how
they are in the occupied state.
We have to recognize this.
Lowest energy to highest energy and
it is a function of occupancy.
And we will see a few examples of
that. That is the second part of
the Aufbau Principle.
And the third one talks about the
question of degeneracy,
and it is called the Hund's Rule.
The Hund's Rule talks about
degeneracy. What do I
mean by degeneracy?
Degeneracy is where you have
orbitals of equivalent energy.
How do we put electrons into such a
system? Orbitals of equivalent
energy, we strive for unpaired
electrons. And what do I mean by
that? Let's give a simple example.
A simple example is,
say, let's look at carbon.
If we look at carbon, if you look
on your Periodic Table you will see
this notation next to carbon,
1s2, 2s2, 2p2. What are we learning
here? This first number here gives
the value of n.
So this tells me that this is n
equals one.
And s is this lowercase notation
that the spectroscopists use in
place of the l number.
This is s. S, according to
spectroscopy, means that l equals
zero. N equals one.
L equals zero. And the superscript
two is an indication of electron
occupancy. In plain English,
this says that there are two
electrons in the 1s orbital.
And then we can continue. This is
two electrons in the 2s orbital,
and there are two electrons in the
2p orbital.
Another way of displaying this is in
a box notation.
Here is 1s, if you like,
k shell. Here is 2s, l shell.
And then there is 2p. And we know
from over here that 2p has the
possibility of three
different m numbers.
And I showed you last day that this
is one case where trying to go in a
Cartesian space makes sense,
m equals one, zero and minus one
indicates the three principle
coordinate directions.
And, indeed, people refer to these
as the 2px orbital,
the 2py orbital and the 2pz orbital.
So now let's fill the orbitals.
Let's occupy the orbitals.
And we are going to use the Pauli
Exclusion Principle and we are going
to use the fact that we fill from
lower energy to higher energy.
So one goes spin-up and the other
goes spin-down.
Why spin-up and spin-down?
Because the fourth quantum number
is s. And Pauli says no two
electrons in a given system can have
the entire set of quantum numbers
identical. So this one is spin-up
plus a half, this is spin-down minus
a half. So it is compact notation.
Likewise with 2s, spin-up,
spin-down. I have two electrons now
to occupy the p-orbitals.
But these three p orbitals are
degenerate. That is to say they are
of equivalent energy.
So the Hund's Rule tells me how to
put them in. And the Hund Rule says
try to go for unpaired electrons.
I will put one up here. And I
cannot have an unpaired electron in
the same orbital.
That would violate Pauli.
So the only way I can get an
unpaired is to put it alone in
another orbital.
And I don't care if you want to put
it over here. I am not going to
fight you over it.
So the Hund's Rule is telling us to
put the two electrons in separate
orbitals unpaired as opposed to we
should not put them in like this,
like some librarian might want to
fill nicely from left to right,
up and down. No. No librarian rule.
Bad. Very bad.
Somebody might be sort of a
disorganized librarian and might
want to put this one upside down but
over here. Very bad.
You violate the Hund Rule.
If you violate the Hund Rule,
you will have to talk about the
appropriate physics punishment.
This is the Hund Rule and it
explains electron occupancy.
Now what we can do is look at an
energy level diagram.
And this is the energy level
diagram for multi-electron atoms.
And this is posted at the website.
Just to get you oriented, this is
energy decreasing from zero to
progressively negative numbers.
Stability is at the bottom. N
equals one, l equals s, can
only be an s-orbital.
L can only equal zero.
So we have two electrons here.
Here is two plus six is eight. And
now here is two plus six.
And what we see is that 4s lies
below 3d, according to this set of
rules. And you don't have to know
this by memory.
I would never ask you to tell me
what the various energies are from
heart. Instead,
I would ask you given that this is
the electron occupancy what is the
explanation for it?
So then you could explain in terms
of these levels.
You can see 4s, then 3d,
then 4p and then 5s. Let's see if
that happens. Sure enough there is
the 3s and then 3p.
And now 4s jumps in before the 3d.
Now these are the 3d elements. And
now 4p. And now this is 5s.
5s lies below 4d.
And, likewise, we get over to here
and finally, look,
here is 5d and 5f almost at the same
level. And this is an example where
unoccupied 5d lies below 4f.
As soon as lanthanum takes that
electron, this level rises ever so
slightly, and then we jump down here
and fill in the lanthanide.
But you notice up here how close
these levels are,
so we are looking at very,
very fine distinctions.
And here d, of course,
can take the values from plus three
down to minus three.
That is seven different m values.
Seven times two is 14. Hence, you
have the 14 lanthanide,
14 actinide elements. In other
words, everything makes sense here.
And now we can rationalize what is
going on in the Periodic Table.
Not on the basis of simple filling
in ascending order,
but a modified rate of filling as
according to the Aufbau Principle.
Things are looking pretty good.
Now what I want to do is take you
exactly to this place where we are
right now, where we see energy
levels and electron filling,
but I want to start all over again
and I want to use a totally
different approach.
I want to get to this same place
now by wave mechanics.
I want to get there by wave
mechanics to arrive at the same
place. And what is the same place?
The same place is that energy is a
function of these four quantum
numbers. And for that I need to
invoke the names of three giants.
And we are going to look at each of
them and their contributions
in turn.
De Broglie, Heisenberg and
Schrodinger. Let's go one by one.
We are going to start with de
Broglie. De Broglie was the child
of French aristocracy.
He was thinking about doing a PhD
in political science and pursuing a
diplomatic career,
but he was also drawn to science.
And finally he decided to pursue a
PhD in physics.
This was in Paris in the early ‘20s.
And he was very literary.
The thing about his thesis was it a
very short thesis.
And what makes it great is that he
began with an interesting question.
You know the old adage. If you ask
a dumb question there is only a dumb
answer. And the thesis is your
statement. It is what you profess.
Well, the thesis is also,
another way to think of it,
an answer to a question. So de
Broglie's genius was in asking a
really, really good question.
And then he answered it elegantly
in a very small number of pages.
I want to read sort of my
encapsulation of his question.
If a photon, which has no mass,
can behave as a particle --
And we know this.
Sometimes we model light as beams
of light, as rays of light.
And sometimes we model light as a
wave. And that doesn't bother us at
all. Light can behave as a particle.
In some cases it helps us to think
of it that way.
Other times it behaves as a wave,
and that is the way it helps us to
think and rationalize what we
observe. If a photon which has no
mass can behave as a particle,
does it follow then an electron
which has mass can behave as a wave?
1920. Bohr started talking about
quantization. Now de Broglie is
saying I am going to go further.
I am not only going to say that we
have quantization in the motion.
We are going to say that the
electron behaves as a wave.
One more time. It is beautiful.
This is tautology. Ask not what
you can do for your country,
etc. If a proton which has no mass
can behave as a particle,
does it follow that an electron
which has mass can behave as a wave?
And, if so, he says this is what
that electron wavelength would be.
Let's first of all associate him
with this. De Broglie,
1924, in his PhD thesis says if an
electron has wavelike properties
this would be its wavelength.
There is no hc over lambda. The
electron is not moving
at the speed of light.
He has to have some other expression
for it. It is the ratio of the
Planck constant to its momentum
**lambda = h/p**.
And this is just a Newtonian
expression of momentum,
the product of the mass of the
electron times its instant velocity.
That is what he said. What are the
consequences of that?
How can we go somewhere with that?
Now, you recall in Bohr the quantum
condition.
Bohr expressed the quantum condition
by the angular momentum,
quantum condition in the following
manner. Bohr said that the angular
momentum, mvr is quantized where n
is this integer counter h over 2 pi.
This is the proportionality that is
multiplied by the quantum.
If de Broglie is correct, we could
then model the electron in its orbit
not moving as a particle,
but let's model it as a wave.
So what kind of a wave is it?
There are only two kinds of waves.
There are traveling waves and there
are standing waves.
Well, this is in a stationary orbit
so we need to have a standing wave.
Not to scale. Let's imagine this
is the electron in its orbit.
It is in a standing wave
configuration. There is
a geometric constraint.
In order to have an electron in a
stationary orbit this implies
standing wave.
And a standing wave means there is
a geometric constraint which is what?
I have to have a whole number of
wavelengths to get around the
circumference.
So the circumference is 2 pi r.
And if it is going to be a standing
wave then this must be an integral
number of wavelengths.
But de Broglie has told us that the
wavelength is related to the instant
velocity through this formula.
So let's substitute in. That will
give us 2 pi r goes as nh over mv.
And I can cross-multiply here.
That will give me mvr equals nh over
2 pi **mvr = nh/(2*pi)**.
The quantum condition of Bohr falls
out of the three line derivation if
you accept de Broglie's hypothesis
that the electron in this set of
circumstances can be
modeled as a wave.
So he got his PhD thesis,
and in 1929 he gets the Nobel prize.
He is off to a flying start.
Einstein read the thesis and loved
it. He loved it.
But what do we care what Einstein
says? Einstein is only a
theoretician and de Broglie is a
theoretician, so one theoretician
propping up another theoretician,
this is a mutual admiration society.
We need what? We need evidence.
We need experimental evidence.
Let's go back to high school for a
moment, just by way of background.
Remember these water tanks where
you could put a vibrator in a water
tank. And this is top view of a
water tank. And if you had a motor
here that was causing a paddle to
move up and down you could cause
waves to form at one end of the tank
and move from left to right.
And, furthermore,
you can do little experiments.
For example, you could put a dam in
the middle of the tank.
And if the distance between the
wall and the dam was large in
comparison to the wavelength then
what happens? If this distance is
large in comparison to the
wavelength, this dam simply
casts a shadow.
It is like light casting a shadow.
So this is simple. When d is large
in comparison to lambda the
obstacles cast shadows.
And this is equivalent to modeling
the wave fronts as particles.
This is ray optics. This is the
equivalent to ray optics, isn't it?
You don't have to know anything
about wave-like behavior.
You have a front coming here.
You have an obstacle. Where it can
get by it moves through.
But in the other case where the gap,
this d spacing,
when the d spacing was small in
comparison to the wavelength the
obstacles did not cast a shadow.
The obstacles did not cast a shadow.
You did not see a tiny, tiny little
stream coming out of each of these
orifices. Instead you saw this.
Recall. You weren't sick that day,
right? You were there. You
remember this.
Now, this is diffraction.
And there is no way to explain this
phenomenon if you model the water as
having particle-like properties.
You have to invoke wave-like
properties. Only by using wave-like
properties as an explanation can you
describe diffraction.
Invoke wave-like properties to
explain.
De Broglie has said electrons,
under certain circumstances, can be
modeled as behaving as though they
are waves. What is going to be the
experiment? We need to come up with
an experiment in which the electron
is going to be forced to behave as a
wave. We are going to go to the
United States,
1927, Bell Labs.
This is fast. Look,
de Broglie was 1924.
The thesis is written in French over
in Paris. Somebody is reading the
literature. They are top of this.
And they do a beautiful experiment.
What do they do as their experiment?
People have been working with
x-rays for about 30 years by this
time. And there is a technique
known as x-ray diffraction.
And x-ray diffraction allows us to
characterize solid matter.
It is so important, in fact,
we are going to study it in 3.
91 in detail next month.
But for now let's simply say that we
have a diffraction which we know
occurs only by wave-like behavior
and it involves the use of x-rays.
Why x-rays? Well, the wavelength
of x-rays is on the order of about
an angstrom. If there is
diffraction and the wavelength is
about an angstrom,
clearly the feature that the x-rays
are diffracting from must
be, likewise, dimension.
It has got to be small.
What is really small that would fit
this? Atomic spacing.
Interatomic spacing is also on the
order of about an angstrom.
What Davisson and Germer did was
constructed an elegant experiment.
They took a crystal,
this is a single crystal of nickel
that has regular planes of atoms,
and those planes are spaced on the
order of an angstrom or less apart,
and they irradiated this with x-rays.
This is h nu.
X-rays coming in.
And they looked at the result.
And the result is a diffraction
pattern that produces
a series of rings.
You have a single beam coming in,
and you end up with rings. And
these rings are indicative of the
crystal structure of nickel and can
be explained only by invoking
wavelike properties of light.
Here is the critical experiment.
If de Broglie is correct, let's
irradiate the nickel crystal with a
beam of electrons where the
wavelength of the beam of electrons
is identical to the wavelength of
the x-rays that were used
and see what we get.
So we use the de Broglie formula.
And now we repeat the experiment,
only this time we come with an
electron beam.
And the wavelength of the e beam is
equal to the wavelength of the
x-rays that were used previously.
The moment of truth.
Photographic plate.
What do they see? Circles.
Electron diffraction was born.
Electron. This is the e beam.
There is no way that you could
irradiate a crystal of nickel with a
single beam of x-rays and get that
circular ring pattern if the
electron beam were behaving as a
particle beam only.
So this validates de Broglie and it
also validates the whole concept of
wave-particle duality.
Wave particle duality had been
applied to photons.
People are very comfortable talking
about individual photons,
but now we can talk about waves of
matter. And so if you say de
Broglie's name to many physicists,
the tagline they associate with de
Broglie is "matter waves" as
distinct from light waves.
We have waves of light.
We have waves of mater.
That is important. That is number
one. I have to get you over to
where we were earlier this morning.
Number 2, we are going to Werner
Heisenberg. What did he do?
Well, he studied with Pauli.
He and Pauli were buddies in
graduate school.
They worked for Sommerfeld.
You know, Sommerfeld never won a
Nobel prize.
But we should count how many Nobel
prize winners he educated in his
group. There was something very,
very important going on in the
mentoring process there.
Heisenberg gets his PhD with
Sommerfeld in Munich and then goes
to Copenhagen to work with Bohr.
And he was getting burnt out so he
decided to take a vacation.
He went up North from Denmark to a
deserted island of the coast of
Norway and stayed there
for about three weeks.
He came back, and with him he
brought the Uncertainty Principle.
And what he said was that when you
get down to atomic length scales
there are limits,
there are constraints on our ability
to observe. At atomic length scale,
limits on our ability to observe.
And this is important because it
means that since we cannot observe,
we cannot talk about individual
electrons. We are going to have to
stop doing that because we cannot
find an individual electron.
Every observation we make involves
the exchange of energy.
When I am looking at that wall,
its photons are hitting the wall and
coming back to me.
And so the photons hitting the wall
actually put pressure on the wall,
but the pressure is so low. Who
cares? But you might say why does
that make any difference?
Well, suppose I want to look at
something like an electron in orbit
here. We know that this dimension
is roughly one angstrom unit,
right? The Bohr radius, for
hydrogen is 0.
29 angstroms. Round numbers,
this is one angstrom.
Well, if I want to look at this,
I want to have very accurate
capabilities in terms of the energy
that I use, the light that I use.
So I would choose something that
has a wavelength of roughly an
angstrom. You wouldn't measure the
dimension of a human hair with a
yardstick, so I need something on
this order. If I take lambda equals
one angstrom, go through hc over
lambda, you will discover that the
energy of a photon with one angstrom
as its wavelength is on the order of
12,400 electron volts.
Well, I am looking for this electron.
What is the binding energy of the
ground state electron in hydrogen?
It is 13.6 electron volts. So I am
going to go and I am going to look
for something that has an energy of
13.6 electron volts holding it in
place, and I am going to use a
flashlight with 12,
00 electron volts. I think I am
liable to disturb the very thing I
am going to measure.
You say, well,
why are you doing something so
foolish? Why don't you use
something that has much lower energy?
OK, let's choose something that has,
say, a hundred times less than that.
This will be 13.6 over 100 electron
volts. Well, if I go through that,
the wavelength of such a photon is
going to be so big that I am back to
measuring the dimension of the human
hair with the yardstick.
So now you see the dilemma.
If I want to get down to very,
very fine dimensions, I have to
bring huge energy and disturb the
very thing I am going to measure.
So this is the Uncertainty
Principle. It leads to uncertainty
in the measure.
Suppose I asked you to time a 100
meter spring in the Olympics,
but I give you a stopwatch that is
measured in units of 15 seconds?
Well, these people are blasting
through it, nine seconds and change.
You have a whole bunch of runners
and five of them have all run it in
under 15 seconds.
I guess the whole batch is running
under 15 seconds.
And all you have is either zero or
15 because that is your resolution.
What does that tell you? It tells
you that you cannot resolve what is
going on at that level.
In fact, there is something else
you cannot resolve.
You cannot resolve causality.
When the uncertainty gets that big,
you don't know which came first.
Cause and effect below a certain
level of definition is blurred.
Cause and effect,
that becomes uncertain,
too. This is philosophical,
as a lot of this physics should be.
What does it mean? It means that
we better get away from these
deterministic models where we have a
little electron here with its
potential energy and its kinetic
energy. And it is moving around and
we are sitting on that electron
watching it go about
its daily affairs.
Those models are not going to work
anymore. What we see as a result of
Heisenberg is the shift from
deterministic models.
In other words, we cannot talk
about individual particles.
Deterministic models are now giving
way to probabilistic models.
Now, instead of talking about
individual particles,
we talk about ensembles of particles.
I don't know where any one of them
is, but I can tell you on average
where any one of them is likely to
be. But I don't know where any one
of them is. This means there is no
more chicken and egg.
It is chickenality and eggness,
understand? Everything is blurred.
And to just give you a sense of what
this is, the analytical expression
that Heisenberg gave in one of its
forms is that the uncertainty in the
momentum of a particle times the
uncertainty in its position,
this is the uncertainty. Not the
value of the position.
It is the value plus or minus
something. This is the plus or
minus, so the product of the
uncertainty and the momentum.
And since we are not expecting the
mass of the particle to change,
what we really are saying is the
uncertainty in its velocity times
the uncertainty in its position is
greater than the ratio of the Planck
constant divided by 2 pi.
He put a quantitative limit on that.
And so I went through the math on
this and said suppose I wanted to be
really sloppy and I wanted to say if
the delta X, the uncertainty in
position is on the order
of one angstrom.
In other words,
I just want to know where the
electron is somewhere within the
shell radius of the ground state of
atomic hydrogen anywhere.
I don't even care if you try to put
it at the heart of the nucleus.
Anywhere from plus 0.5 to minus 0.
angstroms. All around. If you
plug in the numbers,
you will end up with an uncertainty
in the momentum as being
on the order of 15%.
A 100% uncertainty in the position
gives rise to a 15% uncertainty in
velocity. Well,
what does that tell you?
It tells you that any attempt to
try to specify,
to localize the position of an
electron in such a system is futile.
You are not going to be able to do
it. And that is even without
relativity. It gets even worse.
It is impossible to specify
accurately the position
of the electron.
And the second thing is that the
observer is part of the observation
through that interaction.
The way he described is when you
try to get down a quantum dimensions
and you are standing there with your
camera, just remember the sun is at
your back and your shadow is always
in the picture.
You disturb the thing that you try
to measure. That is number two.
Now number three. Let's go to
Erwin Schrodinger.
Erwin Schrodinger also an Austrian.
He was the University of Zürich and
was getting burned out,
too. It was Christmastime 1925.
He said I had it, so he headed off
to a villa in the mountains,
Christmastime 1925. Came back and
just after New Year's 1926 and gave
us wave mechanics.
That is what he did over his
Christmas vacation, wave mechanics.
So Christmas 1925 to New Year's 1926,
what did you do for your Christmas
vacation? Oh,
I got wave mechanics.
That is what I did. Wave mechanics.
What he did was took de Broglie
motion of the electron as a wave and
developed a wave equation
for such matter waves.
And the thing about what Schrodinger
did was all he imposed on a system
was the electron behaves as a wave
and is bound. That is to say it is
confined to the atom.
He did not invoke the quantum
condition, but he gets to the
quantum condition.
That is the brilliance of
Schršdinger. I want to give you an
analogy. I am going to show the
equation, but I don't expect
you to solve it.
You will have to have a little more
math, a little more physics to do
that, but I am going to talk about
something that you are familiar with.
And that is a violin string.
Let's take the situation where we
take a violin string and we pluck it.
I am going to graph this.
This is the y coordinate and this
is the x coordinate.
The violin string extends from x
equals zero to x equals L,
and we pluck the violin string.
What do we have here?
This is a bound system.
It is bound. And so there are only
two kinds of waves that can exist on
this violin string,
standing or traveling.
Well, if this thing is going to
vibrate for any length of time it
better be a standing wave.
And what do we have here? Well,
we know one solution to this is the
following. I am going to solve it
graphically and I am going to solve
it analytically.
You know, there is another solution.
It could do this.
The ends are fixed.
That is all I am doing.
But there is another solution.
I could do this. I could call this
n equals one, the fundamental
frequency. And I could call these
others n equals two,
n equals three. These are harmonics.
You know them as harmonics or
overtones. And the mathematics of
that equation involved a double
derivative in time of x plus some
constant times x equals zero with
some constraints on it.
**x'' + k^2 x = 0**
And the solution to this equation is
y equals A cosine kx plus B sine kx
**y = A cos kx + B sin kx**.
All I have done is said I have a
fixed system here.
What comes out of the math is that
the value of k,
there are multiple values,
there are multiple solutions.
I just showed you that you can have
a single wave, a double,
a triple and so on.
It turns out it is n pi over l.
**k = n*pi / l** The modes of
vibration in that string are
quantized. There is a plurality of
solutions that conform to the
boundary conditions,
which is that the system must be
bound.
What did Schrodinger do?
Schršdinger said I am going to take
that same idea,
and I am going to apply it to a
larger scale system.
And what he does is writes this
equation here.
This is the Schršdinger equation.
It is a wave equation. It is in
complex notation and so on.
And the solution to this equation
looks like this where it is written
in terms of a quantity
called a wavefunction.
C is the wavefunction.
And the wavefunction is clearly
nonphysical, but what we do know is
that we can take the product of the
wavefunction times its complex
conjugate. And this is proportional
to the probability of finding an
electron. And this is
a function of space.
Finding an electron in a region of
space. This actually gives rise to
a set of orbitals.
The second thing is,
just as in the case of the violin
string, the wave equation,
as posed by Schršdinger, has a
plurality of solutions.
And the plurality of solutions have
discrete values associated
with them.
But when you solve the Schršdinger
equation, you don't get just a set
of solutions that are dependent upon
one number. You get a set of
solutions that are dependent upon --
These quantum states fall out of the
solution to this equation.
No condition imposed from the
beginning that there must
be quantized states.
Instead, we have a certain energy,
we have certain geometric
constraints, we have a certain
energy constraint and these fall out.
We have multiple solutions.
These are termed the eigenvalues.
In math, when you see multiple
value solutions,
these are eigenvalues.
And these wavefunctions are the
eigenfunctions.
Now we are back to where we started.
We are back over to E as a function
of n, l, m and s,
only we got there through this other
torturous route.
And here are the samples of the
wavefunctions.
And you can see,
we will take this up next day and we
will have a look at what the orbital
shapes look like as a result of
plotting this. See you Wednesday.
