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All right. Today, in particular the second
half of the lecture today, we're going to do some of the
hardest concepts that we've done so far, and also some of the
most important concepts that we've done so far.
I just want to give you a heads-up on that.
Last time we had we had solved the Schrödinger equation for the
binding energies of the electron to the nucleus.
And that means if I write the generic Schrödinger equation
down here, what we found was what this E was.
And we found that it had a nice analytical form,
that the allowed binding energies were quantized.
They were quantized by that principle quantum number n.
And so we can write those allowed energies as E sub n is
minus R sub H over n squared.
So, n is this principle quantum number.
And n runs from 1, 2 all the way up to infinity.
Today what we're going to do is we're going to look at the
solutions to the Schrödinger equation for this,
for the wave function, for the wave.
And remember we said that the wave, in some way,
represents our electron. And today we are going to talk
about exactly what is the meaning of the wave function.
But, before we do that, I have to tell you that when
you go to solve the Schrödinger equation for the wave function
here, what happens is that two more quantum numbers pop out.
And one of the quantum numbers that pop out in the solution for
the wave function is a quantum number we call l.
l is the angular momentum quantum number.
It's called the angular momentum quantum number because
it, in fact, determines how much angular momentum the electron
has. It is a quantum number,
which means it has allowed values, very specific values.
The allowed values of l are the following.
They are 0. Unlike n, the lowest value that
l can have is 0. 0, 1, 2, 3 all the way up,
but it has got a limit. l has got a limit.
The limit is n minus one.
So l can never be larger than n minus one.
It's tied to the principal quantum number.
Why is it tied to the principal quantum number?
Well, notice that only the principle quantum number tells
you about the binding energy. It tells you about the total
energy of this electron interacting with the nucleus.
Whereas, the quantum number l, the angular momentum quantum
number, it determines how much angular momentum the electron
has. And, in a classical picture,
you may be familiar with the fact that if something has got
angular momentum, it has rotational kinetic
energy. And since the total energy is
that rotational kinetic energy plus potential energy,
well, then l is never going to be as large as n,
because otherwise then the electron would only have
rotational kinetic energy and no potential energy.
And that wouldn't work, right?
So, that's why there is a limit here to l that is as large as n
minus 1. Because you have to leave some
energy to be potential energy here.
And then, finally, we have the third quantum
number. And that third quantum number
we call m. It is the magnetic quantum
number. And it is called magnetic
because, in fact, it does determine the behavior
of an atom in a magnetic field. Although, we're actually not
going to take a look at that. But more specifically what m
is, is the z-component of the angular momentum.
And its allowed values are the following.
It's allowed values are n equals 0,
so just like l, it can have a value of zero,
plus 1, plus 2 all the way up to plus l.
Look at this. The magnetic quantum number is
tied to the angular momentum quantum number.
Why? Well, because this is the
z-component of the angular momentum, or it dictates the
z-component of the angular momentum.
l dictated the total angular momentum.
Well, the z-component of the angular momentum is never going
to be bigger than the total angular momentum,
so that is why n is tied to l. And, of course,
when we talk about a component, we're going to have a direction
here, too. And so these quantum numbers m
can be 0 plus or minus 1, plus or minus 2 all the way
down to plus or minus l. So, that is it.
We've got to have three quantum numbers here to really explain
our system. The reason we have three
quantum numbers is because we have a 3-dimensional system.
So it a little bit makes sense that you've got to have three
quantum numbers to really describe the hydrogen atom here.
And then the consequence of having these three quantum
numbers, when you solve that differential equation,
remember we talked last time where those quantum numbers come
from when you solve a differential equation,
as you will when you do 18.03. Those quantum numbers come from
imposing these boundary conditions on your differential
equation. Those boundary conditions make
a generic or general differential equation specific
to the problem you're working on.
And when you do that, those quantum numbers fall
right out. And, again, we believe them to
be true because they agree with our observations of nature.
Now we've got to deal with three quantum numbers that are
going to have to then describe these states that we talked
about last time. For example,
we talked about the ground state, n is equal to 1
here. But what I'm saying to you now
is that there are a few more quantum numbers that we have to
add to label that ground state correctly.
Or, to label it completely. And, as I just said,
if n is equal to 1, well, then we learned about l
here. l, the largest value it can
have, is n minus 1. So, we have a quantum number l
that is tied here. And that quantum number is 0.
l is equal to 0. And we just learned that we
have a third quantum number, m.
And the greatest value that m can have is plus l.
So, if l is 0 then m is 0. And so, therefore,
our ground state, the label that we have got to
put to our ground state is the (1, 0, 0) state.
So, we're expanding our definition of the label of that
state. Last time it was n equals 1.
Now it is (1, 0, 0).
And, if we have an electron in that (1, 0, 0) state,
then we're going to have a wave function that describes that
electron. That electron is going to be
represented by a wave function. That wave function is psi(1,
0, 0). The energy of that (1,
0, 0) state is minus the Rydberg constant over 1 squared.
Just like we had last time.
Now, suppose we had the first excited state,
which was n equal 2, as we looked at last time.
If n is equal to 2, what is the smallest value that
l can have? Zero.
And if l is equal to 0, what is the only value that m
can have? Zero.
We have now a state that is the (2, 0, 0) state.
And, if you have an electron in the (2, 0, 0) state,
we're going to describe that electron by a wave function that
we're going to label psi (2, 0, 0).
And the energy of that state, we saw last time,
minus the Rydberg constant over 4.
But if n is equal to 2, we also just saw that the
quantum number l can have another value.
And that is one. And if l is equal to 1,
what is the largest value that m can have?
One. And so we've got another state
here. It's the (2,
1, 1) state. And, if you've got an electron
in the (2, 1, 1) state, that electron is
going to be described by a wave function that we're going to
call the psi(2, 1, 1) state,
the psi(2, 1, 1) wave function.
Notice that the energy of this state is still minus the Rydberg
constant over 4. It is the
same as the (2, 0, 0) state.
These two states are said to be degenerate.
Degenerate means have the same energy.
If n is equal to 2, then l of course can be equal
to 1. What's the next smaller value
that m can have? Zero.
So, we have a (2, 1, 0) state.
If an electron is in that state, the wave function
describing that electron is (2, 1, 0).
Again, the energy here is minus the Rydberg constant over 4.
Same energy because the energy is only dictated by the n
quantum number. And then, finally,
what's the smallest value m can be if you have n equal 2,
l equal 1? Negative 1.
So, we have a (2, 1, -1) state.
The wave function describing that electron in that state is
the psi(2, 1, -1) wave function.
Now, I described the electrons in these states by these wave
functions psi. But there is another language,
as I alluded to last time. Another language that is used
to describe the wave functions. And that language is this
orbital language. We talked last time how an
orbital is nothing other than a wave function.
It is the solution to the Schrödinger equation for the
hydrogen atom. It is actually the spatial part
of the wave function, meaning there is another part
of the wave function called the spin part.
But we will look at that a little bit later.
And how are we going to label these wave functions in the
orbital language? Well, we're going to label them
by the principle quantum number n.
And, of course, the angular momentum quantum
number l and then a subscript here m for the magnetic quantum
number. But what we do is instead of
using the actual value of l, we have a letter code.
For example, if l is equal to 0,
we call that the s orbital. In the case of the (1,
0, 0) state, if you have an electron in that
state, the wave function that describes that electron is the
1s orbital or the psi(1, 0, 0) wave function.
This is the principal quantum number.
This is the code s for l is equal to 0.
And then, likewise, in the (2, 0,
0) state, the wave function that describes that electron
will be 2s. Because here is the n equals 2
and then our code for l equals 0 is s.
Now, when l is equal 1, again, we don't use l equals 1,
we use a letter. That letter is p.
We call that a p orbital. This state here,
the (2, 1, 1) state is going to be a 2p wave function,
or the electron in it is going to be represented by this 2p
wave function. The same thing for this state.
They're all 2p's because l is equal to 1 for all these three
states and n is equal to 2. Now, if I had here an angular
momentum quantum number of 2, we'd call that a d orbital.
If I had one that l equals 3, I would call that an f orbital,
and so on and so forth. But now, what I haven't said
anything about yet, is the m quantum number here.
Again, we've got a code for this.
We're actually not going to use the number m.
We are going to use some letters.
And the code is that when m is equal to 0 we are going to put a
z subscript here on this p orbital designation,
when m is equal to 0. When m is equal to 1 we're
going to be an x subscript on this p orbital designation,
and when m is equal to minus 1 we're going to put a y subscript
on this p orbital designation. Now, I should tell you that
what I just did here is not actually correct.
In other words, m is equal to 1 isn't really
the x subscript and m is equal to minus 1 isn't really the y
subscript. The reason is because these two
p wave functions, when m is 1 and m is minus 1,
they're complex functions. And what we do is we do some
linear combination of the m equals 1 and m equals minus 1
wave functions to get what we call the px wave function and
the py wave function. One is a plus linear
combination. One is a negative linear
combination. You're not responsible for
that. Therefore, you are not
responsible for knowing when m is equal to 1,
we have an x and when m is equal to minus 1,
we have a y. You are responsible for knowing
when m is equal to 0, you have a z subscript here.
Now, I should make one other point here.
And that is, you see I don't have a
subscript here for m on the s orbitals.
Well, in an s state, when l is equal to 0,
there is only one possible value of m, which is zero.
And so we just leave out that subscript.
We don't put a z here on the s designation.
What does this mean, the fact that we've got these
extra quantum numbers? Well, it means we have extra
states. And these states,
actually, though have the same energies that we saw last time.
But let's try to expand that a little bit and draw an energy
level diagram so that we really understand what is going on.
Here is my energy level diagram, energy going up this
way. And when n is equal to 1,
then, our quantum numbers, the most complete description
here is the (1, 0, 0) state.
And the electron in that state has the wave function that is
represents that electron. We call that the 1s wave
function. That is the energy.
When n is equal to 2, you see now we have now four
different states that are all the same energy,
2s, 2py, 2pz, 2px.
They have all the same energy. They are degenerate.
Again, degenerate means having the same energy.
And, in general, what we've got is n squared
degenerate states at each end. So, these have all the same
energy. They differ in how much angular
momentum the electron has and how much angular momentum the
electron has in the z direction if it were in the presence of a
magnetic field. But they have the same energy
because energy is only dictated by the n quantum number.
Now, when n is equal to 3, how many degenerate states do
we have? Nine.
Here they are. The energy is minus 1 ninth the
Rydberg constant. Here are the nine degenerate
states. 3s, 3py, 3pz,
3px. That pattern just reproduces
2s, 2py, 2pz, 2px, so we won't go through
those. But now we've got some extra
states here. Again, their quantum numbers
are all 3 for n, but when n is 3,
the largest value of l is 2. And so all of these states here
have l is equal to 2. And our code for l is equal to
2 were these d states. If you have an electron in
those states, it's represented by a 3d wave
function. But now, the reason these 3d
wave functions differ is because they have different amounts of
angular momentum in the z component.
They have different values of m.
If n is equal to 3, l is equal 2.
This smallest value that m can have is minus 2,
right? And then I am going to just
call, when m is equal to minus 2, xy here in the subscript.
When m is equal to minus 1, I am going to put yz in this
subscript. When m is equal to 0,
I am going to put z squared in this subscript.
That is going to be my code for m is equal to 0.
In the case of the 3d orbitals, the 3d wave functions,
I am going to put a z squared in that subscript.
Then when m is equal to 1, I am putting an xz.
And when m is equal to 2, an x squared minus y squared.
Again, you are now responsible
for knowing when m is minus 2, this is xy, when m is minus 1,
this is yz, because that isn't strictly correct.
Again, we've complex functions. We're making them real.
And we're taking linear combinations.
You are responsible for knowing that when m is equal to 0 in a
3d state, this subscript here is z squared.
When m is equal to 0 in a 2p state, that's a z subscript.
When m is equal to 0 in a 3d state, that's a z squared
subscript. You will look at these 3d
states more in detail when we talk about transition metals in
the second half of the course with Professor Drennan.
So, that's our energy level diagram.
Everybody okay with this? Questions?
Okay.
We've found these two other quantum numbers.
We've got three quantum numbers.
We know how to label the states, and we know how to label
the wave functions that describe the electrons in these states,
but now it's time to talk about what a wave function really
means, or how exactly does a wave function represent this
particle, this electron. How do you interpret the wave
function? And the answer is,
you don't. The answer is that that wave
function is the wave function. It is one of these concepts
that you cannot or you don't have a ready classical analogy.
It's one of these concepts that you don't experience in your
everyday life, and so you cannot say it's like
this, because within your everyday world there is nothing
to draw a correct analogy to. However, the wave function
squared does have a physical interpretation that is
meaningful in your environment. And that is that this wave
function squared, and let me just write it out
here now in its full glory. Remember, this wave function,
we're going to denote by three quantum numbers,
n, l and m, and it's a function of r, theta, and phi.
If I take this wave function and I square it,
the interpretation of that wave function squared is that it is
the probability density --
-- that is it is a probability per unit volume.
A density is always per unit volume.
So, it's a probability per unit volume.
Where did that come from? Good question.
Soon after Schrödinger wrote down his wave equations,
there was a lot of discussion in the scientific community
about what does psi really mean. What is the interpretation of
psi? And there were a lot of
interpretations given, a lot of explanations given,
but finally, one very cleaver scientist by
the name of Max Born said, if I think of psi squared as a
probability density, then I can understand the
predictions that the Schrödinger equation makes.
So, psi squared is an interpretation.
He said, if I think of psi squared as the probability per
unit volume and then I go and analyze all of the results of
solving the Schrödinger equation in terms of that concept,
that it is a probability density, then things make sense.
And those predictions agree with my observations.
You know what? That's it.
There is no derivation here of psi squared.
Psi squared is just that. It is an assumption.
It is an interpretation. But, you know what,
you use that interpretation and that makes predictions that
agree with our observations. And there are no observations
that even hint that that interpretation isn't correct.
But that is it. You don't go any further.
There's no derivation of this. This is an interpretation.
It is a probability density. Well, this gentleman,
Max Born, was a really clever scientist.
Max Born is the same Born of something called the
Born-Oppenheimer approximation that maybe some of you know
about. It is the same Born of the
Distorted-Wave Born Approximation,
which probably nobody knows about.
But I've got to tell you that despite the fact that Max Born
has all of these accomplishments,
the interpretation of psi squared is a probability
density, the Born-Oppenheimer approximation,
the Distorted-Wave Born Approximation,
despite all of those accomplishments,
he is best known for being the grandfather of Olivia
Newton-John. [LAUGHTER] That is true.
Actually, a couple of weeks ago in the Boston Globe,
and I guess papers throughout the country, has a magazine
called Parade Magazine. And it actually had an article
about Olivia Newton-John, if somebody is going who's
Olivia Newton-John, that's just an indication of
how old I am. Olivia Newton-John is a singer.
Grease.
It was interesting. The Boston Globe a few weeks
ago in Parade Magazine, there was an article on her.
And it mentioned that her grandfather was Max Born.
Didn't say anything about Max Born after that,
but did mention that. All right.
Now you know what psi squared is.
It's a probability density. You and I, we can understand
that. Psi squared,
we can understand that. It's a probability density.
But what I want to really emphasize here is that it is a
probability density. It is not probability.
It is probability per unit volume.
That's important. In a moment,
we're going to look at a quantity that is a probability.
But this is a probability density.
You've got to get the units correct here.
How can we use that to understand something about where
the electron is in an atom? Well, let's do that on the
slide here. Kill those lights.
Probability density. What I am going to do is I am
going to represent the probability density here by a
density dot diagram. I am going to take my 1s
orbital, my 1s wave function, or my psi(1,
0, 0) wave function, and I am going to square it.
And then I'm going to take the results of that squared function
and plot it as a dot diagram, here, where the density of the
dots is proportional to the probability density.
So the darker the dots, the higher the probability
density. And, as the dots get more and
more dispersed, the probability density is
getting smaller. I did that for my 1s wave
function, squared the 1s wave function and plotted it here on
x, y, z. Oh, I got the x and the y
turned around. That's all right.
I plotted it here. What you can see is that the
probability density is a maximum at the origin,
at the nucleus, and that it decays equally in
all directions. It is isotropic.
There is no angular dependence. It's spherically symmetric.
And that you can also see nicely, if you look at the
actual plot of the wave function.
Not psi squared now. The wave function,
which I am plotting here. This is the wave function and I
am plotting it versus r. Actually, what I am plotting it
here is as r over this something called the "a nought."
a nought is a constant. a nought is called the Bohr
radius. a nought is about a half an
angstrom. I will explain to you where a
nought comes from in a little bit, but right now it's just a
constant. Here is the actual form of that
wave function. It's pretty simple.
This stuff right in here, this is all a constant.
Just one over pi a nought cubed.
**1 / (pi (a0)^3)** I said this was about a half an angstrom.
And then the important part is right here.
It's a decaying exponential in r, e to the minus r over a
nought. And so the wave
function starts out at some high finite value at r is equal to 0
and it drops off exponentially as r goes to infinity.
That's what we're plotting. And you can see that reflected
here. When you square this you can
see that reflected in that probability density.
What does it tell me? It tells me that the
probability density is highest at the nucleus and drops off in
an exponential fashion equally in all directions.
This is spherically symmetric. We're going to say some more
about where is the electron in a moment after we go through not
only 1s, but 2s and 3s. Here I plotted the 2s wave
function. I took the 2s wave function,
I squared it, so I got values as a function
of r. And, again, the density of the
dots is proportional to that wave function squared.
And, again, what do you see? You see that the probability
density is largest at the origin.
And that it decays again equally in all directions
because it's spherically symmetric.
And it decays so much that all of a sudden you get to some
value of r, and there is no probability density.
And then, as you get to the large value of r,
well, the probability density increases again right here.
And then it finally decays as you get to larger values of r.
What this wave function has in it, here, is something called a
node. There is a value of r right in
here where there is no probability density.
That value of r, in terms of this constant,
a nought, is 2 times a nought.
We call that a radial node. You can see that radial node
very easily if you look at a plot of the actual wave
function. We're not squaring the wave.
This is the wave function. Here it is.
Here is the form for it. At r is equal to zero
it's some finite large value, and it decreases.
It decreases until we get to a value of zero,
psi is equal to 0. Let me just draw that here on
the board. I am plotting here the psi(2,
0, 0) wave function, the 2s wave function.
I am plotting it as a function of r.
What it looks like is that at r is equal to 0,
it starts out as some high finite value and it drops.
And it actually goes through a zero.
And it drops and it keeps dropping until some point and
becomes negative. And then it turns around and it
approaches a zero again. And so at very large r the wave
function is zero. right here, where this wave
function becomes zero, that's what we call a radial
node, r equals 2 a nought.
This is a radial node. It is a value of r that makes
the wave function be equal to zero.
If you wanted to solve for the radial node, you'd simply take
the functional form for 2s, which is 1 over 4 (1 over 2 pi
a nought cubed) to the one half times (2 minus (r over a
nought)) times e to the (minus r over 2 a nought).
If you wanted to solve
for the value of r at which this node occurs,
you set everything equal to zero.
You solve for r. This goes away.
This stuff is just a constant. It goes away.
You have 2 minus r over a nought,
and that's equal to zero. You can see,
when you solve this, that r is equal to 2 a nought.
That is the value of r at which
you have a node. Now, notice this.
At r equal to a nought, the wave function changes
sign. It was positive up here.
It is negative down here. That is a general
characteristic. When the wave function goes
through a node, the wave function changes sign.
Why is that important? Well, have you maybe in high
school -- We haven't talked about p
orbitals yet. But remember you put sometimes
a plus sign on a p orbital and a minus sign on a p orbital?
Did you do that? Yeah?
Well, that plus and the minus represented the sign of the
amplitude of the wave function. Plus meant it had a positive
amplitude. Minus meant it had a negative
amplitude. That's what it meant.
Sometimes you may have called it the phase of the wave
function. You know why that's important,
the phase of the wave function? It's important because when you
go to do chemical bonding, when you bring up two atoms and
you are overlapping two atoms, and you're overlapping the wave
functions that have the same sign, you've got constructive
interference and you have chemical bonding.
If you come in and you overlap two wave functions that have
opposite signs, you've got destructive
interference and you've got no chemical bonding.
We're going to see that later, but that's why that's so
important, is the amplitude of the wave function here and these
nodes. Let's look at the 3s wave
function. Here we are back up here.
Plotted the 3s wave function. The probability density of the
3s wave function. Again, what do you see?
You see right in the center, probability density is the
largest. And then, as r increases,
that probability density drops off.
However, you get to a value of r, right here.
where you've got another node, another value of r that makes
the wave function be equal to zero.
And then, as you increase r, the wave function again,
the probability density increases.
It increases because look right here.
You're getting a more and more negative value.
If you square a negative value you're going to get a large
positive value. That is what this high
probability density reflects. It reflects the fact that the
wave function right in this area has a large negative value.
You square it. You're going to get a large
positive value. And then, as you increase r
again, well, again you get another value of r where you
have a node. The wave function goes to zero.
Here's that point right there. Here is psi equals 0.
Wave function zero. And then, as you increase r,
well, again as you increase r, the probability density goes
up. Why?
Because the wave function increased right here.
See that? Square it.
It's going to go up. And then it turns around and
goes to zero and this decays. If you wanted to solve for this
radial node and this radial node, which I will let you do,
you take the functional form, set it equal to zero and solve
for r. That is easy.
All of this stuff is a constant.
This goes away. It's the quadratic and you're
solving for the roots. So what have we learned?
We've learned the that 1s wave function, 2s wave function,
3s wave function, and in fact all s wave
functions or l=0 wave functions, all of them are spherically
symmetric. They have no angular
dependence. Remember, we said that in
general the wave function depends on r,
theta and phi. s wave functions have no theta
and phi dependence in them, unlike p wave functions,
which we will look at very soon.
Also, very important here, the probability density of all
s wave functions is a finite value at r equals 0.
Those are the two important things.
And you know how to solve now for a radial node.
You set the wave function equal to zero, solve for the value of
r that makes the wave function zero.
Now, we have to continue this discussion so we can start to
talk about the probability of finding an electron some
distance from the nucleus. We just talked about
probability density. We haven't talked about
probability yet. And here comes probability.
We're going to plot something and we're going to talk about
something called the radial probability distribution.
What is that? Well, the radial probability
distribution is the probability of finding an electron in a
shell. You know what a shell is.
A shell is a sphere, but the sphere is hollow,
except that at the diameter or at the circumference,
you have a thin coating. That thinness has got a dr to
it. It has a thickness to it.
I'm sorry. It's a probability.
This radial probability distribution is the probability
of finding an electron in a shell of radius r and a
thickness dr. Let me try to explain that some
more. Here is this gray part.
That's my probability density that I drew before,
the dot diagram. This is a 1s wave function.
This is a probability density for the 1s wave function,
psi(1, 0, 0) squared. This blue thing here is a
spherical shell. It has some radius here,
r. It also has some thickness,
dr. You see I cut away here,
this thickness dr? The radial probability
distribution is the probability of finding this electron in this
spherical shell that is of radius r and thickness dr.
So, it's the probability of finding the electron at some
distance r to r plus dr. That's the radial probability
distribution. Now, I want you to notice,
what is the volume, here, of this spherical shell?
Well, that volume is just 4 pi r squared, the surface area of
that spherical shell, times the thickness which we're
going to call dr. That is the
volume. Now, if I wanted a probability
of finding my electron in an s wave function some distance from
the nucleus, some distance r to r plus dr, what am I going to
do? Well, I'm going to take my
probability density, which I said was psi squared
probability per unit volume. I am going to multiply it by a
volume. Then I am going to get a
probability of finding my electron some distance r to r
plus dr from the nucleus. And that is what we're going to
talk about on Monday, is the actual probability of
finding the electron from the nucleus.
See you on Monday.