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And your equation is equal to E
Psi.
00:00:18.000 --> 00:00:25.000
And we saw that these energies,
the binding energies of the
00:00:25.000 --> 00:00:30.000
electron to the nucleus,
were quantized.
00:00:35.000 --> 00:00:41.000
Well, these energy levels then
were given by minus this Rydberg
00:00:41.000 --> 00:00:48.000
constant R sub H over n squared.
And this n
00:00:48.000 --> 00:00:53.000
was our quantum number.
It is a principle quantum
00:00:53.000 --> 00:00:56.000
number.
And we saw that n,
00:00:56.000 --> 00:01:01.000
the smallest value was 1,
2 and ran all the way up to
00:01:01.000 --> 00:01:08.000
infinity.
Well, today what we are going
00:01:08.000 --> 00:01:16.000
to do is solve this equation,
or we are going to look at the
00:01:16.000 --> 00:01:24.000
results of solving this equation
for the wavefunction Psi.
00:01:24.000 --> 00:01:30.000
Now, Psi, in general,
is the function of r,
00:01:30.000 --> 00:01:38.000
theta, phi, and also time.
But we are going to be looking
00:01:38.000 --> 00:01:42.000
at problems in which time does
not have an effect.
00:01:42.000 --> 00:01:47.000
In other words,
the wave functions that we are
00:01:47.000 --> 00:01:53.000
going to be looking at are what
are called stationary waves.
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We actually are not going to be
looking at the wave function as
00:01:59.000 --> 00:02:04.000
a chemical reaction is
happening.
00:02:04.000 --> 00:02:08.000
We are either going to look at
it before or after,
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but not during.
And, in those cases,
00:02:11.000 --> 00:02:14.000
we are looking at a wave
function.
00:02:14.000 --> 00:02:18.000
And the atom is just stable and
is sitting there.
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The time dependence here does
not have a consequence.
00:02:23.000 --> 00:02:27.000
And so, therefore,
the wave functions that we are
00:02:27.000 --> 00:02:31.000
going to be looking at are just
a function of r,
00:02:31.000 --> 00:02:36.000
theta and phi.
And we are looking at what is
00:02:36.000 --> 00:02:40.000
called time-independent quantum
mechanics.
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Later on, actually,
if you are in chemistry,
00:02:45.000 --> 00:02:50.000
in a graduate course in
chemistry, is when you look at
00:02:50.000 --> 00:02:55.000
time-dependent wave functions.
We are going to look at
00:02:55.000 --> 00:03:02.000
time-independent quantum
mechanics, the stationary wave.
00:03:02.000 --> 00:03:06.000
Now, it turns out that when we
go and solve the Schrˆdinger
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equation here for Psi,
what happens is that two more
00:03:10.000 --> 00:03:15.000
quantum numbers drop out of that
solution to the differential
00:03:15.000 --> 00:03:19.000
equation.
Remember, we said last time how
00:03:19.000 --> 00:03:22.000
quantum numbers arise.
They arise from imposing
00:03:22.000 --> 00:03:26.000
boundary conditions on a
differential equation,
00:03:26.000 --> 00:03:30.000
making that differential
equation applicable to your
00:03:30.000 --> 00:03:36.000
actual physical problem.
And so, when we do that,
00:03:36.000 --> 00:03:40.000
we get a new quantum number
called l.
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And l is, I think you already
know, the angular momentum
00:03:47.000 --> 00:03:50.000
quantum number.
Absolutely.
00:03:50.000 --> 00:03:56.000
It is called the angular
momentum quantum number because
00:03:56.000 --> 00:04:03.000
it indeed dictates how much
angular momentum the electron
00:04:03.000 --> 00:04:08.000
has.
It has allowed values.
00:04:08.000 --> 00:04:14.000
The allowed values of l,
now, are zero.
00:04:14.000 --> 00:04:22.000
Zero is the smallest value of
l, the lowest value of l
00:04:22.000 --> 00:04:28.000
allowed.
1, 2, all the way up to n minus
00:04:32.000 --> 00:04:36.000
can have.
It cannot be any larger than n
00:04:36.000 --> 00:04:39.000
minus 1.
Why can't it be larger than n
00:04:39.000 --> 00:04:42.000
minus 1?
Well, it cannot because the
00:04:42.000 --> 00:04:48.000
angular momentum quantum number,
at least if you want to think
00:04:48.000 --> 00:04:53.000
classically for a moment,
dictates how much rotational
00:04:53.000 --> 00:04:58.000
kinetic energy you have.
And remember that this energy,
00:04:58.000 --> 00:05:03.000
here, is dependent only on n.
This energy is the sum of
00:05:03.000 --> 00:05:06.000
kinetic energy plus potential
energy.
00:05:06.000 --> 00:05:10.000
If l had the same value as n,
well, essentially,
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that would mean that we would
have only rotational kinetic
00:05:15.000 --> 00:05:18.000
energy and we would have no
potential energy.
00:05:18.000 --> 00:05:22.000
But that is not right.
We have potential energy.
00:05:22.000 --> 00:05:25.000
We have potential energy of
interaction.
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So, physically,
that is why l is tied to n.
00:05:30.000 --> 00:05:34.000
And it cannot be larger than n
minus 1.
00:05:34.000 --> 00:05:41.000
And then, we have a third
quantum number that drops out of
00:05:41.000 --> 00:05:45.000
that solution,
which is called M.
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It is the magnetic quantum
number.
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It is called that because
indeed it dictates how an atom
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moves in a magnetic field.
Or, how it behaves in a
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magnetic field.
But, more precisely,
00:06:07.000 --> 00:06:13.000
m is the z-component of the
angular momentum.
00:06:13.000 --> 00:06:21.000
l is the total angular moment.
m dictates the z-component of
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the angular momentum.
And the allowed values of m are
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m equals zero.
You can have no angular
00:06:33.000 --> 00:06:38.000
momentum in the z-component.
Or, plus one,
00:06:38.000 --> 00:06:44.000
plus two, plus three,
all the way up to plus l.
00:06:44.000 --> 00:06:50.000
Again, m is tied to l.
It cannot be larger than l,
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because if it were then you
would have more angular momentum
00:06:58.000 --> 00:07:06.000
in the z-component than you had
total angular momentum.
00:07:06.000 --> 00:07:10.000
And that is a no-no.
So, m is tied to l.
00:07:10.000 --> 00:07:14.000
The largest value you can have
is l.
00:07:14.000 --> 00:07:21.000
But, since this is a
z-component and we've got some
00:07:21.000 --> 00:07:26.000
direction, m could also be minus
1, minus two,
00:07:26.000 --> 00:07:30.000
minus three,
minus l.
00:07:30.000 --> 00:07:33.000
We have three quantum numbers,
now.
00:07:33.000 --> 00:07:39.000
That kind of makes sense
because we have a 3-dimensional
00:07:39.000 --> 00:07:43.000
problem.
We are going to have to have
00:07:43.000 --> 00:07:49.000
three quantum numbers to
completely describe our system.
00:07:49.000 --> 00:07:54.000
The consequence,
here, of having three quantum
00:07:54.000 --> 00:08:00.000
numbers is that we now have more
states.
00:08:11.000 --> 00:08:14.000
For example,
our n equals 1 state that we
00:08:14.000 --> 00:08:18.000
talked about last time,
more completely we have to
00:08:18.000 --> 00:08:21.000
describe that state by two other
quantum numbers.
00:08:21.000 --> 00:08:25.000
When n is equal to one,
what is the only value that l
00:08:25.000 --> 00:08:27.000
can have?
Zero.
00:08:27.000 --> 00:08:30.000
And if l is zero,
what is the only value m can
00:08:30.000 --> 00:08:32.000
have?
Zero.
00:08:32.000 --> 00:08:37.000
And so, more appropriately,
or more completely,
00:08:37.000 --> 00:08:42.000
that n equals 1 state is the
(1, 0, 0) state.
00:08:42.000 --> 00:08:48.000
And, if we have an electron in
that (1, 0, 0) state,
00:08:48.000 --> 00:08:55.000
we are going to describe that
electron by the wave function
00:08:55.000 --> 00:09:00.000
Psi(1, 0, 0).
Now, what I have not told you
00:09:00.000 --> 00:09:05.000
yet is exactly how Psi
represents the electron.
00:09:05.000 --> 00:09:09.000
I am just telling you right now
that Psi does.
00:09:09.000 --> 00:09:14.000
Exactly how it does is
something I haven't told you
00:09:14.000 --> 00:09:17.000
yet.
And we will do that later in
00:09:17.000 --> 00:09:20.000
the lecture today,
sort of.
00:09:20.000 --> 00:09:24.000
That is the energy,
minus the Rydberg constant.
00:09:24.000 --> 00:09:30.000
But now, if n is equal two,
what is the smallest value that
00:09:30.000 --> 00:09:33.000
l can have?
Zero.
00:09:33.000 --> 00:09:36.000
And the value of m in that
case?
00:09:36.000 --> 00:09:39.000
Zero.
And so our n equals 2 state is
00:09:39.000 --> 00:09:42.000
more completely the (2,
0, 0) state.
00:09:42.000 --> 00:09:47.000
And the wave function that
describes the electron in that
00:09:47.000 --> 00:09:51.000
state is the Psi(2,
0, 0) wave function.
00:09:51.000 --> 00:09:55.000
And then, of course,
here is the energy of that
00:09:55.000 --> 00:09:58.000
state.
Now, when n is equal to 2,
00:09:58.000 --> 00:10:03.000
what is the next larger value
of l?
00:10:03.000 --> 00:10:05.000
One.
And if l is equal to one,
00:10:05.000 --> 00:10:09.000
what is the largest value that
m can be?
00:10:09.000 --> 00:10:12.000
One.
And so we have another state
00:10:12.000 --> 00:10:14.000
here, the (2,
1, 1) state.
00:10:14.000 --> 00:10:19.000
And if you have an electron in
that state, it is described by
00:10:19.000 --> 00:10:22.000
the Psi(2, 1,
1) wave function.
00:10:22.000 --> 00:10:27.000
However, the energy here is
also minus one-quarter the
00:10:27.000 --> 00:10:32.000
Rydberg constant.
It has the same energy as the
00:10:32.000 --> 00:10:36.000
(2, 0, 0) state.
Now, if n is equal two and l is
00:10:36.000 --> 00:10:40.000
equal to one,
what is the next larger value
00:10:40.000 --> 00:10:40.000
of m?
Zero.
And so we have a (2,
1, 0) state.
00:10:43.000 --> 00:10:48.000
Again, that wave function,
for an electron in that state,
00:10:48.000 --> 00:10:50.000
we label as Psi(2,
1, 0).
00:10:50.000 --> 00:10:53.000
And then, finally,
when n is equal to two,
00:10:53.000 --> 00:10:57.000
l is equal to one,
what is the final possible
00:10:57.000 --> 00:11:01.000
value of m?
Minus one.
00:11:01.000 --> 00:11:04.000
We have a (2,
1, -1) state and a wave
00:11:04.000 --> 00:11:09.000
function that is the (2,
1, -1) wave function.
00:11:09.000 --> 00:11:15.000
Notice that the energy of all
of these four states is the
00:11:15.000 --> 00:11:20.000
same, minus one-quarter the
Rydberg constant.
00:11:20.000 --> 00:11:24.000
These states are what we call
degenerate.
00:11:24.000 --> 00:11:30.000
Degenerate means having the
same energy.
00:11:30.000 --> 00:11:34.000
That is important.
Now, this is the way we label
00:11:34.000 --> 00:11:38.000
wave functions,
but we also have a different
00:11:38.000 --> 00:11:42.000
scheme for talking about wave
functions.
00:11:42.000 --> 00:11:46.000
That is, we have an orbital
scheme.
00:11:46.000 --> 00:11:50.000
And, as I said,
or alluded to the other day,
00:11:50.000 --> 00:11:55.000
an orbital is nothing other
than a wave function.
00:11:55.000 --> 00:12:01.000
It is a solution to the
Schrˆdinger equation.
00:12:01.000 --> 00:12:04.000
That is what an orbital is.
It is actually the spatial part
00:12:04.000 --> 00:12:08.000
of the wave function.
There is another part called
00:12:08.000 --> 00:12:11.000
the spin part,
which we will deal with later,
00:12:11.000 --> 00:12:14.000
but an orbital is essentially a
wave function.
00:12:14.000 --> 00:12:17.000
We have a different language
for describing orbitals.
00:12:17.000 --> 00:12:20.000
The way we do this,
I think you are already
00:12:20.000 --> 00:12:23.000
familiar with,
but we are going to describe it
00:12:23.000 --> 00:12:26.000
by the principle quantum number
and, of course,
00:12:26.000 --> 00:12:30.000
the angular momentum quantum
number and the magnetic quantum
00:12:30.000 --> 00:12:35.000
number.
Except that we have a scheme or
00:12:35.000 --> 00:12:40.000
a code for l and m.
That code uses letters instead
00:12:40.000 --> 00:12:42.000
of numbers.
For example,
00:12:42.000 --> 00:12:48.000
if l is equal to zero here,
we call that an s wave function
00:12:48.000 --> 00:12:51.000
or an s orbital.
For example,
00:12:51.000 --> 00:12:55.000
in the orbital language,
instead of Psi(1,
00:12:55.000 --> 00:13:00.000
0, 0), we call this a 1s
orbital.
00:13:00.000 --> 00:13:02.000
Here is the principle quantum
number.
00:13:02.000 --> 00:13:05.000
Here is s.
l is equal to zero.
00:13:05.000 --> 00:13:08.000
That is just our code for l is
equal to zero.
00:13:08.000 --> 00:13:12.000
This state, (2,
0, 0), you have an electron in
00:13:12.000 --> 00:13:15.000
that state.
Well, it can describe that
00:13:15.000 --> 00:13:19.000
wavefunction by the 2s orbital.
Here is the principle quantum
00:13:19.000 --> 00:13:22.000
number.
He is s, our code for l equals
00:13:22.000 --> 00:13:25.000
zero.
Now, when n is equal to 1,
00:13:25.000 --> 00:13:30.000
we call that a p wave function
or a p orbital.
00:13:30.000 --> 00:13:33.000
This state right here,
it is the 2p state.
00:13:33.000 --> 00:13:39.000
Because n is two and l is one,
this state is also the 2p wave
00:13:39.000 --> 00:13:44.000
function and this state,
it is also the 2p wave
00:13:44.000 --> 00:13:47.000
function.
Now, if I had a wave function
00:13:47.000 --> 00:13:52.000
up here where l was equal to
two, we would call that d
00:13:52.000 --> 00:13:56.000
orbital.
And, if I had a wave function
00:13:56.000 --> 00:14:00.000
up here where l was three,
we would call that an f
00:14:00.000 --> 00:14:05.000
orbital.
But now, all of these wave
00:14:05.000 --> 00:14:09.000
functions in the orbital
language have the same
00:14:09.000 --> 00:14:15.000
designation, and that is because
we have not taken care of this
00:14:15.000 --> 00:14:19.000
yet, the m quantum number,
magnetic quantum number.
00:14:19.000 --> 00:14:24.000
And we also have an alphabet
scheme for those quantum
00:14:24.000 --> 00:14:27.000
numbers.
The bottom line is that when m
00:14:27.000 --> 00:14:33.000
is equal to zero,
we put a z subscript on the p.
00:14:33.000 --> 00:14:39.000
When m is equal to zero,
that is our 2pz wave function.
00:14:39.000 --> 00:14:44.000
When m is equal to one,
we are going to put an x
00:14:44.000 --> 00:14:47.000
subscript on the p wave
function.
00:14:47.000 --> 00:14:54.000
And when m is equal to minus
one, we are going to put a y
00:14:54.000 --> 00:14:59.000
subscription on the wave
function.
00:14:59.000 --> 00:15:04.000
However, I have to tell you
that in the case of m is equal
00:15:04.000 --> 00:15:08.000
to one and m is equal to minus
one, that is not strictly
00:15:08.000 --> 00:15:12.000
correct.
And the reason it is not is
00:15:12.000 --> 00:15:16.000
because when you solve
Schrˆdinger equation for the (2,
00:15:16.000 --> 00:15:21.000
1, 1) wave function and the (2,
1, -1) wave function,
00:15:21.000 --> 00:15:25.000
the solutions are complex wave
functions.
00:15:25.000 --> 00:15:29.000
They are not real wave
function.
00:15:29.000 --> 00:15:35.000
And so, in order for us to be
able to draw and think about the
00:15:35.000 --> 00:15:42.000
px and the py wave functions,
what we do is we take a linear
00:15:42.000 --> 00:15:47.000
combination of the px and the py
wave functions.
00:15:47.000 --> 00:15:54.000
We take a positive linear
combination, in the case of the
00:15:54.000 --> 00:15:59.000
px wave function.
The px wave function is really
00:15:59.000 --> 00:16:03.000
this wave function plus this
wave function.
00:16:03.000 --> 00:16:08.000
And the py wave function is
really this wave function minus
00:16:08.000 --> 00:16:12.000
this wave function.
Then we will get a real
00:16:12.000 --> 00:16:15.000
function, and that is easier to
deal with.
00:16:15.000 --> 00:16:18.000
That is strictly what px and py
are.
00:16:18.000 --> 00:16:23.000
px and py are these linear
combinations of the actual
00:16:23.000 --> 00:16:29.000
functions that come out of the
Schrˆdinger equation.
00:16:29.000 --> 00:16:32.000
pz is exactly correct.
pz, M is equal to zero,
00:16:32.000 --> 00:16:36.000
in the pz wavefunction.
Now, you are not responsible
00:16:36.000 --> 00:16:40.000
for knowing that.
I just wanted to let you know.
00:16:40.000 --> 00:16:45.000
You don't have to remember that
m equal to one gives you the x
00:16:45.000 --> 00:16:50.000
subscript, and m equal to minus
one gives you the y subscript.
00:16:50.000 --> 00:16:54.000
You do have to know that when m
is equal to zero,
00:16:54.000 --> 00:17:00.000
you get a z subscript,
because that is exactly right.
00:17:00.000 --> 00:17:03.000
Now, just one other comment
here.
00:17:03.000 --> 00:17:08.000
That is, you see we did not put
a subscript on the s wave
00:17:08.000 --> 00:17:13.000
functions.
Well, that is because for an s
00:17:13.000 --> 00:17:17.000
wave function,
the only choice you have for m
00:17:17.000 --> 00:17:20.000
is zero.
And so we leave out that
00:17:20.000 --> 00:17:24.000
subscript.
We never put a z on there or a
00:17:24.000 --> 00:17:27.000
z on there.
It is always 1s,
00:17:27.000 --> 00:17:32.000
2s, or 3s.
Well, in order to understand
00:17:32.000 --> 00:17:37.000
that just a little more,
let's draw an energy level
00:17:37.000 --> 00:17:41.000
diagram.
Here is the energy again.
00:17:41.000 --> 00:17:45.000
And for n equals 1,
we now see that the more
00:17:45.000 --> 00:17:50.000
complete description is three
quantum numbers,
00:17:50.000 --> 00:17:53.000
(1, 0, 0).
That gives us the (1,
00:17:53.000 --> 00:17:57.000
0, 0) state,
or sometimes we say the 1s
00:17:57.000 --> 00:18:00.000
state.
There is one state at that
00:18:00.000 --> 00:18:05.000
energy.
But, in the case of n equals 2,
00:18:05.000 --> 00:18:11.000
we already saw that we have
four different states for the
00:18:11.000 --> 00:18:14.000
quantum number.
Four different states.
00:18:14.000 --> 00:18:19.000
They all have the same energy.
They are degenerate.
00:18:19.000 --> 00:18:23.000
Degenerate means having the
same energy.
00:18:23.000 --> 00:18:30.000
They differ in how much angular
momentum the electron has.
00:18:30.000 --> 00:18:35.000
Or, how much angular momentum
the electron would have in the z
00:18:35.000 --> 00:18:38.000
component if it were in a
magnetic field.
00:18:38.000 --> 00:18:43.000
The total energy is the same
because only n determines the
00:18:43.000 --> 00:18:46.000
energy.
It is just that the amount of
00:18:46.000 --> 00:18:50.000
angular momentum,
or the z component of the
00:18:50.000 --> 00:18:52.000
angular momentum,
differs in 2s,
00:18:52.000 --> 00:18:56.000
2py, 2pz, 2px,
but they all have the same
00:18:56.000 --> 00:19:01.000
energy.
In general, for any value of n,
00:19:01.000 --> 00:19:08.000
there are n squared degenerate
states at each value of n.
00:19:08.000 --> 00:19:14.000
If we have n is equal to three,
then the energy is minus
00:19:14.000 --> 00:19:21.000
one-ninth the Rydberg constant.
But how many states do we have
00:19:21.000 --> 00:19:24.000
at n equal three?
Nine.
00:19:24.000 --> 00:19:28.000
Here they are.
Just like for n equal two,
00:19:28.000 --> 00:19:34.000
we have the 3s,
3py, 3pz, and 3px.
00:19:34.000 --> 00:19:38.000
And there are the associated
quantum numbers for that.
00:19:38.000 --> 00:19:44.000
But now we have some states
here that I have labeled 3d
00:19:44.000 --> 00:19:47.000
states.
Well, when you have a d state,
00:19:47.000 --> 00:19:52.000
that means l is equal to two.
All of these states have
00:19:52.000 --> 00:19:58.000
principle quantum number three
and angular momentum quantum
00:19:58.000 --> 00:20:03.000
number two.
And they differ also by the
00:20:03.000 --> 00:20:08.000
amount of angular momentum in
the z direction.
00:20:08.000 --> 00:20:12.000
They differ in the quantum
number m.
00:20:12.000 --> 00:20:16.000
For example,
we are going to call the m
00:20:16.000 --> 00:20:20.000
equal minus two state the
3d(xy).
00:20:20.000 --> 00:20:24.000
We are going to put xy as a
subscript.
00:20:24.000 --> 00:20:30.000
For m equal minus one,
we are going to put a subscript
00:20:30.000 --> 00:20:33.000
yz.
For m equals zero,
00:20:33.000 --> 00:20:38.000
we are going to put a subscript
z squared.
00:20:38.000 --> 00:20:40.000
For m equal one,
d(xz).
00:20:40.000 --> 00:20:44.000
And for m equal two,
x squared minus y squared.
00:20:44.000 --> 00:20:49.000
Again, for m equal minus two,
00:20:49.000 --> 00:20:54.000
minus one, one and two,
those wave functions,
00:20:54.000 --> 00:20:59.000
when you solve Schrˆdinger
equations, are complex wave
00:20:59.000 --> 00:21:04.000
functions.
And what we do to talk about
00:21:04.000 --> 00:21:08.000
the wave functions is we take
linear combinations of them to
00:21:08.000 --> 00:21:12.000
make them real.
And so, when I say m is minus
00:21:12.000 --> 00:21:16.000
two, is the 3dxy wave function,
it is not strictly correct.
00:21:16.000 --> 00:21:20.000
Therefore, again,
you don't need to know m minus
00:21:20.000 --> 00:21:23.000
two.
You don't need to know that
00:21:23.000 --> 00:21:24.000
subscript.
Or m equal one,
00:21:24.000 --> 00:21:29.000
you don't need to know that
subscript.
00:21:29.000 --> 00:21:32.000
But for m equal zero,
this is a z squared.
00:21:32.000 --> 00:21:36.000
Absolutely.
These wave functions are linear
00:21:36.000 --> 00:21:38.000
combinations.
This one is not.
00:21:38.000 --> 00:21:43.000
It is a real function when it
comes out of the Schrˆdinger
00:21:43.000 --> 00:21:47.000
equation.
You will talk about these 3d
00:21:47.000 --> 00:21:51.000
wave functions a lot with
Professor Cummins in the
00:21:51.000 --> 00:21:56.000
second-half of the course.
You will actually look at the
00:21:56.000 --> 00:22:01.000
shapes of those wave functions
in detail.
00:22:01.000 --> 00:22:04.000
So, that's the energy level
diagram, here.
00:22:04.000 --> 00:22:08.000
All right.
Now let's actually talk about
00:22:08.000 --> 00:22:11.000
the shapes of the wave
functions.
00:22:11.000 --> 00:22:16.000
I am going to raise this
screen, I think.
00:22:25.000 --> 00:22:40.000
What do these wave functions
actually look like?
00:22:40.000 --> 00:22:45.000
Well, for a hydrogen atom,
our wave function here,
00:22:45.000 --> 00:22:52.000
given by three quantum numbers,
n, l and m, function of r,
00:22:52.000 --> 00:22:57.000
theta and phi,
it turns out that those wave
00:22:57.000 --> 00:23:04.000
functions are factorable into a
function that is only in r and a
00:23:04.000 --> 00:23:10.000
function that is only in the
angles.
00:23:10.000 --> 00:23:13.000
You can write that,
no approximation,
00:23:13.000 --> 00:23:16.000
this is just the way it turns
out.
00:23:16.000 --> 00:23:22.000
The function that is a function
only of r, R of r,
00:23:22.000 --> 00:23:27.000
is called the radial function.
We will call it capital R,
00:23:27.000 --> 00:23:31.000
radial function of r.
It is labeled by only two
00:23:31.000 --> 00:23:35.000
quantum numbers,
n and l.
00:23:35.000 --> 00:23:40.000
The function that is a function
only of the angles,
00:23:40.000 --> 00:23:43.000
theta and phi,
we are going to call Y.
00:23:43.000 --> 00:23:48.000
This is the angular part of the
wave function.
00:23:48.000 --> 00:23:52.000
And it labeled by only two
quantum numbers,
00:23:52.000 --> 00:23:58.000
but they are l and m.
Sometimes we call this angular
00:23:58.000 --> 00:24:02.000
part, for short,
the Y(lm)'s.
00:24:02.000 --> 00:24:06.000
There is a radial part,
and there is an angular part.
00:24:06.000 --> 00:24:12.000
The actual functional forms are
what I show you here on the side
00:24:12.000 --> 00:24:15.000
screen.
And, in this case,
00:24:15.000 --> 00:24:20.000
what I did is to separate the
radial part from the angular
00:24:20.000 --> 00:24:22.000
part.
This first part,
00:24:22.000 --> 00:24:28.000
here, is the radial part of the
wave function.
00:24:28.000 --> 00:24:32.000
And here on the right is the
angular part of the wave
00:24:32.000 --> 00:24:35.000
function.
And I have written them down
00:24:35.000 --> 00:24:39.000
for the 1s, the 2s,
and the bottom one is the 3s,
00:24:39.000 --> 00:24:44.000
although I left the label off
in order to get the whole wave
00:24:44.000 --> 00:24:48.000
function in there.
I want you to notice,
00:24:48.000 --> 00:24:52.000
here, that the angular part,
the Y(lm) for the s wave
00:24:52.000 --> 00:24:56.000
functions, it has no theta and
phi in it.
00:24:56.000 --> 00:25:02.000
There is no angular dependence.
They are spherically symmetric.
00:25:02.000 --> 00:25:08.000
That is going to be different
for the p wave functions.
00:25:08.000 --> 00:25:13.000
The angular part is just one
over four pi to the one-half
00:25:13.000 --> 00:25:16.000
power. And
00:25:16.000 --> 00:25:21.000
it is the radial part,
here, that we actually are
00:25:21.000 --> 00:25:24.000
going to take a look at right
now.
00:25:24.000 --> 00:25:30.000
Let's start with that 1s wave
function, up there.
00:25:30.000 --> 00:25:33.000
If I plot that wavefunction,
this is Psi(1,
00:25:33.000 --> 00:25:37.000
0, 0), or the 1s wave function
versus r.
00:25:37.000 --> 00:25:42.000
Oh, I should tell you one other
thing that I didn't tell you.
00:25:42.000 --> 00:25:46.000
That is that throughout these
wave functions,
00:25:46.000 --> 00:25:50.000
you see this thing called a
nought.
00:25:50.000 --> 00:25:54.000
a nought is a constant.
It is called the Bohr radius.
00:25:54.000 --> 00:26:00.000
I will explain to you where
that comes from in a little bit
00:26:00.000 --> 00:26:04.000
later.
But it has a value of about
00:26:04.000 --> 00:26:08.000
0.529 angstroms.
That is just a constant.
00:26:08.000 --> 00:26:12.000
Let's plot here the (1,
0, 0) wave function.
00:26:12.000 --> 00:26:17.000
If I went and plotted it,
what I would find is simply
00:26:17.000 --> 00:26:22.000
that the wave function at r is
equal to zero,
00:26:22.000 --> 00:26:27.000
here, would start out at some
high finite value,
00:26:27.000 --> 00:26:33.000
and there would just be an
exponential decay.
00:26:33.000 --> 00:26:36.000
Because if you look here at the
functional form,
00:26:36.000 --> 00:26:40.000
what do you have?
Well, you have all this stuff,
00:26:40.000 --> 00:26:45.000
but that is just a constant.
And the only thing you have is
00:26:45.000 --> 00:26:49.000
an e to the minus r over a
nought
00:26:49.000 --> 00:26:52.000
dependence.
That is what gives you this
00:26:52.000 --> 00:26:57.000
exponential drop in the wave
function.
00:26:57.000 --> 00:27:04.000
What this says is that the wave
function at all values of r has
00:27:04.000 --> 00:27:10.000
a positive value.
Now, what about the Psi(2,
00:27:10.000 --> 00:27:15.000
0, 0) wave function?
Let's look at that.
00:27:15.000 --> 00:27:22.000
Psi(2,0,0), or the 2s wave
function as a function of r.
00:27:22.000 --> 00:27:27.000
What happens here?
Well, we are plotting
00:27:27.000 --> 00:27:32.000
essentially this.
All of this stuff is a
00:27:32.000 --> 00:27:36.000
constant.
And we have a two minus r over
00:27:36.000 --> 00:27:41.000
a nought times an e to the minus
r over 2 a nought.
00:27:41.000 --> 00:27:45.000
That is what we are really
00:27:45.000 --> 00:27:48.000
plotting here.
And, if I did that,
00:27:48.000 --> 00:27:51.000
it would look something like
this.
00:27:51.000 --> 00:27:54.000
We start at some large,
positive value here.
00:27:54.000 --> 00:28:00.000
And you see that the wave
function decreases.
00:28:00.000 --> 00:28:06.000
And it gets to a value of r
where Psi is equal to zero.
00:28:06.000 --> 00:28:12.000
That is a radial node.
And in the case of the 2s wave
00:28:12.000 --> 00:28:18.000
function, that radial node
occurs at r equals 2 a nought.
00:28:18.000 --> 00:28:22.000
And then the wave function
00:28:22.000 --> 00:28:27.000
becomes negative,
increases, and gets more and
00:28:27.000 --> 00:28:32.000
more negative,
until you get to a point where
00:28:32.000 --> 00:28:40.000
it starts increasing again and
then approaches zero.
00:28:40.000 --> 00:28:44.000
This part, here,
of the wave function is really
00:28:44.000 --> 00:28:49.000
dictated by the exponential
term, the e to the minus r over
00:28:49.000 --> 00:28:54.000
2 a nought.
This part of the wave function
00:28:54.000 --> 00:29:00.000
is dictated by this polynomial
here, two minus r over ao.
00:29:00.000 --> 00:29:04.000
If you wanted to solve for that
00:29:04.000 --> 00:29:09.000
radial node, what would you do?
You would take that functional
00:29:09.000 --> 00:29:12.000
form, set it equal to zero and
solve for r.
00:29:12.000 --> 00:29:16.000
And so you can see that 2 minus
r over a nought set equal to
00:29:16.000 --> 00:29:20.000
zero,
that when r is 2 a nought,
00:29:20.000 --> 00:29:24.000
that the wave function would
have a value of zero.
00:29:24.000 --> 00:29:29.000
That is
how you solve for the value of r
00:29:29.000 --> 00:29:34.000
at which you have a node.
Now, this is really important
00:29:34.000 --> 00:29:37.000
here.
That is, at the radial node,
00:29:37.000 --> 00:29:42.000
the wave function changes sign.
The amplitude of the wave
00:29:42.000 --> 00:29:45.000
function goes from positive to
negative.
00:29:45.000 --> 00:29:50.000
That is important because at
all nodes, for all wave
00:29:50.000 --> 00:29:53.000
functions, the wave function
changes sign.
00:29:53.000 --> 00:29:59.000
And the reason the sign of the
wave function is so important is
00:29:59.000 --> 00:30:03.000
in chemical bonding.
But let me back up for a
00:30:03.000 --> 00:30:06.000
moment.
Many of you have talked about p
00:30:06.000 --> 00:30:09.000
orbitals or have seen p orbitals
before.
00:30:09.000 --> 00:30:14.000
Sometimes on a lobe of a p
orbital, you put a plus sign and
00:30:14.000 --> 00:30:16.000
sometimes you put a negative
sign.
00:30:16.000 --> 00:30:18.000
You have seen that,
right?
00:30:18.000 --> 00:30:20.000
Okay.
Well, what that is just
00:30:20.000 --> 00:30:25.000
referring to is the sign of the
amplitude of the wave function.
00:30:25.000 --> 00:30:29.000
It means that in that area the
amplitude is positive,
00:30:29.000 --> 00:30:34.000
and in the other area the
amplitude is negative.
00:30:34.000 --> 00:30:37.000
And the reason the amplitudes
are so important,
00:30:37.000 --> 00:30:42.000
or the sign of the amplitudes
are so important is because in a
00:30:42.000 --> 00:30:46.000
chemical reaction,
when you are bringing two atoms
00:30:46.000 --> 00:30:51.000
together and your electrons that
are represented by waves are
00:30:51.000 --> 00:30:56.000
overlapping, if you are bringing
in two wave functions that have
00:30:56.000 --> 00:30:59.000
the same sign,
well, then you are going to
00:30:59.000 --> 00:31:05.000
have constructive interference.
And you are going to have
00:31:05.000 --> 00:31:09.000
chemical bonding.
If you bring in two atoms,
00:31:09.000 --> 00:31:14.000
and the wave functions are
overlapping and they have
00:31:14.000 --> 00:31:17.000
opposite signs,
you have destructive
00:31:17.000 --> 00:31:22.000
interference and you are not
going to have any chemical
00:31:22.000 --> 00:31:25.000
bonding.
That is why the sign of those
00:31:25.000 --> 00:31:30.000
wave functions is so important.
So, that is Psi(2,
00:31:30.000 --> 00:31:34.000
0, 0).
What about Psi(3,
00:31:34.000 --> 00:31:38.000
0, 0)?
That is the last function,
00:31:38.000 --> 00:31:45.000
here, on the side walls.
And let me just write down the
00:31:45.000 --> 00:31:53.000
radial part, 27 minus 18(r over
a nought) plus 2 times (r over a
00:31:53.000 --> 00:32:01.000
nought) quantity squared times e
to the minus r over 3 a nought.
00:32:09.000 --> 00:32:14.000
And now, if I plotted that
function, Psi(3,0,0),
00:32:14.000 --> 00:32:19.000
3s wave function,
I would find out that r equals
00:32:19.000 --> 00:32:26.000
zero, large value of psi finite,
it drops, it crosses zero,
00:32:26.000 --> 00:32:30.000
gets negative,
then gets less negative,
00:32:30.000 --> 00:32:36.000
crosses zero again,
and then drops off.
00:32:36.000 --> 00:32:40.000
In the case of the 3s wave
function, we have two radial
00:32:40.000 --> 00:32:43.000
nodes.
We have a radial node right in
00:32:43.000 --> 00:32:47.000
here, and we have a radial node
right in there.
00:32:47.000 --> 00:32:51.000
And, if you want to know what
those radial nodes are,
00:32:51.000 --> 00:32:56.000
you set the wave function equal
to zero and solve for the values
00:32:56.000 --> 00:33:01.000
of r that make that wave
function zero.
00:33:01.000 --> 00:33:04.000
And, if you do that,
you would find this would come
00:33:04.000 --> 00:33:08.000
out to be, in terms of units of
a nought,
00:33:08.000 --> 00:33:10.000
1.9 a nought.
And right here,
00:33:10.000 --> 00:33:13.000
it would be 7.1 a nought.
The wave function,
00:33:13.000 --> 00:33:17.000
in the case of the 3s,
has a positive value for r less
00:33:17.000 --> 00:33:20.000
than 1.9 a nought,
00:33:20.000 --> 00:33:25.000
has a negative value from 1.9 a
nought to 7.1 a nought,
00:33:25.000 --> 00:33:29.000
and then a positive value again
00:33:29.000 --> 00:33:33.000
from 7.1 a nought to infinity.
00:33:33.000 --> 00:33:37.000
So, those are the wave
00:33:37.000 --> 00:33:45.000
functions, the functional forms,
what they actually look like.
00:33:45.000 --> 00:33:53.000
Now, it is time to talk about
what the wave function actually
00:33:53.000 --> 00:34:00.000
means, and how does it represent
the electron.
00:34:12.000 --> 00:34:16.000
Well, this was,
of course, a very puzzling
00:34:16.000 --> 00:34:20.000
question to the scientific
community.
00:34:20.000 --> 00:34:25.000
As soon as S wrote down is
Schrˆdinger equation,
00:34:25.000 --> 00:34:30.000
hmm, somehow these waves
represent the particles,
00:34:30.000 --> 00:34:35.000
but exactly how do they
represent where the particles
00:34:35.000 --> 00:34:40.000
are?
And the answer to that question
00:34:40.000 --> 00:34:46.000
is essentially there is no
answer to that question.
00:34:46.000 --> 00:34:50.000
Wave functions are wave
functions.
00:34:50.000 --> 00:34:57.000
It is one of these concepts
that you cannot draw a classical
00:34:57.000 --> 00:35:01.000
analogy to.
You want to say,
00:35:01.000 --> 00:35:05.000
well, a wave function does
this.
00:35:05.000 --> 00:35:10.000
But the only way you can
describe it is in terms of
00:35:10.000 --> 00:35:17.000
language that is something that
you experience everyday in your
00:35:17.000 --> 00:35:22.000
world, so you cannot.
A wave function is a wave
00:35:22.000 --> 00:35:26.000
function.
I cannot draw a correct analogy
00:35:26.000 --> 00:35:32.000
to a classical world.
Really, that is the case.
00:35:32.000 --> 00:35:39.000
However, it took a very smart
gentleman by the name of Max
00:35:39.000 --> 00:35:46.000
Born to look at this problem.
He said, "If I take the wave
00:35:46.000 --> 00:35:52.000
function and I square it,
if I interpret that as a
00:35:52.000 --> 00:35:57.000
probability density,
then I can understand all the
00:35:57.000 --> 00:36:04.000
predictions made by the
Schrˆdinger equation within that
00:36:04.000 --> 00:36:09.000
framework."
In other words,
00:36:09.000 --> 00:36:16.000
he said, let me take Psi and l
and m as a function r,
00:36:16.000 --> 00:36:23.000
theta, and phi and square it.
Let me interpret that as a
00:36:23.000 --> 00:36:27.000
probability density.
00:36:32.000 --> 00:36:36.000
Probability density is not a
probability.
00:36:36.000 --> 00:36:41.000
It is a density.
Density is always per unit
00:36:41.000 --> 00:36:45.000
volume.
Probability density is a
00:36:45.000 --> 00:36:49.000
probability per unit volume.
00:36:54.000 --> 00:36:57.000
It is a probability per unit
volume.
00:36:57.000 --> 00:37:00.000
Well, if I use that
interpretation,
00:37:00.000 --> 00:37:04.000
then I can understand all the
predictions made by the
00:37:04.000 --> 00:37:09.000
Schrˆdinger equation.
It makes sense.
00:37:09.000 --> 00:37:13.000
And, you know what,
that is it.
00:37:13.000 --> 00:37:19.000
Because that interpretation
does agree with our
00:37:19.000 --> 00:37:25.000
observations,
it is therefore believed to be
00:37:25.000 --> 00:37:32.000
correct.
But it is just an assumption.
00:37:32.000 --> 00:37:37.000
It is an interpretation.
There is no derivation for it.
00:37:37.000 --> 00:37:41.000
It is just that the
interpretation works.
00:37:41.000 --> 00:37:45.000
If it works,
we therefore believe it to be
00:37:45.000 --> 00:37:48.000
accurate.
There is no indication,
00:37:48.000 --> 00:37:54.000
there are no data that seem to
contradict that interpretation,
00:37:54.000 --> 00:38:00.000
so we think it is right.
That is what Max Born said.
00:38:00.000 --> 00:38:03.000
Now, Max Born was really
something in terms of his
00:38:03.000 --> 00:38:08.000
scientific accomplishments.
Not only did he recognize or
00:38:08.000 --> 00:38:12.000
have the insight to realize what
Psi squared was,
00:38:12.000 --> 00:38:16.000
but he is also the Born of the
Born-Oppenheimer Approximation
00:38:16.000 --> 00:38:20.000
that maybe some of you have
heard about before.
00:38:20.000 --> 00:38:24.000
He is also the Born in the
Distorted-Wave Born
00:38:24.000 --> 00:38:27.000
Approximation,
which probably none of you have
00:38:27.000 --> 00:38:32.000
heard before.
But, despite all of those
00:38:32.000 --> 00:38:37.000
accomplishments,
psi squared interpretation,
00:38:37.000 --> 00:38:42.000
Born-Oppenheimer Approximation,
Distorted-Wave Born
00:38:42.000 --> 00:38:46.000
Approximation,
he is best known for being the
00:38:46.000 --> 00:38:50.000
grandfather of Olivia
Newton-John.
00:38:50.000 --> 00:38:54.000
That's right.
Oliver Newton-John is a singer
00:38:54.000 --> 00:38:59.000
in Grease.
Two weeks ago in the Boston
00:38:59.000 --> 00:39:04.000
Globe Parade Magazine,
which I actually think is a
00:39:04.000 --> 00:39:09.000
magazine that goes throughout
the country in all the Sunday
00:39:09.000 --> 00:39:15.000
newspapers, there is a long
article on Olivia Newton-John
00:39:15.000 --> 00:39:19.000
and a short sentence about her
grandfather, Max Born.
00:39:19.000 --> 00:39:24.000
So, that is our interpretation,
thanks to Max Born.
00:39:24.000 --> 00:39:29.000
Now, how are we going to use
that?
00:39:29.000 --> 00:39:35.000
Well, first of all,
let's take our functional forms
00:39:35.000 --> 00:39:40.000
for Psi, here,
and square it and plot those
00:39:40.000 --> 00:39:47.000
probability densities for the
individual wave functions and
00:39:47.000 --> 00:39:50.000
see what we get.
00:39:55.000 --> 00:40:00.000
The way I am going to plot the
probability density is by using
00:40:00.000 --> 00:40:04.000
some dots.
And the density of the dots is
00:40:04.000 --> 00:40:08.000
going to reflect the probability
density.
00:40:08.000 --> 00:40:13.000
The more dense the dots,
the larger the probability
00:40:13.000 --> 00:40:16.000
density.
If I take that functional form
00:40:16.000 --> 00:40:21.000
for the 1s wave function and
square it and then plot the
00:40:21.000 --> 00:40:27.000
value of that function squared
with this density dot diagram,
00:40:27.000 --> 00:40:33.000
then you can see that the dots
here are most dense right at the
00:40:33.000 --> 00:40:37.000
origin, and that they
exponentially decay in all
00:40:37.000 --> 00:40:41.000
directions.
The probability density here
00:40:41.000 --> 00:40:45.000
for 1s wave function is greatest
at the origin,
00:40:45.000 --> 00:40:48.000
r equals 0, and it decays
exponentially in all directions.
00:40:48.000 --> 00:40:52.000
It is spherically symmetric.
That is what you would expect
00:40:52.000 --> 00:40:56.000
because that is what the wave
function looks like.
00:40:56.000 --> 00:40:59.000
You square that,
you get another exponential,
00:40:59.000 --> 00:41:03.000
and it decays exponentially in
all directions.
00:41:03.000 --> 00:41:08.000
That is a probability density,
probability of finding the
00:41:08.000 --> 00:41:12.000
electron per unit volume at some
value r, theta,
00:41:12.000 --> 00:41:15.000
and phi.
And it turns out it doesn't
00:41:15.000 --> 00:41:20.000
matter what theta and phi are
because this is spherically
00:41:20.000 --> 00:41:23.000
symmetric.
What about the 2s wave
00:41:23.000 --> 00:41:26.000
function?
Well, here is the 2s
00:41:26.000 --> 00:41:30.000
probability density.
Again, you can see the
00:41:30.000 --> 00:41:34.000
probability density is a maximum
at the origin,
00:41:34.000 --> 00:41:37.000
at the nucleus.
That probability density decays
00:41:37.000 --> 00:41:41.000
uniformly in all directions.
And it decays so much that at
00:41:41.000 --> 00:41:45.000
some point, you have no
probability density.
00:41:45.000 --> 00:41:47.000
Why?
Because that is the node.
00:41:47.000 --> 00:41:50.000
If you square zero,
you still get zero.
00:41:50.000 --> 00:41:52.000
r equals 2 a nought.
00:41:52.000 --> 00:41:57.000
You can see that in the
probability density.
00:41:57.000 --> 00:42:00.000
But then again,
as you move up this way,
00:42:00.000 --> 00:42:03.000
as you increase r,
the probability density
00:42:03.000 --> 00:42:04.000
increases again.
Why?
00:42:04.000 --> 00:42:08.000
Remember the wave function?
It has changed sign.
00:42:08.000 --> 00:42:11.000
But in this area,
here, where it is negative,
00:42:11.000 --> 00:42:15.000
if I square it,
well, the probability density
00:42:15.000 --> 00:42:19.000
still is going to be large.
Square a negative number,
00:42:19.000 --> 00:42:22.000
you are going to have a large
positive number.
00:42:22.000 --> 00:42:27.000
That is why the probability
density increases right in here,
00:42:27.000 --> 00:42:32.000
and then, again,
it decays towards zero.
00:42:32.000 --> 00:42:37.000
You can see the radial node not
only in the wave function,
00:42:37.000 --> 00:42:41.000
but also in the probability
density.
00:42:41.000 --> 00:42:47.000
Here is the probability density
for the 3s wave function.
00:42:47.000 --> 00:42:52.000
Once again, probability density
is a maximum at r equals 0,
00:42:52.000 --> 00:42:58.000
and it decays uniformly in all
directions.
00:42:58.000 --> 00:43:01.000
It decays so much that at some
value of r, right here,
00:43:01.000 --> 00:43:03.000
the probability density is
zero.
00:43:03.000 --> 00:43:05.000
Why?
Because the wave function is
00:43:05.000 --> 00:43:07.000
zero.
You square it,
00:43:07.000 --> 00:43:11.000
and you are going to get a zero
for the probability density.
00:43:11.000 --> 00:43:14.000
And then the probability
density increases again.
00:43:14.000 --> 00:43:16.000
Why?
Because you are getting a more
00:43:16.000 --> 00:43:20.000
and more negative value for the
wave function right in this
00:43:20.000 --> 00:43:21.000
area.
Square that,
00:43:21.000 --> 00:43:25.000
and it is going to increase.
And then, as you continue to
00:43:25.000 --> 00:43:30.000
increase r, probability density
decreases.
00:43:30.000 --> 00:43:33.000
It decreases again,
so that you get a zero.
00:43:33.000 --> 00:43:37.000
You get a zero because the wave
function is zero right there.
00:43:37.000 --> 00:43:41.000
This is our second radial node.
But then, the probability
00:43:41.000 --> 00:43:45.000
density increases again.
It increases because the wave
00:43:45.000 --> 00:43:48.000
function increases.
Square that,
00:43:48.000 --> 00:43:51.000
we are going to get a high
probability density,
00:43:51.000 --> 00:43:55.000
and then it tapers off.
So, the important point here is
00:43:55.000 --> 00:44:00.000
the interpretation of the
probability density.
00:44:00.000 --> 00:44:05.000
Probability per unit volume.
The fact that the s wave
00:44:05.000 --> 00:44:08.000
functions are all spherically
symmetric.
00:44:08.000 --> 00:44:13.000
They do not have an angular
dependence to them.
00:44:13.000 --> 00:44:18.000
And what a radial node is.
If you want to get a radial
00:44:18.000 --> 00:44:23.000
node, you take the wave
function, set it equal to zero,
00:44:23.000 --> 00:44:30.000
solve for the value of r,
and that gives you a zero.
00:44:30.000 --> 00:44:34.000
Now, so far,
we have talked only about the
00:44:34.000 --> 00:44:38.000
probability density and this
interpretation.
00:44:38.000 --> 00:44:43.000
We have not talked about any
probabilities yet.
00:44:43.000 --> 00:44:48.000
And, to do so,
we are going to talk about this
00:44:48.000 --> 00:44:52.000
function, here.
It is called a radial
00:44:52.000 --> 00:44:58.000
probability distribution.
The radial probability
00:44:58.000 --> 00:45:04.000
distribution is the probability
of finding an electron in a
00:45:04.000 --> 00:45:08.000
spherical shell.
That spherical shell will be
00:45:08.000 --> 00:45:12.000
some distance r away from the
nucleus.
00:45:12.000 --> 00:45:18.000
That spherical shell will have
a radius r and will have a
00:45:18.000 --> 00:45:20.000
thickness.
And the thickness,
00:45:20.000 --> 00:45:26.000
we are going to call dr.
This is not a solid sphere.
00:45:26.000 --> 00:45:30.000
This is a shell.
This is a sphere,
00:45:30.000 --> 00:45:35.000
but the thickness of that
sphere is very small.
00:45:35.000 --> 00:45:41.000
The thickness of it is dr.
And, to try to represent that a
00:45:41.000 --> 00:45:45.000
little bit better,
I show you here a picture of
00:45:45.000 --> 00:45:50.000
the probability density for the
(1, 0, 0) state.
00:45:50.000 --> 00:45:54.000
This is kind of my density dot
diagram.
00:45:54.000 --> 00:46:00.000
And then, this blue thing is my
spherical shell.
00:46:00.000 --> 00:46:04.000
This blue thing,
here, has a radius r,
00:46:04.000 --> 00:46:08.000
and this blue thing has a
thickness dr.
00:46:08.000 --> 00:46:14.000
And so, I am saying that the
radial probability distribution
00:46:14.000 --> 00:46:21.000
is going to be the probability
of finding the electron in this
00:46:21.000 --> 00:46:26.000
spherical shell.
That spherical shell is a
00:46:26.000 --> 00:46:33.000
distance r from the nucleus and
has a thickness dr.
00:46:33.000 --> 00:46:37.000
Now, I want to point out that
the volume of the spherical
00:46:37.000 --> 00:46:41.000
shell is just the surface area,
here, 4 pi r squared,
00:46:41.000 --> 00:46:44.000
times the thickness,
which is dr.
00:46:44.000 --> 00:46:49.000
Not a very thick spherical
00:46:49.000 --> 00:46:51.000
shell.
The radial probability
00:46:51.000 --> 00:46:56.000
distribution is the probability
of finding that electron in that
00:46:56.000 --> 00:47:02.000
spherical shell.
It is like the probability of
00:47:02.000 --> 00:47:10.000
finding the electron a distance
r to r plus dr
00:47:10.000 --> 00:47:14.000
from the nucleus.
Why is that important?
00:47:14.000 --> 00:47:21.000
Well, because if I want to
calculate a probability,
00:47:21.000 --> 00:47:27.000
what I can do then is take the
probability density here,
00:47:27.000 --> 00:47:34.000
Psi squared for an s orbital,
which is probability per unit
00:47:34.000 --> 00:47:43.000
volume, and I can then multiply
it by that unit volume.
00:47:43.000 --> 00:47:48.000
In this case it was the 4pi r
squared dr.
00:47:48.000 --> 00:47:53.000
And the result will be a
probability, because I have
00:47:53.000 --> 00:47:57.000
probability density,
probability per unit volume
00:47:57.000 --> 00:48:02.000
times a volume,
and that is a probability.
00:48:02.000 --> 00:48:09.000
Now we are getting somewhere in
terms of figuring out what the
00:48:09.000 --> 00:48:15.000
probability is of finding the
electron some distance r to r
00:48:15.000 --> 00:48:22.000
plus dr from the nucleus.
In the case of the s orbitals,
00:48:22.000 --> 00:48:28.000
I said that the Psi was a
product of the radial part and
00:48:28.000 --> 00:48:34.000
the Y(lm) angular part.
Remember that the Y(lm) for the
00:48:34.000 --> 00:48:40.000
s orbitals was always one over
the square-root of one over 4pi.
00:48:40.000 --> 00:48:46.000
The Y(lm) squared is going to
00:48:46.000 --> 00:48:50.000
cancel with 4pi,
and you are just going to have
00:48:50.000 --> 00:48:54.000
r squared times the radial part
squared.
00:48:54.000 --> 00:48:57.000
For a 1s orbital,
if you want to actually
00:48:57.000 --> 00:49:01.000
calculate the probability at
some value r,
00:49:01.000 --> 00:49:07.000
you just have to take Psi
squared and multiply it by 4pi r
00:49:07.000 --> 00:49:12.000
squared dr.
00:49:12.000 --> 00:49:16.000
However, in the case of all
other orbitals,
00:49:16.000 --> 00:49:22.000
you cannot do that because they
are not spherically symmetric.
00:49:22.000 --> 00:49:28.000
And so, for all other orbitals,
you have to take the radial
00:49:28.000 --> 00:49:34.000
part and multiply it by r
squared dr.
00:49:34.000 --> 00:49:38.000
I will explain that a little
bit more next time.
00:49:38.000 --> 00:49:41.000
Okay.
See you on Monday.