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That is the largest value it

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And your equation is equal to E
Psi.

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And we saw that these energies,
the binding energies of the

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electron to the nucleus,
were quantized.

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Well, these energy levels then
were given by minus this Rydberg

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constant R sub H over n squared.
And this n

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was our quantum number.
It is a principle quantum

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number.
And we saw that n,

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the smallest value was 1,
2 and ran all the way up to

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infinity.
Well, today what we are going

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to do is solve this equation,
or we are going to look at the

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results of solving this equation
for the wavefunction Psi.

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Now, Psi, in general,
is the function of r,

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theta, phi, and also time.
But we are going to be looking

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at problems in which time does
not have an effect.

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In other words,
the wave functions that we are

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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

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a chemical reaction is
happening.

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We are either going to look at
it before or after,

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but not during.
And, in those cases,

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we are looking at a wave
function.

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And the atom is just stable and
is sitting there.

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The time dependence here does
not have a consequence.

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And so, therefore,
the wave functions that we are

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going to be looking at are just
a function of r,

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theta and phi.
And we are looking at what is

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called time-independent quantum
mechanics.

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Later on, actually,
if you are in chemistry,

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in a graduate course in
chemistry, is when you look at

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time-dependent wave functions.
We are going to look at

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time-independent quantum
mechanics, the stationary wave.

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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

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quantum numbers drop out of that
solution to the differential

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equation.
Remember, we said last time how

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quantum numbers arise.
They arise from imposing

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boundary conditions on a
differential equation,

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making that differential
equation applicable to your

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actual physical problem.
And so, when we do that,

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we get a new quantum number
called l.

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And l is, I think you already
know, the angular momentum

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quantum number.
Absolutely.

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It is called the angular
momentum quantum number because

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it indeed dictates how much
angular momentum the electron

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has.
It has allowed values.

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The allowed values of l,
now, are zero.

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Zero is the smallest value of
l, the lowest value of l

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allowed.
1, 2, all the way up to n minus

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can have.
It cannot be any larger than n

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minus 1.
Why can't it be larger than n

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minus 1?
Well, it cannot because the

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angular momentum quantum number,
at least if you want to think

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classically for a moment,
dictates how much rotational

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kinetic energy you have.
And remember that this energy,

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here, is dependent only on n.
This energy is the sum of

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kinetic energy plus potential
energy.

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If l had the same value as n,
well, essentially,

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that would mean that we would
have only rotational kinetic

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energy and we would have no
potential energy.

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But that is not right.
We have potential energy.

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We have potential energy of
interaction.

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So, physically,
that is why l is tied to n.

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And it cannot be larger than n
minus 1.

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And then, we have a third
quantum number that drops out of

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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,

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m is the z-component of the
angular momentum.

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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

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momentum in the z-component.
Or, plus one,

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plus two, plus three,
all the way up to plus l.

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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

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in the z-component than you had
total angular momentum.

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And that is a no-no.
So, m is tied to l.

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The largest value you can have
is l.

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But, since this is a
z-component and we've got some

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direction, m could also be minus
1, minus two,

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minus three,
minus l.

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We have three quantum numbers,
now.

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That kind of makes sense
because we have a 3-dimensional

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problem.
We are going to have to have

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three quantum numbers to
completely describe our system.

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The consequence,
here, of having three quantum

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numbers is that we now have more
states.

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For example,
our n equals 1 state that we

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talked about last time,
more completely we have to

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describe that state by two other
quantum numbers.

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When n is equal to one,
what is the only value that l

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can have?
Zero.

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And if l is zero,
what is the only value m can

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have?
Zero.

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And so, more appropriately,
or more completely,

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that n equals 1 state is the
(1, 0, 0) state.

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And, if we have an electron in
that (1, 0, 0) state,

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we are going to describe that
electron by the wave function

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Psi(1, 0, 0).
Now, what I have not told you

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yet is exactly how Psi
represents the electron.

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I am just telling you right now
that Psi does.

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Exactly how it does is
something I haven't told you

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yet.
And we will do that later in

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the lecture today,
sort of.

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That is the energy,
minus the Rydberg constant.

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But now, if n is equal two,
what is the smallest value that

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l can have?
Zero.

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And the value of m in that
case?

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Zero.
And so our n equals 2 state is

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more completely the (2,
0, 0) state.

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And the wave function that
describes the electron in that

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state is the Psi(2,
0, 0) wave function.

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And then, of course,
here is the energy of that

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state.
Now, when n is equal to 2,

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what is the next larger value
of l?

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00:10:03,000 --> 00:10:05,000
One.
And if l is equal to one,

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what is the largest value that
m can be?

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One.
And so we have another state

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00:10:12,000 --> 00:10:14,000
here, the (2,
1, 1) state.

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And if you have an electron in
that state, it is described by

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00:10:19,000 --> 00:10:22,000
the Psi(2, 1,
1) wave function.

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00:10:22,000 --> 00:10:27,000
However, the energy here is
also minus one-quarter the

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00:10:27,000 --> 00:10:32,000
Rydberg constant.
It has the same energy as the

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00:10:32,000 --> 00:10:36,000
(2, 0, 0) state.
Now, if n is equal two and l is

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equal to one,
what is the next larger value

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00:10:40,000 --> 00:10:40,000
of m?
Zero.
And so we have a (2,
1, 0) state.

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00:10:43,000 --> 00:10:48,000
Again, that wave function,
for an electron in that state,

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00:10:48,000 --> 00:10:50,000
we label as Psi(2,
1, 0).

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00:10:50,000 --> 00:10:53,000
And then, finally,
when n is equal to two,

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00:10:53,000 --> 00:10:57,000
l is equal to one,
what is the final possible

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00:10:57,000 --> 00:11:01,000
value of m?
Minus one.

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00:11:01,000 --> 00:11:04,000
We have a (2,
1, -1) state and a wave

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00:11:04,000 --> 00:11:09,000
function that is the (2,
1, -1) wave function.

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00:11:09,000 --> 00:11:15,000
Notice that the energy of all
of these four states is the

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00:11:15,000 --> 00:11:20,000
same, minus one-quarter the
Rydberg constant.

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00:11:20,000 --> 00:11:24,000
These states are what we call
degenerate.

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Degenerate means having the
same energy.

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00:11:30,000 --> 00:11:34,000
That is important.
Now, this is the way we label

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wave functions,
but we also have a different

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00:11:38,000 --> 00:11:42,000
scheme for talking about wave
functions.

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00:11:42,000 --> 00:11:46,000
That is, we have an orbital
scheme.

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00:11:46,000 --> 00:11:50,000
And, as I said,
or alluded to the other day,

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00:11:50,000 --> 00:11:55,000
an orbital is nothing other
than a wave function.

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It is a solution to the
Schrˆdinger equation.

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00:12:01,000 --> 00:12:04,000
That is what an orbital is.
It is actually the spatial part

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00:12:04,000 --> 00:12:08,000
of the wave function.
There is another part called

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00:12:08,000 --> 00:12:11,000
the spin part,
which we will deal with later,

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but an orbital is essentially a
wave function.

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We have a different language
for describing orbitals.

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The way we do this,
I think you are already

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familiar with,
but we are going to describe it

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by the principle quantum number
and, of course,

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00:12:26,000 --> 00:12:30,000
the angular momentum quantum
number and the magnetic quantum

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00:12:30,000 --> 00:12:35,000
number.
Except that we have a scheme or

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00:12:35,000 --> 00:12:40,000
a code for l and m.
That code uses letters instead

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of numbers.
For example,

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if l is equal to zero here,
we call that an s wave function

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or an s orbital.
For example,

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00:12:51,000 --> 00:12:55,000
in the orbital language,
instead of Psi(1,

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00:12:55,000 --> 00:13:00,000
0, 0), we call this a 1s
orbital.

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00:13:00,000 --> 00:13:02,000
Here is the principle quantum
number.

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00:13:02,000 --> 00:13:05,000
Here is s.
l is equal to zero.

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00:13:05,000 --> 00:13:08,000
That is just our code for l is
equal to zero.

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00:13:08,000 --> 00:13:12,000
This state, (2,
0, 0), you have an electron in

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00:13:12,000 --> 00:13:15,000
that state.
Well, it can describe that

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00:13:15,000 --> 00:13:19,000
wavefunction by the 2s orbital.
Here is the principle quantum

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00:13:19,000 --> 00:13:22,000
number.
He is s, our code for l equals

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00:13:22,000 --> 00:13:25,000
zero.
Now, when n is equal to 1,

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00:13:25,000 --> 00:13:30,000
we call that a p wave function
or a p orbital.

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00:13:30,000 --> 00:13:33,000
This state right here,
it is the 2p state.

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00:13:33,000 --> 00:13:39,000
Because n is two and l is one,
this state is also the 2p wave

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00:13:39,000 --> 00:13:44,000
function and this state,
it is also the 2p wave

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00:13:44,000 --> 00:13:47,000
function.
Now, if I had a wave function

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00:13:47,000 --> 00:13:52,000
up here where l was equal to
two, we would call that d

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00:13:52,000 --> 00:13:56,000
orbital.
And, if I had a wave function

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00:13:56,000 --> 00:14:00,000
up here where l was three,
we would call that an f

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00:14:00,000 --> 00:14:05,000
orbital.
But now, all of these wave

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00:14:05,000 --> 00:14:09,000
functions in the orbital
language have the same

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00:14:09,000 --> 00:14:15,000
designation, and that is because
we have not taken care of this

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00:14:15,000 --> 00:14:19,000
yet, the m quantum number,
magnetic quantum number.

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00:14:19,000 --> 00:14:24,000
And we also have an alphabet
scheme for those quantum

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00:14:24,000 --> 00:14:27,000
numbers.
The bottom line is that when m

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00:14:27,000 --> 00:14:33,000
is equal to zero,
we put a z subscript on the p.

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00:14:33,000 --> 00:14:39,000
When m is equal to zero,
that is our 2pz wave function.

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00:14:39,000 --> 00:14:44,000
When m is equal to one,
we are going to put an x

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00:14:44,000 --> 00:14:47,000
subscript on the p wave
function.

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00:14:47,000 --> 00:14:54,000
And when m is equal to minus
one, we are going to put a y

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00:14:54,000 --> 00:14:59,000
subscription on the wave
function.

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00:14:59,000 --> 00:15:04,000
However, I have to tell you
that in the case of m is equal

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00:15:04,000 --> 00:15:08,000
to one and m is equal to minus
one, that is not strictly

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00:15:08,000 --> 00:15:12,000
correct.
And the reason it is not is

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00:15:12,000 --> 00:15:16,000
because when you solve
Schrˆdinger equation for the (2,

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00:15:16,000 --> 00:15:21,000
1, 1) wave function and the (2,
1, -1) wave function,

200
00:15:21,000 --> 00:15:25,000
the solutions are complex wave
functions.

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00:15:25,000 --> 00:15:29,000
They are not real wave
function.

202
00:15:29,000 --> 00:15:35,000
And so, in order for us to be
able to draw and think about the

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00:15:35,000 --> 00:15:42,000
px and the py wave functions,
what we do is we take a linear

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00:15:42,000 --> 00:15:47,000
combination of the px and the py
wave functions.

205
00:15:47,000 --> 00:15:54,000
We take a positive linear
combination, in the case of the

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00:15:54,000 --> 00:15:59,000
px wave function.
The px wave function is really

207
00:15:59,000 --> 00:16:03,000
this wave function plus this
wave function.

208
00:16:03,000 --> 00:16:08,000
And the py wave function is
really this wave function minus

209
00:16:08,000 --> 00:16:12,000
this wave function.
Then we will get a real

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00:16:12,000 --> 00:16:15,000
function, and that is easier to
deal with.

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00:16:15,000 --> 00:16:18,000
That is strictly what px and py
are.

212
00:16:18,000 --> 00:16:23,000
px and py are these linear
combinations of the actual

213
00:16:23,000 --> 00:16:29,000
functions that come out of the
Schrˆdinger equation.

214
00:16:29,000 --> 00:16:32,000
pz is exactly correct.
pz, M is equal to zero,

215
00:16:32,000 --> 00:16:36,000
in the pz wavefunction.
Now, you are not responsible

216
00:16:36,000 --> 00:16:40,000
for knowing that.
I just wanted to let you know.

217
00:16:40,000 --> 00:16:45,000
You don't have to remember that
m equal to one gives you the x

218
00:16:45,000 --> 00:16:50,000
subscript, and m equal to minus
one gives you the y subscript.

219
00:16:50,000 --> 00:16:54,000
You do have to know that when m
is equal to zero,

220
00:16:54,000 --> 00:17:00,000
you get a z subscript,
because that is exactly right.

221
00:17:00,000 --> 00:17:03,000
Now, just one other comment
here.

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00:17:03,000 --> 00:17:08,000
That is, you see we did not put
a subscript on the s wave

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00:17:08,000 --> 00:17:13,000
functions.
Well, that is because for an s

224
00:17:13,000 --> 00:17:17,000
wave function,
the only choice you have for m

225
00:17:17,000 --> 00:17:20,000
is zero.
And so we leave out that

226
00:17:20,000 --> 00:17:24,000
subscript.
We never put a z on there or a

227
00:17:24,000 --> 00:17:27,000
z on there.
It is always 1s,

228
00:17:27,000 --> 00:17:32,000
2s, or 3s.
Well, in order to understand

229
00:17:32,000 --> 00:17:37,000
that just a little more,
let's draw an energy level

230
00:17:37,000 --> 00:17:41,000
diagram.
Here is the energy again.

231
00:17:41,000 --> 00:17:45,000
And for n equals 1,
we now see that the more

232
00:17:45,000 --> 00:17:50,000
complete description is three
quantum numbers,

233
00:17:50,000 --> 00:17:53,000
(1, 0, 0).
That gives us the (1,

234
00:17:53,000 --> 00:17:57,000
0, 0) state,
or sometimes we say the 1s

235
00:17:57,000 --> 00:18:00,000
state.
There is one state at that

236
00:18:00,000 --> 00:18:05,000
energy.
But, in the case of n equals 2,

237
00:18:05,000 --> 00:18:11,000
we already saw that we have
four different states for the

238
00:18:11,000 --> 00:18:14,000
quantum number.
Four different states.

239
00:18:14,000 --> 00:18:19,000
They all have the same energy.
They are degenerate.

240
00:18:19,000 --> 00:18:23,000
Degenerate means having the
same energy.

241
00:18:23,000 --> 00:18:30,000
They differ in how much angular
momentum the electron has.

242
00:18:30,000 --> 00:18:35,000
Or, how much angular momentum
the electron would have in the z

243
00:18:35,000 --> 00:18:38,000
component if it were in a
magnetic field.

244
00:18:38,000 --> 00:18:43,000
The total energy is the same
because only n determines the

245
00:18:43,000 --> 00:18:46,000
energy.
It is just that the amount of

246
00:18:46,000 --> 00:18:50,000
angular momentum,
or the z component of the

247
00:18:50,000 --> 00:18:52,000
angular momentum,
differs in 2s,

248
00:18:52,000 --> 00:18:56,000
2py, 2pz, 2px,
but they all have the same

249
00:18:56,000 --> 00:19:01,000
energy.
In general, for any value of n,

250
00:19:01,000 --> 00:19:08,000
there are n squared degenerate
states at each value of n.

251
00:19:08,000 --> 00:19:14,000
If we have n is equal to three,
then the energy is minus

252
00:19:14,000 --> 00:19:21,000
one-ninth the Rydberg constant.
But how many states do we have

253
00:19:21,000 --> 00:19:24,000
at n equal three?
Nine.

254
00:19:24,000 --> 00:19:28,000
Here they are.
Just like for n equal two,

255
00:19:28,000 --> 00:19:34,000
we have the 3s,
3py, 3pz, and 3px.

256
00:19:34,000 --> 00:19:38,000
And there are the associated
quantum numbers for that.

257
00:19:38,000 --> 00:19:44,000
But now we have some states
here that I have labeled 3d

258
00:19:44,000 --> 00:19:47,000
states.
Well, when you have a d state,

259
00:19:47,000 --> 00:19:52,000
that means l is equal to two.
All of these states have

260
00:19:52,000 --> 00:19:58,000
principle quantum number three
and angular momentum quantum

261
00:19:58,000 --> 00:20:03,000
number two.
And they differ also by the

262
00:20:03,000 --> 00:20:08,000
amount of angular momentum in
the z direction.

263
00:20:08,000 --> 00:20:12,000
They differ in the quantum
number m.

264
00:20:12,000 --> 00:20:16,000
For example,
we are going to call the m

265
00:20:16,000 --> 00:20:20,000
equal minus two state the
3d(xy).

266
00:20:20,000 --> 00:20:24,000
We are going to put xy as a
subscript.

267
00:20:24,000 --> 00:20:30,000
For m equal minus one,
we are going to put a subscript

268
00:20:30,000 --> 00:20:33,000
yz.
For m equals zero,

269
00:20:33,000 --> 00:20:38,000
we are going to put a subscript
z squared.

270
00:20:38,000 --> 00:20:40,000
For m equal one,
d(xz).

271
00:20:40,000 --> 00:20:44,000
And for m equal two,
x squared minus y squared.

272
00:20:44,000 --> 00:20:49,000
Again, for m equal minus two,

273
00:20:49,000 --> 00:20:54,000
minus one, one and two,
those wave functions,

274
00:20:54,000 --> 00:20:59,000
when you solve Schrˆdinger
equations, are complex wave

275
00:20:59,000 --> 00:21:04,000
functions.
And what we do to talk about

276
00:21:04,000 --> 00:21:08,000
the wave functions is we take
linear combinations of them to

277
00:21:08,000 --> 00:21:12,000
make them real.
And so, when I say m is minus

278
00:21:12,000 --> 00:21:16,000
two, is the 3dxy wave function,
it is not strictly correct.

279
00:21:16,000 --> 00:21:20,000
Therefore, again,
you don't need to know m minus

280
00:21:20,000 --> 00:21:23,000
two.
You don't need to know that

281
00:21:23,000 --> 00:21:24,000
subscript.
Or m equal one,

282
00:21:24,000 --> 00:21:29,000
you don't need to know that
subscript.

283
00:21:29,000 --> 00:21:32,000
But for m equal zero,
this is a z squared.

284
00:21:32,000 --> 00:21:36,000
Absolutely.
These wave functions are linear

285
00:21:36,000 --> 00:21:38,000
combinations.
This one is not.

286
00:21:38,000 --> 00:21:43,000
It is a real function when it
comes out of the Schrˆdinger

287
00:21:43,000 --> 00:21:47,000
equation.
You will talk about these 3d

288
00:21:47,000 --> 00:21:51,000
wave functions a lot with
Professor Cummins in the

289
00:21:51,000 --> 00:21:56,000
second-half of the course.
You will actually look at the

290
00:21:56,000 --> 00:22:01,000
shapes of those wave functions
in detail.

291
00:22:01,000 --> 00:22:04,000
So, that's the energy level
diagram, here.

292
00:22:04,000 --> 00:22:08,000
All right.
Now let's actually talk about

293
00:22:08,000 --> 00:22:11,000
the shapes of the wave
functions.

294
00:22:11,000 --> 00:22:16,000
I am going to raise this
screen, I think.

295
00:22:25,000 --> 00:22:40,000
What do these wave functions
actually look like?

296
00:22:40,000 --> 00:22:45,000
Well, for a hydrogen atom,
our wave function here,

297
00:22:45,000 --> 00:22:52,000
given by three quantum numbers,
n, l and m, function of r,

298
00:22:52,000 --> 00:22:57,000
theta and phi,
it turns out that those wave

299
00:22:57,000 --> 00:23:04,000
functions are factorable into a
function that is only in r and a

300
00:23:04,000 --> 00:23:10,000
function that is only in the
angles.

301
00:23:10,000 --> 00:23:13,000
You can write that,
no approximation,

302
00:23:13,000 --> 00:23:16,000
this is just the way it turns
out.

303
00:23:16,000 --> 00:23:22,000
The function that is a function
only of r, R of r,

304
00:23:22,000 --> 00:23:27,000
is called the radial function.
We will call it capital R,

305
00:23:27,000 --> 00:23:31,000
radial function of r.
It is labeled by only two

306
00:23:31,000 --> 00:23:35,000
quantum numbers,
n and l.

307
00:23:35,000 --> 00:23:40,000
The function that is a function
only of the angles,

308
00:23:40,000 --> 00:23:43,000
theta and phi,
we are going to call Y.

309
00:23:43,000 --> 00:23:48,000
This is the angular part of the
wave function.

310
00:23:48,000 --> 00:23:52,000
And it labeled by only two
quantum numbers,

311
00:23:52,000 --> 00:23:58,000
but they are l and m.
Sometimes we call this angular

312
00:23:58,000 --> 00:24:02,000
part, for short,
the Y(lm)'s.

313
00:24:02,000 --> 00:24:06,000
There is a radial part,
and there is an angular part.

314
00:24:06,000 --> 00:24:12,000
The actual functional forms are
what I show you here on the side

315
00:24:12,000 --> 00:24:15,000
screen.
And, in this case,

316
00:24:15,000 --> 00:24:20,000
what I did is to separate the
radial part from the angular

317
00:24:20,000 --> 00:24:22,000
part.
This first part,

318
00:24:22,000 --> 00:24:28,000
here, is the radial part of the
wave function.

319
00:24:28,000 --> 00:24:32,000
And here on the right is the
angular part of the wave

320
00:24:32,000 --> 00:24:35,000
function.
And I have written them down

321
00:24:35,000 --> 00:24:39,000
for the 1s, the 2s,
and the bottom one is the 3s,

322
00:24:39,000 --> 00:24:44,000
although I left the label off
in order to get the whole wave

323
00:24:44,000 --> 00:24:48,000
function in there.
I want you to notice,

324
00:24:48,000 --> 00:24:52,000
here, that the angular part,
the Y(lm) for the s wave

325
00:24:52,000 --> 00:24:56,000
functions, it has no theta and
phi in it.

326
00:24:56,000 --> 00:25:02,000
There is no angular dependence.
They are spherically symmetric.

327
00:25:02,000 --> 00:25:08,000
That is going to be different
for the p wave functions.

328
00:25:08,000 --> 00:25:13,000
The angular part is just one
over four pi to the one-half

329
00:25:13,000 --> 00:25:16,000
power. And

330
00:25:16,000 --> 00:25:21,000
it is the radial part,
here, that we actually are

331
00:25:21,000 --> 00:25:24,000
going to take a look at right
now.

332
00:25:24,000 --> 00:25:30,000
Let's start with that 1s wave
function, up there.

333
00:25:30,000 --> 00:25:33,000
If I plot that wavefunction,
this is Psi(1,

334
00:25:33,000 --> 00:25:37,000
0, 0), or the 1s wave function
versus r.

335
00:25:37,000 --> 00:25:42,000
Oh, I should tell you one other
thing that I didn't tell you.

336
00:25:42,000 --> 00:25:46,000
That is that throughout these
wave functions,

337
00:25:46,000 --> 00:25:50,000
you see this thing called a
nought.

338
00:25:50,000 --> 00:25:54,000
a nought is a constant.
It is called the Bohr radius.

339
00:25:54,000 --> 00:26:00,000
I will explain to you where
that comes from in a little bit

340
00:26:00,000 --> 00:26:04,000
later.
But it has a value of about

341
00:26:04,000 --> 00:26:08,000
0.529 angstroms.
That is just a constant.

342
00:26:08,000 --> 00:26:12,000
Let's plot here the (1,
0, 0) wave function.

343
00:26:12,000 --> 00:26:17,000
If I went and plotted it,
what I would find is simply

344
00:26:17,000 --> 00:26:22,000
that the wave function at r is
equal to zero,

345
00:26:22,000 --> 00:26:27,000
here, would start out at some
high finite value,

346
00:26:27,000 --> 00:26:33,000
and there would just be an
exponential decay.

347
00:26:33,000 --> 00:26:36,000
Because if you look here at the
functional form,

348
00:26:36,000 --> 00:26:40,000
what do you have?
Well, you have all this stuff,

349
00:26:40,000 --> 00:26:45,000
but that is just a constant.
And the only thing you have is

350
00:26:45,000 --> 00:26:49,000
an e to the minus r over a
nought

351
00:26:49,000 --> 00:26:52,000
dependence.
That is what gives you this

352
00:26:52,000 --> 00:26:57,000
exponential drop in the wave
function.

353
00:26:57,000 --> 00:27:04,000
What this says is that the wave
function at all values of r has

354
00:27:04,000 --> 00:27:10,000
a positive value.
Now, what about the Psi(2,

355
00:27:10,000 --> 00:27:15,000
0, 0) wave function?
Let's look at that.

356
00:27:15,000 --> 00:27:22,000
Psi(2,0,0), or the 2s wave
function as a function of r.

357
00:27:22,000 --> 00:27:27,000
What happens here?
Well, we are plotting

358
00:27:27,000 --> 00:27:32,000
essentially this.
All of this stuff is a

359
00:27:32,000 --> 00:27:36,000
constant.
And we have a two minus r over

360
00:27:36,000 --> 00:27:41,000
a nought times an e to the minus
r over 2 a nought.

361
00:27:41,000 --> 00:27:45,000
That is what we are really

362
00:27:45,000 --> 00:27:48,000
plotting here.
And, if I did that,

363
00:27:48,000 --> 00:27:51,000
it would look something like
this.

364
00:27:51,000 --> 00:27:54,000
We start at some large,
positive value here.

365
00:27:54,000 --> 00:28:00,000
And you see that the wave
function decreases.

366
00:28:00,000 --> 00:28:06,000
And it gets to a value of r
where Psi is equal to zero.

367
00:28:06,000 --> 00:28:12,000
That is a radial node.
And in the case of the 2s wave

368
00:28:12,000 --> 00:28:18,000
function, that radial node
occurs at r equals 2 a nought.

369
00:28:18,000 --> 00:28:22,000
And then the wave function

370
00:28:22,000 --> 00:28:27,000
becomes negative,
increases, and gets more and

371
00:28:27,000 --> 00:28:32,000
more negative,
until you get to a point where

372
00:28:32,000 --> 00:28:40,000
it starts increasing again and
then approaches zero.

373
00:28:40,000 --> 00:28:44,000
This part, here,
of the wave function is really

374
00:28:44,000 --> 00:28:49,000
dictated by the exponential
term, the e to the minus r over

375
00:28:49,000 --> 00:28:54,000
2 a nought.
This part of the wave function

376
00:28:54,000 --> 00:29:00,000
is dictated by this polynomial
here, two minus r over ao.

377
00:29:00,000 --> 00:29:04,000
If you wanted to solve for that

378
00:29:04,000 --> 00:29:09,000
radial node, what would you do?
You would take that functional

379
00:29:09,000 --> 00:29:12,000
form, set it equal to zero and
solve for r.

380
00:29:12,000 --> 00:29:16,000
And so you can see that 2 minus
r over a nought set equal to

381
00:29:16,000 --> 00:29:20,000
zero,
that when r is 2 a nought,

382
00:29:20,000 --> 00:29:24,000
that the wave function would
have a value of zero.

383
00:29:24,000 --> 00:29:29,000
That is
how you solve for the value of r

384
00:29:29,000 --> 00:29:34,000
at which you have a node.
Now, this is really important

385
00:29:34,000 --> 00:29:37,000
here.
That is, at the radial node,

386
00:29:37,000 --> 00:29:42,000
the wave function changes sign.
The amplitude of the wave

387
00:29:42,000 --> 00:29:45,000
function goes from positive to
negative.

388
00:29:45,000 --> 00:29:50,000
That is important because at
all nodes, for all wave

389
00:29:50,000 --> 00:29:53,000
functions, the wave function
changes sign.

390
00:29:53,000 --> 00:29:59,000
And the reason the sign of the
wave function is so important is

391
00:29:59,000 --> 00:30:03,000
in chemical bonding.
But let me back up for a

392
00:30:03,000 --> 00:30:06,000
moment.
Many of you have talked about p

393
00:30:06,000 --> 00:30:09,000
orbitals or have seen p orbitals
before.

394
00:30:09,000 --> 00:30:14,000
Sometimes on a lobe of a p
orbital, you put a plus sign and

395
00:30:14,000 --> 00:30:16,000
sometimes you put a negative
sign.

396
00:30:16,000 --> 00:30:18,000
You have seen that,
right?

397
00:30:18,000 --> 00:30:20,000
Okay.
Well, what that is just

398
00:30:20,000 --> 00:30:25,000
referring to is the sign of the
amplitude of the wave function.

399
00:30:25,000 --> 00:30:29,000
It means that in that area the
amplitude is positive,

400
00:30:29,000 --> 00:30:34,000
and in the other area the
amplitude is negative.

401
00:30:34,000 --> 00:30:37,000
And the reason the amplitudes
are so important,

402
00:30:37,000 --> 00:30:42,000
or the sign of the amplitudes
are so important is because in a

403
00:30:42,000 --> 00:30:46,000
chemical reaction,
when you are bringing two atoms

404
00:30:46,000 --> 00:30:51,000
together and your electrons that
are represented by waves are

405
00:30:51,000 --> 00:30:56,000
overlapping, if you are bringing
in two wave functions that have

406
00:30:56,000 --> 00:30:59,000
the same sign,
well, then you are going to

407
00:30:59,000 --> 00:31:05,000
have constructive interference.
And you are going to have

408
00:31:05,000 --> 00:31:09,000
chemical bonding.
If you bring in two atoms,

409
00:31:09,000 --> 00:31:14,000
and the wave functions are
overlapping and they have

410
00:31:14,000 --> 00:31:17,000
opposite signs,
you have destructive

411
00:31:17,000 --> 00:31:22,000
interference and you are not
going to have any chemical

412
00:31:22,000 --> 00:31:25,000
bonding.
That is why the sign of those

413
00:31:25,000 --> 00:31:30,000
wave functions is so important.
So, that is Psi(2,

414
00:31:30,000 --> 00:31:34,000
0, 0).
What about Psi(3,

415
00:31:34,000 --> 00:31:38,000
0, 0)?
That is the last function,

416
00:31:38,000 --> 00:31:45,000
here, on the side walls.
And let me just write down the

417
00:31:45,000 --> 00:31:53,000
radial part, 27 minus 18(r over
a nought) plus 2 times (r over a

418
00:31:53,000 --> 00:32:01,000
nought) quantity squared times e
to the minus r over 3 a nought.

419
00:32:09,000 --> 00:32:14,000
And now, if I plotted that
function, Psi(3,0,0),

420
00:32:14,000 --> 00:32:19,000
3s wave function,
I would find out that r equals

421
00:32:19,000 --> 00:32:26,000
zero, large value of psi finite,
it drops, it crosses zero,

422
00:32:26,000 --> 00:32:30,000
gets negative,
then gets less negative,

423
00:32:30,000 --> 00:32:36,000
crosses zero again,
and then drops off.

424
00:32:36,000 --> 00:32:40,000
In the case of the 3s wave
function, we have two radial

425
00:32:40,000 --> 00:32:43,000
nodes.
We have a radial node right in

426
00:32:43,000 --> 00:32:47,000
here, and we have a radial node
right in there.

427
00:32:47,000 --> 00:32:51,000
And, if you want to know what
those radial nodes are,

428
00:32:51,000 --> 00:32:56,000
you set the wave function equal
to zero and solve for the values

429
00:32:56,000 --> 00:33:01,000
of r that make that wave
function zero.

430
00:33:01,000 --> 00:33:04,000
And, if you do that,
you would find this would come

431
00:33:04,000 --> 00:33:08,000
out to be, in terms of units of
a nought,

432
00:33:08,000 --> 00:33:10,000
1.9 a nought.
And right here,

433
00:33:10,000 --> 00:33:13,000
it would be 7.1 a nought.
The wave function,

434
00:33:13,000 --> 00:33:17,000
in the case of the 3s,
has a positive value for r less

435
00:33:17,000 --> 00:33:20,000
than 1.9 a nought,

436
00:33:20,000 --> 00:33:25,000
has a negative value from 1.9 a
nought to 7.1 a nought,

437
00:33:25,000 --> 00:33:29,000
and then a positive value again

438
00:33:29,000 --> 00:33:33,000
from 7.1 a nought to infinity.

439
00:33:33,000 --> 00:33:37,000
So, those are the wave

440
00:33:37,000 --> 00:33:45,000
functions, the functional forms,
what they actually look like.

441
00:33:45,000 --> 00:33:53,000
Now, it is time to talk about
what the wave function actually

442
00:33:53,000 --> 00:34:00,000
means, and how does it represent
the electron.

443
00:34:12,000 --> 00:34:16,000
Well, this was,
of course, a very puzzling

444
00:34:16,000 --> 00:34:20,000
question to the scientific
community.

445
00:34:20,000 --> 00:34:25,000
As soon as S wrote down is
Schrˆdinger equation,

446
00:34:25,000 --> 00:34:30,000
hmm, somehow these waves
represent the particles,

447
00:34:30,000 --> 00:34:35,000
but exactly how do they
represent where the particles

448
00:34:35,000 --> 00:34:40,000
are?
And the answer to that question

449
00:34:40,000 --> 00:34:46,000
is essentially there is no
answer to that question.

450
00:34:46,000 --> 00:34:50,000
Wave functions are wave
functions.

451
00:34:50,000 --> 00:34:57,000
It is one of these concepts
that you cannot draw a classical

452
00:34:57,000 --> 00:35:01,000
analogy to.
You want to say,

453
00:35:01,000 --> 00:35:05,000
well, a wave function does
this.

454
00:35:05,000 --> 00:35:10,000
But the only way you can
describe it is in terms of

455
00:35:10,000 --> 00:35:17,000
language that is something that
you experience everyday in your

456
00:35:17,000 --> 00:35:22,000
world, so you cannot.
A wave function is a wave

457
00:35:22,000 --> 00:35:26,000
function.
I cannot draw a correct analogy

458
00:35:26,000 --> 00:35:32,000
to a classical world.
Really, that is the case.

459
00:35:32,000 --> 00:35:39,000
However, it took a very smart
gentleman by the name of Max

460
00:35:39,000 --> 00:35:46,000
Born to look at this problem.
He said, "If I take the wave

461
00:35:46,000 --> 00:35:52,000
function and I square it,
if I interpret that as a

462
00:35:52,000 --> 00:35:57,000
probability density,
then I can understand all the

463
00:35:57,000 --> 00:36:04,000
predictions made by the
Schrˆdinger equation within that

464
00:36:04,000 --> 00:36:09,000
framework."
In other words,

465
00:36:09,000 --> 00:36:16,000
he said, let me take Psi and l
and m as a function r,

466
00:36:16,000 --> 00:36:23,000
theta, and phi and square it.
Let me interpret that as a

467
00:36:23,000 --> 00:36:27,000
probability density.

468
00:36:32,000 --> 00:36:36,000
Probability density is not a
probability.

469
00:36:36,000 --> 00:36:41,000
It is a density.
Density is always per unit

470
00:36:41,000 --> 00:36:45,000
volume.
Probability density is a

471
00:36:45,000 --> 00:36:49,000
probability per unit volume.

472
00:36:54,000 --> 00:36:57,000
It is a probability per unit
volume.

473
00:36:57,000 --> 00:37:00,000
Well, if I use that
interpretation,

474
00:37:00,000 --> 00:37:04,000
then I can understand all the
predictions made by the

475
00:37:04,000 --> 00:37:09,000
Schrˆdinger equation.
It makes sense.

476
00:37:09,000 --> 00:37:13,000
And, you know what,
that is it.

477
00:37:13,000 --> 00:37:19,000
Because that interpretation
does agree with our

478
00:37:19,000 --> 00:37:25,000
observations,
it is therefore believed to be

479
00:37:25,000 --> 00:37:32,000
correct.
But it is just an assumption.

480
00:37:32,000 --> 00:37:37,000
It is an interpretation.
There is no derivation for it.

481
00:37:37,000 --> 00:37:41,000
It is just that the
interpretation works.

482
00:37:41,000 --> 00:37:45,000
If it works,
we therefore believe it to be

483
00:37:45,000 --> 00:37:48,000
accurate.
There is no indication,

484
00:37:48,000 --> 00:37:54,000
there are no data that seem to
contradict that interpretation,

485
00:37:54,000 --> 00:38:00,000
so we think it is right.
That is what Max Born said.

486
00:38:00,000 --> 00:38:03,000
Now, Max Born was really
something in terms of his

487
00:38:03,000 --> 00:38:08,000
scientific accomplishments.
Not only did he recognize or

488
00:38:08,000 --> 00:38:12,000
have the insight to realize what
Psi squared was,

489
00:38:12,000 --> 00:38:16,000
but he is also the Born of the
Born-Oppenheimer Approximation

490
00:38:16,000 --> 00:38:20,000
that maybe some of you have
heard about before.

491
00:38:20,000 --> 00:38:24,000
He is also the Born in the
Distorted-Wave Born

492
00:38:24,000 --> 00:38:27,000
Approximation,
which probably none of you have

493
00:38:27,000 --> 00:38:32,000
heard before.
But, despite all of those

494
00:38:32,000 --> 00:38:37,000
accomplishments,
psi squared interpretation,

495
00:38:37,000 --> 00:38:42,000
Born-Oppenheimer Approximation,
Distorted-Wave Born

496
00:38:42,000 --> 00:38:46,000
Approximation,
he is best known for being the

497
00:38:46,000 --> 00:38:50,000
grandfather of Olivia
Newton-John.

498
00:38:50,000 --> 00:38:54,000
That's right.
Oliver Newton-John is a singer

499
00:38:54,000 --> 00:38:59,000
in Grease.
Two weeks ago in the Boston

500
00:38:59,000 --> 00:39:04,000
Globe Parade Magazine,
which I actually think is a

501
00:39:04,000 --> 00:39:09,000
magazine that goes throughout
the country in all the Sunday

502
00:39:09,000 --> 00:39:15,000
newspapers, there is a long
article on Olivia Newton-John

503
00:39:15,000 --> 00:39:19,000
and a short sentence about her
grandfather, Max Born.

504
00:39:19,000 --> 00:39:24,000
So, that is our interpretation,
thanks to Max Born.

505
00:39:24,000 --> 00:39:29,000
Now, how are we going to use
that?

506
00:39:29,000 --> 00:39:35,000
Well, first of all,
let's take our functional forms

507
00:39:35,000 --> 00:39:40,000
for Psi, here,
and square it and plot those

508
00:39:40,000 --> 00:39:47,000
probability densities for the
individual wave functions and

509
00:39:47,000 --> 00:39:50,000
see what we get.

510
00:39:55,000 --> 00:40:00,000
The way I am going to plot the
probability density is by using

511
00:40:00,000 --> 00:40:04,000
some dots.
And the density of the dots is

512
00:40:04,000 --> 00:40:08,000
going to reflect the probability
density.

513
00:40:08,000 --> 00:40:13,000
The more dense the dots,
the larger the probability

514
00:40:13,000 --> 00:40:16,000
density.
If I take that functional form

515
00:40:16,000 --> 00:40:21,000
for the 1s wave function and
square it and then plot the

516
00:40:21,000 --> 00:40:27,000
value of that function squared
with this density dot diagram,

517
00:40:27,000 --> 00:40:33,000
then you can see that the dots
here are most dense right at the

518
00:40:33,000 --> 00:40:37,000
origin, and that they
exponentially decay in all

519
00:40:37,000 --> 00:40:41,000
directions.
The probability density here

520
00:40:41,000 --> 00:40:45,000
for 1s wave function is greatest
at the origin,

521
00:40:45,000 --> 00:40:48,000
r equals 0, and it decays
exponentially in all directions.

522
00:40:48,000 --> 00:40:52,000
It is spherically symmetric.
That is what you would expect

523
00:40:52,000 --> 00:40:56,000
because that is what the wave
function looks like.

524
00:40:56,000 --> 00:40:59,000
You square that,
you get another exponential,

525
00:40:59,000 --> 00:41:03,000
and it decays exponentially in
all directions.

526
00:41:03,000 --> 00:41:08,000
That is a probability density,
probability of finding the

527
00:41:08,000 --> 00:41:12,000
electron per unit volume at some
value r, theta,

528
00:41:12,000 --> 00:41:15,000
and phi.
And it turns out it doesn't

529
00:41:15,000 --> 00:41:20,000
matter what theta and phi are
because this is spherically

530
00:41:20,000 --> 00:41:23,000
symmetric.
What about the 2s wave

531
00:41:23,000 --> 00:41:26,000
function?
Well, here is the 2s

532
00:41:26,000 --> 00:41:30,000
probability density.
Again, you can see the

533
00:41:30,000 --> 00:41:34,000
probability density is a maximum
at the origin,

534
00:41:34,000 --> 00:41:37,000
at the nucleus.
That probability density decays

535
00:41:37,000 --> 00:41:41,000
uniformly in all directions.
And it decays so much that at

536
00:41:41,000 --> 00:41:45,000
some point, you have no
probability density.

537
00:41:45,000 --> 00:41:47,000
Why?
Because that is the node.

538
00:41:47,000 --> 00:41:50,000
If you square zero,
you still get zero.

539
00:41:50,000 --> 00:41:52,000
r equals 2 a nought.

540
00:41:52,000 --> 00:41:57,000
You can see that in the
probability density.

541
00:41:57,000 --> 00:42:00,000
But then again,
as you move up this way,

542
00:42:00,000 --> 00:42:03,000
as you increase r,
the probability density

543
00:42:03,000 --> 00:42:04,000
increases again.
Why?

544
00:42:04,000 --> 00:42:08,000
Remember the wave function?
It has changed sign.

545
00:42:08,000 --> 00:42:11,000
But in this area,
here, where it is negative,

546
00:42:11,000 --> 00:42:15,000
if I square it,
well, the probability density

547
00:42:15,000 --> 00:42:19,000
still is going to be large.
Square a negative number,

548
00:42:19,000 --> 00:42:22,000
you are going to have a large
positive number.

549
00:42:22,000 --> 00:42:27,000
That is why the probability
density increases right in here,

550
00:42:27,000 --> 00:42:32,000
and then, again,
it decays towards zero.

551
00:42:32,000 --> 00:42:37,000
You can see the radial node not
only in the wave function,

552
00:42:37,000 --> 00:42:41,000
but also in the probability
density.

553
00:42:41,000 --> 00:42:47,000
Here is the probability density
for the 3s wave function.

554
00:42:47,000 --> 00:42:52,000
Once again, probability density
is a maximum at r equals 0,

555
00:42:52,000 --> 00:42:58,000
and it decays uniformly in all
directions.

556
00:42:58,000 --> 00:43:01,000
It decays so much that at some
value of r, right here,

557
00:43:01,000 --> 00:43:03,000
the probability density is
zero.

558
00:43:03,000 --> 00:43:05,000
Why?
Because the wave function is

559
00:43:05,000 --> 00:43:07,000
zero.
You square it,

560
00:43:07,000 --> 00:43:11,000
and you are going to get a zero
for the probability density.

561
00:43:11,000 --> 00:43:14,000
And then the probability
density increases again.

562
00:43:14,000 --> 00:43:16,000
Why?
Because you are getting a more

563
00:43:16,000 --> 00:43:20,000
and more negative value for the
wave function right in this

564
00:43:20,000 --> 00:43:21,000
area.
Square that,

565
00:43:21,000 --> 00:43:25,000
and it is going to increase.
And then, as you continue to

566
00:43:25,000 --> 00:43:30,000
increase r, probability density
decreases.

567
00:43:30,000 --> 00:43:33,000
It decreases again,
so that you get a zero.

568
00:43:33,000 --> 00:43:37,000
You get a zero because the wave
function is zero right there.

569
00:43:37,000 --> 00:43:41,000
This is our second radial node.
But then, the probability

570
00:43:41,000 --> 00:43:45,000
density increases again.
It increases because the wave

571
00:43:45,000 --> 00:43:48,000
function increases.
Square that,

572
00:43:48,000 --> 00:43:51,000
we are going to get a high
probability density,

573
00:43:51,000 --> 00:43:55,000
and then it tapers off.
So, the important point here is

574
00:43:55,000 --> 00:44:00,000
the interpretation of the
probability density.

575
00:44:00,000 --> 00:44:05,000
Probability per unit volume.
The fact that the s wave

576
00:44:05,000 --> 00:44:08,000
functions are all spherically
symmetric.

577
00:44:08,000 --> 00:44:13,000
They do not have an angular
dependence to them.

578
00:44:13,000 --> 00:44:18,000
And what a radial node is.
If you want to get a radial

579
00:44:18,000 --> 00:44:23,000
node, you take the wave
function, set it equal to zero,

580
00:44:23,000 --> 00:44:30,000
solve for the value of r,
and that gives you a zero.

581
00:44:30,000 --> 00:44:34,000
Now, so far,
we have talked only about the

582
00:44:34,000 --> 00:44:38,000
probability density and this
interpretation.

583
00:44:38,000 --> 00:44:43,000
We have not talked about any
probabilities yet.

584
00:44:43,000 --> 00:44:48,000
And, to do so,
we are going to talk about this

585
00:44:48,000 --> 00:44:52,000
function, here.
It is called a radial

586
00:44:52,000 --> 00:44:58,000
probability distribution.
The radial probability

587
00:44:58,000 --> 00:45:04,000
distribution is the probability
of finding an electron in a

588
00:45:04,000 --> 00:45:08,000
spherical shell.
That spherical shell will be

589
00:45:08,000 --> 00:45:12,000
some distance r away from the
nucleus.

590
00:45:12,000 --> 00:45:18,000
That spherical shell will have
a radius r and will have a

591
00:45:18,000 --> 00:45:20,000
thickness.
And the thickness,

592
00:45:20,000 --> 00:45:26,000
we are going to call dr.
This is not a solid sphere.

593
00:45:26,000 --> 00:45:30,000
This is a shell.
This is a sphere,

594
00:45:30,000 --> 00:45:35,000
but the thickness of that
sphere is very small.

595
00:45:35,000 --> 00:45:41,000
The thickness of it is dr.
And, to try to represent that a

596
00:45:41,000 --> 00:45:45,000
little bit better,
I show you here a picture of

597
00:45:45,000 --> 00:45:50,000
the probability density for the
(1, 0, 0) state.

598
00:45:50,000 --> 00:45:54,000
This is kind of my density dot
diagram.

599
00:45:54,000 --> 00:46:00,000
And then, this blue thing is my
spherical shell.

600
00:46:00,000 --> 00:46:04,000
This blue thing,
here, has a radius r,

601
00:46:04,000 --> 00:46:08,000
and this blue thing has a
thickness dr.

602
00:46:08,000 --> 00:46:14,000
And so, I am saying that the
radial probability distribution

603
00:46:14,000 --> 00:46:21,000
is going to be the probability
of finding the electron in this

604
00:46:21,000 --> 00:46:26,000
spherical shell.
That spherical shell is a

605
00:46:26,000 --> 00:46:33,000
distance r from the nucleus and
has a thickness dr.

606
00:46:33,000 --> 00:46:37,000
Now, I want to point out that
the volume of the spherical

607
00:46:37,000 --> 00:46:41,000
shell is just the surface area,
here, 4 pi r squared,

608
00:46:41,000 --> 00:46:44,000
times the thickness,
which is dr.

609
00:46:44,000 --> 00:46:49,000
Not a very thick spherical

610
00:46:49,000 --> 00:46:51,000
shell.
The radial probability

611
00:46:51,000 --> 00:46:56,000
distribution is the probability
of finding that electron in that

612
00:46:56,000 --> 00:47:02,000
spherical shell.
It is like the probability of

613
00:47:02,000 --> 00:47:10,000
finding the electron a distance
r to r plus dr

614
00:47:10,000 --> 00:47:14,000
from the nucleus.
Why is that important?

615
00:47:14,000 --> 00:47:21,000
Well, because if I want to
calculate a probability,

616
00:47:21,000 --> 00:47:27,000
what I can do then is take the
probability density here,

617
00:47:27,000 --> 00:47:34,000
Psi squared for an s orbital,
which is probability per unit

618
00:47:34,000 --> 00:47:43,000
volume, and I can then multiply
it by that unit volume.

619
00:47:43,000 --> 00:47:48,000
In this case it was the 4pi r
squared dr.

620
00:47:48,000 --> 00:47:53,000
And the result will be a
probability, because I have

621
00:47:53,000 --> 00:47:57,000
probability density,
probability per unit volume

622
00:47:57,000 --> 00:48:02,000
times a volume,
and that is a probability.

623
00:48:02,000 --> 00:48:09,000
Now we are getting somewhere in
terms of figuring out what the

624
00:48:09,000 --> 00:48:15,000
probability is of finding the
electron some distance r to r

625
00:48:15,000 --> 00:48:22,000
plus dr from the nucleus.
In the case of the s orbitals,

626
00:48:22,000 --> 00:48:28,000
I said that the Psi was a
product of the radial part and

627
00:48:28,000 --> 00:48:34,000
the Y(lm) angular part.
Remember that the Y(lm) for the

628
00:48:34,000 --> 00:48:40,000
s orbitals was always one over
the square-root of one over 4pi.

629
00:48:40,000 --> 00:48:46,000
The Y(lm) squared is going to

630
00:48:46,000 --> 00:48:50,000
cancel with 4pi,
and you are just going to have

631
00:48:50,000 --> 00:48:54,000
r squared times the radial part
squared.

632
00:48:54,000 --> 00:48:57,000
For a 1s orbital,
if you want to actually

633
00:48:57,000 --> 00:49:01,000
calculate the probability at
some value r,

634
00:49:01,000 --> 00:49:07,000
you just have to take Psi
squared and multiply it by 4pi r

635
00:49:07,000 --> 00:49:12,000
squared dr.

636
00:49:12,000 --> 00:49:16,000
However, in the case of all
other orbitals,

637
00:49:16,000 --> 00:49:22,000
you cannot do that because they
are not spherically symmetric.

638
00:49:22,000 --> 00:49:28,000
And so, for all other orbitals,
you have to take the radial

639
00:49:28,000 --> 00:49:34,000
part and multiply it by r
squared dr.

640
00:49:34,000 --> 00:49:38,000
I will explain that a little
bit more next time.

641
00:49:38,000 --> 00:49:41,000
Okay.
See you on Monday.