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And your equation is equal to E
Psi.
6
00:00:18 --> 00:00:25
And we saw that these energies,
the binding energies of the
7
00:00:25 --> 00:00:30
electron to the nucleus,
were quantized.
8
00:00:30 --> 00:00:35
9
00:00:35 --> 00:00:41
Well, these energy levels then
were given by minus this Rydberg
10
00:00:41 --> 00:00:48
constant R sub H over n squared.
And this n
11
00:00:48 --> 00:00:53
was our quantum number.
It is a principle quantum
12
00:00:53 --> 00:00:56
number.
And we saw that n,
13
00:00:56 --> 00:01:01
the smallest value was 1,
2 and ran all the way up to
14
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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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00:01:24 --> 00:01:30
Now, Psi, in general,
is the function of r,
18
00:01:30 --> 00:01:38
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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00:01:42 --> 00:01:47
In other words,
the wave functions that we are
21
00:01:47 --> 00:01:53
going to be looking at are what
are called stationary waves.
22
00:01:53 --> 00:01:59
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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00:02:04 --> 00:02:08
We are either going to look at
it before or after,
25
00:02:08 --> 00:02:11
but not during.
And, in those cases,
26
00:02:11 --> 00:02:14
we are looking at a wave
function.
27
00:02:14 --> 00:02:18
And the atom is just stable and
is sitting there.
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00:02:18 --> 00:02:23
The time dependence here does
not have a consequence.
29
00:02:23 --> 00:02:27
And so, therefore,
the wave functions that we are
30
00:02:27 --> 00:02:31
going to be looking at are just
a function of r,
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00:02:31 --> 00:02:36
theta and phi.
And we are looking at what is
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00:02:36 --> 00:02:40
called time-independent quantum
mechanics.
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00:02:40 --> 00:02:45
Later on, actually,
if you are in chemistry,
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00:02:45 --> 00:02:50
in a graduate course in
chemistry, is when you look at
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00:02:50 --> 00:02:55
time-dependent wave functions.
We are going to look at
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00:02:55 --> 00:03:02
time-independent quantum
mechanics, the stationary wave.
37
00:03:02 --> 00:03:06
Now, it turns out that when we
go and solve the Schrödinger
38
00:03:06 --> 00:03:10
equation here for Psi,
what happens is that two more
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00:03:10 --> 00:03:15
quantum numbers drop out of that
solution to the differential
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00:03:15 --> 00:03:19
equation.
Remember, we said last time how
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00:03:19 --> 00:03:22
quantum numbers arise.
They arise from imposing
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00:03:22 --> 00:03:26
boundary conditions on a
differential equation,
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00:03:26 --> 00:03:30
making that differential
equation applicable to your
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00:03:30 --> 00:03:36
actual physical problem.
And so, when we do that,
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00:03:36 --> 00:03:40
we get a new quantum number
called l.
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00:03:40 --> 00:03:47
And l is, I think you already
know, the angular momentum
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00:03:47 --> 00:03:50
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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00:04:14 --> 00:04:22
Zero is the smallest value of
l, the lowest value of l
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00:04:22 --> 00:04:28
allowed.
1, 2, all the way up to n minus
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00:04:28.602 --> 1.
55
1. --> 00:04:32
That is the largest value it
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00:04:32 --> 00:04:36
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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00:04:39 --> 00:04:42
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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00:05:06 --> 00:05:10
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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00:05:18 --> 00:05:22
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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00:06:27 --> 00:06:33
m equals zero.
You can have no angular
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00:06:33 --> 00:06:38
momentum in the z-component.
Or, plus one,
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00:06:38 --> 00:06:44
plus two, plus three,
all the way up to plus l.
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00:06:44 --> 00:06:50
Again, m is tied to l.
It cannot be larger than l,
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00:06:50 --> 00:06:58
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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00:07:10 --> 00:07:14
The largest value you can have
is l.
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00:07:14 --> 00:07:21
But, since this is a
z-component and we've got some
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00:07:21 --> 00:07:26
direction, m could also be minus
1, minus two,
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minus three,
minus l.
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00:07:30 --> 00:07:33
We have three quantum numbers,
now.
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00:07:33 --> 00:07:39
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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00:07:43 --> 00:07:49
three quantum numbers to
completely describe our system.
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00:07:49 --> 00:07:54
The consequence,
here, of having three quantum
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00:07:54 --> 00:08:00
numbers is that we now have more
states.
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00:08:00 --> 00:08:11
98
00:08:11 --> 00:08:14
For example,
our n equals 1 state that we
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00:08:14 --> 00:08:18
talked about last time,
more completely we have to
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00:08:18 --> 00:08:21
describe that state by two other
quantum numbers.
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00:08:21 --> 00:08:25
When n is equal to one,
what is the only value that l
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00:08:25 --> 00:08:27
can have?
Zero.
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00:08:27 --> 00:08:30
And if l is zero,
what is the only value m can
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00:08:30 --> 00:08:32
have?
Zero.
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00:08:32 --> 00:08:37
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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00:08:42 --> 00:08:48
And, if we have an electron in
that (1, 0, 0) state,
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00:08:48 --> 00:08:55
we are going to describe that
electron by the wave function
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00:08:55 --> 00:09:00
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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00:09:05 --> 00:09:09
I am just telling you right now
that Psi does.
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00:09:09 --> 00:09:14
Exactly how it does is
something I haven't told you
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00:09:14 --> 00:09:17
yet.
And we will do that later in
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00:09:17 --> 00:09:20
the lecture today,
sort of.
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00:09:20 --> 00:09:24
That is the energy,
minus the Rydberg constant.
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00:09:24 --> 00:09:30
But now, if n is equal two,
what is the smallest value that
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00:09:30 --> 00:09:33
l can have?
Zero.
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00:09:33 --> 00:09:36
And the value of m in that
case?
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00:09:36 --> 00:09:39
Zero.
And so our n equals 2 state is
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00:09:39 --> 00:09:42
more completely the (2,
0, 0) state.
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00:09:42 --> 00:09:47
And the wave function that
describes the electron in that
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00:09:47 --> 00:09:51
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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One.
And if l is equal to one,
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00:10:05 --> 00:10:09
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 --> 00:10:14
here, the (2,
1, 1) state.
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00:10:14 --> 00:10:19
And if you have an electron in
that state, it is described by
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00:10:19 --> 00:10:22
the Psi(2, 1,
1) wave function.
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00:10:22 --> 00:10:27
However, the energy here is
also minus one-quarter the
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00:10:27 --> 00:10:32
Rydberg constant.
It has the same energy as the
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00:10:32 --> 00:10:36
(2, 0, 0) state.
Now, if n is equal two and l is
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00:10:36 --> 00:10:40
equal to one,
what is the next larger value
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00:10:40 --> 00:10:40
of m?
Zero.
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00:10:40 --> 00:10:43
And so we have a (2,
1, 0) state.
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00:10:43 --> 00:10:48
Again, that wave function,
for an electron in that state,
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00:10:48 --> 00:10:50
we label as Psi(2,
1, 0).
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00:10:50 --> 00:10:53
And then, finally,
when n is equal to two,
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00:10:53 --> 00:10:57
l is equal to one,
what is the final possible
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00:10:57 --> 00:11:01
value of m?
Minus one.
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00:11:01 --> 00:11:04
We have a (2,
1, -1) state and a wave
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00:11:04 --> 00:11:09
function that is the (2,
1, -1) wave function.
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00:11:09 --> 00:11:15
Notice that the energy of all
of these four states is the
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00:11:15 --> 00:11:20
same, minus one-quarter the
Rydberg constant.
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00:11:20 --> 00:11:24
These states are what we call
degenerate.
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00:11:24 --> 00:11:30
Degenerate means having the
same energy.
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00:11:30 --> 00:11:34
That is important.
Now, this is the way we label
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00:11:34 --> 00:11:38
wave functions,
but we also have a different
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00:11:38 --> 00:11:42
scheme for talking about wave
functions.
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00:11:42 --> 00:11:46
That is, we have an orbital
scheme.
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00:11:46 --> 00:11:50
And, as I said,
or alluded to the other day,
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00:11:50 --> 00:11:55
an orbital is nothing other
than a wave function.
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00:11:55 --> 00:12:01
It is a solution to the
Schrödinger equation.
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00:12:01 --> 00:12:04
That is what an orbital is.
It is actually the spatial part
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00:12:04 --> 00:12:08
of the wave function.
There is another part called
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00:12:08 --> 00:12:11
the spin part,
which we will deal with later,
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00:12:11 --> 00:12:14
but an orbital is essentially a
wave function.
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00:12:14 --> 00:12:17
We have a different language
for describing orbitals.
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00:12:17 --> 00:12:20
The way we do this,
I think you are already
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00:12:20 --> 00:12:23
familiar with,
but we are going to describe it
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00:12:23 --> 00:12:26
by the principle quantum number
and, of course,
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00:12:26 --> 00:12:30
the angular momentum quantum
number and the magnetic quantum
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00:12:30 --> 00:12:35
number.
Except that we have a scheme or
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00:12:35 --> 00:12:40
a code for l and m.
That code uses letters instead
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00:12:40 --> 00:12:42
of numbers.
For example,
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00:12:42 --> 00:12:48
if l is equal to zero here,
we call that an s wave function
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00:12:48 --> 00:12:51
or an s orbital.
For example,
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00:12:51 --> 00:12:55
in the orbital language,
instead of Psi(1,
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00:12:55 --> 00:13:00
0, 0), we call this a 1s
orbital.
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00:13:00 --> 00:13:02
Here is the principle quantum
number.
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00:13:02 --> 00:13:05
Here is s.
l is equal to zero.
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00:13:05 --> 00:13:08
That is just our code for l is
equal to zero.
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00:13:08 --> 00:13:12
This state, (2,
0, 0), you have an electron in
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00:13:12 --> 00:13:15
that state.
Well, it can describe that
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00:13:15 --> 00:13:19
wavefunction by the 2s orbital.
Here is the principle quantum
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00:13:19 --> 00:13:22
number.
He is s, our code for l equals
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00:13:22 --> 00:13:25
zero.
Now, when n is equal to 1,
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00:13:25 --> 00:13:30
we call that a p wave function
or a p orbital.
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00:13:30 --> 00:13:33
This state right here,
it is the 2p state.
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00:13:33 --> 00:13:39
Because n is two and l is one,
this state is also the 2p wave
183
00:13:39 --> 00:13:44
function and this state,
it is also the 2p wave
184
00:13:44 --> 00:13:47
function.
Now, if I had a wave function
185
00:13:47 --> 00:13:52
up here where l was equal to
two, we would call that d
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00:13:52 --> 00:13:56
orbital.
And, if I had a wave function
187
00:13:56 --> 00:14:00
up here where l was three,
we would call that an f
188
00:14:00 --> 00:14:05
orbital.
But now, all of these wave
189
00:14:05 --> 00:14:09
functions in the orbital
language have the same
190
00:14:09 --> 00:14:15
designation, and that is because
we have not taken care of this
191
00:14:15 --> 00:14:19
yet, the m quantum number,
magnetic quantum number.
192
00:14:19 --> 00:14:24
And we also have an alphabet
scheme for those quantum
193
00:14:24 --> 00:14:27
numbers.
The bottom line is that when m
194
00:14:27 --> 00:14:33
is equal to zero,
we put a z subscript on the p.
195
00:14:33 --> 00:14:39
When m is equal to zero,
that is our 2pz wave function.
196
00:14:39 --> 00:14:44
When m is equal to one,
we are going to put an x
197
00:14:44 --> 00:14:47
subscript on the p wave
function.
198
00:14:47 --> 00:14:54
And when m is equal to minus
one, we are going to put a y
199
00:14:54 --> 00:14:59
subscription on the wave
function.
200
00:14:59 --> 00:15:04
However, I have to tell you
that in the case of m is equal
201
00:15:04 --> 00:15:08
to one and m is equal to minus
one, that is not strictly
202
00:15:08 --> 00:15:12
correct.
And the reason it is not is
203
00:15:12 --> 00:15:16
because when you solve
Schrödinger equation for the (2,
204
00:15:16 --> 00:15:21
1, 1) wave function and the (2,
1, -1) wave function,
205
00:15:21 --> 00:15:25
the solutions are complex wave
functions.
206
00:15:25 --> 00:15:29
They are not real wave
function.
207
00:15:29 --> 00:15:35
And so, in order for us to be
able to draw and think about the
208
00:15:35 --> 00:15:42
px and the py wave functions,
what we do is we take a linear
209
00:15:42 --> 00:15:47
combination of the px and the py
wave functions.
210
00:15:47 --> 00:15:54
We take a positive linear
combination, in the case of the
211
00:15:54 --> 00:15:59
px wave function.
The px wave function is really
212
00:15:59 --> 00:16:03
this wave function plus this
wave function.
213
00:16:03 --> 00:16:08
And the py wave function is
really this wave function minus
214
00:16:08 --> 00:16:12
this wave function.
Then we will get a real
215
00:16:12 --> 00:16:15
function, and that is easier to
deal with.
216
00:16:15 --> 00:16:18
That is strictly what px and py
are.
217
00:16:18 --> 00:16:23
px and py are these linear
combinations of the actual
218
00:16:23 --> 00:16:29
functions that come out of the
Schrödinger equation.
219
00:16:29 --> 00:16:32
pz is exactly correct.
pz, M is equal to zero,
220
00:16:32 --> 00:16:36
in the pz wavefunction.
Now, you are not responsible
221
00:16:36 --> 00:16:40
for knowing that.
I just wanted to let you know.
222
00:16:40 --> 00:16:45
You don't have to remember that
m equal to one gives you the x
223
00:16:45 --> 00:16:50
subscript, and m equal to minus
one gives you the y subscript.
224
00:16:50 --> 00:16:54
You do have to know that when m
is equal to zero,
225
00:16:54 --> 00:17:00
you get a z subscript,
because that is exactly right.
226
00:17:00 --> 00:17:03
Now, just one other comment
here.
227
00:17:03 --> 00:17:08
That is, you see we did not put
a subscript on the s wave
228
00:17:08 --> 00:17:13
functions.
Well, that is because for an s
229
00:17:13 --> 00:17:17
wave function,
the only choice you have for m
230
00:17:17 --> 00:17:20
is zero.
And so we leave out that
231
00:17:20 --> 00:17:24
subscript.
We never put a z on there or a
232
00:17:24 --> 00:17:27
z on there.
It is always 1s,
233
00:17:27 --> 00:17:32
2s, or 3s.
Well, in order to understand
234
00:17:32 --> 00:17:37
that just a little more,
let's draw an energy level
235
00:17:37 --> 00:17:41
diagram.
Here is the energy again.
236
00:17:41 --> 00:17:45
And for n equals 1,
we now see that the more
237
00:17:45 --> 00:17:50
complete description is three
quantum numbers,
238
00:17:50 --> 00:17:53
(1, 0, 0).
That gives us the (1,
239
00:17:53 --> 00:17:57
0, 0) state,
or sometimes we say the 1s
240
00:17:57 --> 00:18:00
state.
There is one state at that
241
00:18:00 --> 00:18:05
energy.
But, in the case of n equals 2,
242
00:18:05 --> 00:18:11
we already saw that we have
four different states for the
243
00:18:11 --> 00:18:14
quantum number.
Four different states.
244
00:18:14 --> 00:18:19
They all have the same energy.
They are degenerate.
245
00:18:19 --> 00:18:23
Degenerate means having the
same energy.
246
00:18:23 --> 00:18:30
They differ in how much angular
momentum the electron has.
247
00:18:30 --> 00:18:35
Or, how much angular momentum
the electron would have in the z
248
00:18:35 --> 00:18:38
component if it were in a
magnetic field.
249
00:18:38 --> 00:18:43
The total energy is the same
because only n determines the
250
00:18:43 --> 00:18:46
energy.
It is just that the amount of
251
00:18:46 --> 00:18:50
angular momentum,
or the z component of the
252
00:18:50 --> 00:18:52
angular momentum,
differs in 2s,
253
00:18:52 --> 00:18:56
2py, 2pz, 2px,
but they all have the same
254
00:18:56 --> 00:19:01
energy.
In general, for any value of n,
255
00:19:01 --> 00:19:08
there are n squared degenerate
states at each value of n.
256
00:19:08 --> 00:19:14
If we have n is equal to three,
then the energy is minus
257
00:19:14 --> 00:19:21
one-ninth the Rydberg constant.
But how many states do we have
258
00:19:21 --> 00:19:24
at n equal three?
Nine.
259
00:19:24 --> 00:19:28
Here they are.
Just like for n equal two,
260
00:19:28 --> 00:19:34
we have the 3s,
3py, 3pz, and 3px.
261
00:19:34 --> 00:19:38
And there are the associated
quantum numbers for that.
262
00:19:38 --> 00:19:44
But now we have some states
here that I have labeled 3d
263
00:19:44 --> 00:19:47
states.
Well, when you have a d state,
264
00:19:47 --> 00:19:52
that means l is equal to two.
All of these states have
265
00:19:52 --> 00:19:58
principle quantum number three
and angular momentum quantum
266
00:19:58 --> 00:20:03
number two.
And they differ also by the
267
00:20:03 --> 00:20:08
amount of angular momentum in
the z direction.
268
00:20:08 --> 00:20:12
They differ in the quantum
number m.
269
00:20:12 --> 00:20:16
For example,
we are going to call the m
270
00:20:16 --> 00:20:20
equal minus two state the
3d(xy).
271
00:20:20 --> 00:20:24
We are going to put xy as a
subscript.
272
00:20:24 --> 00:20:30
For m equal minus one,
we are going to put a subscript
273
00:20:30 --> 00:20:33
yz.
For m equals zero,
274
00:20:33 --> 00:20:38
we are going to put a subscript
z squared.
275
00:20:38 --> 00:20:40
For m equal one,
d(xz).
276
00:20:40 --> 00:20:44
And for m equal two,
x squared minus y squared.
277
00:20:44 --> 00:20:49
Again, for m equal minus two,
278
00:20:49 --> 00:20:54
minus one, one and two,
those wave functions,
279
00:20:54 --> 00:20:59
when you solve Schrödinger
equations, are complex wave
280
00:20:59 --> 00:21:04
functions.
And what we do to talk about
281
00:21:04 --> 00:21:08
the wave functions is we take
linear combinations of them to
282
00:21:08 --> 00:21:12
make them real.
And so, when I say m is minus
283
00:21:12 --> 00:21:16
two, is the 3dxy wave function,
it is not strictly correct.
284
00:21:16 --> 00:21:20
Therefore, again,
you don't need to know m minus
285
00:21:20 --> 00:21:23
two.
You don't need to know that
286
00:21:23 --> 00:21:24
subscript.
Or m equal one,
287
00:21:24 --> 00:21:29
you don't need to know that
subscript.
288
00:21:29 --> 00:21:32
But for m equal zero,
this is a z squared.
289
00:21:32 --> 00:21:36
Absolutely.
These wave functions are linear
290
00:21:36 --> 00:21:38
combinations.
This one is not.
291
00:21:38 --> 00:21:43
It is a real function when it
comes out of the Schrödinger
292
00:21:43 --> 00:21:47
equation.
You will talk about these 3d
293
00:21:47 --> 00:21:51
wave functions a lot with
Professor Cummins in the
294
00:21:51 --> 00:21:56
second-half of the course.
You will actually look at the
295
00:21:56 --> 00:22:01
shapes of those wave functions
in detail.
296
00:22:01 --> 00:22:04
So, that's the energy level
diagram, here.
297
00:22:04 --> 00:22:08
All right.
Now let's actually talk about
298
00:22:08 --> 00:22:11
the shapes of the wave
functions.
299
00:22:11 --> 00:22:16
I am going to raise this
screen, I think.
300
00:22:16 --> 00:22:25
301
00:22:25 --> 00:22:40
What do these wave functions
actually look like?
302
00:22:40 --> 00:22:45
Well, for a hydrogen atom,
our wave function here,
303
00:22:45 --> 00:22:52
given by three quantum numbers,
n, l and m, function of r,
304
00:22:52 --> 00:22:57
theta and phi,
it turns out that those wave
305
00:22:57 --> 00:23:04
functions are factorable into a
function that is only in r and a
306
00:23:04 --> 00:23:10
function that is only in the
angles.
307
00:23:10 --> 00:23:13
You can write that,
no approximation,
308
00:23:13 --> 00:23:16
this is just the way it turns
out.
309
00:23:16 --> 00:23:22
The function that is a function
only of r, R of r,
310
00:23:22 --> 00:23:27
is called the radial function.
We will call it capital R,
311
00:23:27 --> 00:23:31
radial function of r.
It is labeled by only two
312
00:23:31 --> 00:23:35
quantum numbers,
n and l.
313
00:23:35 --> 00:23:40
The function that is a function
only of the angles,
314
00:23:40 --> 00:23:43
theta and phi,
we are going to call Y.
315
00:23:43 --> 00:23:48
This is the angular part of the
wave function.
316
00:23:48 --> 00:23:52
And it labeled by only two
quantum numbers,
317
00:23:52 --> 00:23:58
but they are l and m.
Sometimes we call this angular
318
00:23:58 --> 00:24:02
part, for short,
the Y(lm)'s.
319
00:24:02 --> 00:24:06
There is a radial part,
and there is an angular part.
320
00:24:06 --> 00:24:12
The actual functional forms are
what I show you here on the side
321
00:24:12 --> 00:24:15
screen.
And, in this case,
322
00:24:15 --> 00:24:20
what I did is to separate the
radial part from the angular
323
00:24:20 --> 00:24:22
part.
This first part,
324
00:24:22 --> 00:24:28
here, is the radial part of the
wave function.
325
00:24:28 --> 00:24:32
And here on the right is the
angular part of the wave
326
00:24:32 --> 00:24:35
function.
And I have written them down
327
00:24:35 --> 00:24:39
for the 1s, the 2s,
and the bottom one is the 3s,
328
00:24:39 --> 00:24:44
although I left the label off
in order to get the whole wave
329
00:24:44 --> 00:24:48
function in there.
I want you to notice,
330
00:24:48 --> 00:24:52
here, that the angular part,
the Y(lm) for the s wave
331
00:24:52 --> 00:24:56
functions, it has no theta and
phi in it.
332
00:24:56 --> 00:25:02
There is no angular dependence.
They are spherically symmetric.
333
00:25:02 --> 00:25:08
That is going to be different
for the p wave functions.
334
00:25:08 --> 00:25:13
The angular part is just one
over four pi to the one-half
335
00:25:13 --> 00:25:16
power. And
336
00:25:16 --> 00:25:21
it is the radial part,
here, that we actually are
337
00:25:21 --> 00:25:24
going to take a look at right
now.
338
00:25:24 --> 00:25:30
Let's start with that 1s wave
function, up there.
339
00:25:30 --> 00:25:33
If I plot that wavefunction,
this is Psi(1,
340
00:25:33 --> 00:25:37
0, 0), or the 1s wave function
versus r.
341
00:25:37 --> 00:25:42
Oh, I should tell you one other
thing that I didn't tell you.
342
00:25:42 --> 00:25:46
That is that throughout these
wave functions,
343
00:25:46 --> 00:25:50
you see this thing called a
nought.
344
00:25:50 --> 00:25:54
a nought is a constant.
It is called the Bohr radius.
345
00:25:54 --> 00:26:00
I will explain to you where
that comes from in a little bit
346
00:26:00 --> 00:26:04
later.
But it has a value of about
347
00:26:04 --> 00:26:08
0.529 angstroms.
That is just a constant.
348
00:26:08 --> 00:26:12
Let's plot here the (1,
0, 0) wave function.
349
00:26:12 --> 00:26:17
If I went and plotted it,
what I would find is simply
350
00:26:17 --> 00:26:22
that the wave function at r is
equal to zero,
351
00:26:22 --> 00:26:27
here, would start out at some
high finite value,
352
00:26:27 --> 00:26:33
and there would just be an
exponential decay.
353
00:26:33 --> 00:26:36
Because if you look here at the
functional form,
354
00:26:36 --> 00:26:40
what do you have?
Well, you have all this stuff,
355
00:26:40 --> 00:26:45
but that is just a constant.
And the only thing you have is
356
00:26:45 --> 00:26:49
an e to the minus r over a
nought
357
00:26:49 --> 00:26:52
dependence.
That is what gives you this
358
00:26:52 --> 00:26:57
exponential drop in the wave
function.
359
00:26:57 --> 00:27:04
What this says is that the wave
function at all values of r has
360
00:27:04 --> 00:27:10
a positive value.
Now, what about the Psi(2,
361
00:27:10 --> 00:27:15
0, 0) wave function?
Let's look at that.
362
00:27:15 --> 00:27:22
Psi(2,0,0), or the 2s wave
function as a function of r.
363
00:27:22 --> 00:27:27
What happens here?
Well, we are plotting
364
00:27:27 --> 00:27:32
essentially this.
All of this stuff is a
365
00:27:32 --> 00:27:36
constant.
And we have a two minus r over
366
00:27:36 --> 00:27:41
a nought times an e to the minus
r over 2 a nought.
367
00:27:41 --> 00:27:45
That is what we are really
368
00:27:45 --> 00:27:48
plotting here.
And, if I did that,
369
00:27:48 --> 00:27:51
it would look something like
this.
370
00:27:51 --> 00:27:54
We start at some large,
positive value here.
371
00:27:54 --> 00:28:00
And you see that the wave
function decreases.
372
00:28:00 --> 00:28:06
And it gets to a value of r
where Psi is equal to zero.
373
00:28:06 --> 00:28:12
That is a radial node.
And in the case of the 2s wave
374
00:28:12 --> 00:28:18
function, that radial node
occurs at r equals 2 a nought.
375
00:28:18 --> 00:28:22
And then the wave function
376
00:28:22 --> 00:28:27
becomes negative,
increases, and gets more and
377
00:28:27 --> 00:28:32
more negative,
until you get to a point where
378
00:28:32 --> 00:28:40
it starts increasing again and
then approaches zero.
379
00:28:40 --> 00:28:44
This part, here,
of the wave function is really
380
00:28:44 --> 00:28:49
dictated by the exponential
term, the e to the minus r over
381
00:28:49 --> 00:28:54
2 a nought.
This part of the wave function
382
00:28:54 --> 00:29:00
is dictated by this polynomial
here, two minus r over ao.
383
00:29:00 --> 00:29:04
If you wanted to solve for that
384
00:29:04 --> 00:29:09
radial node, what would you do?
You would take that functional
385
00:29:09 --> 00:29:12
form, set it equal to zero and
solve for r.
386
00:29:12 --> 00:29:16
And so you can see that 2 minus
r over a nought set equal to
387
00:29:16 --> 00:29:20
zero,
that when r is 2 a nought,
388
00:29:20 --> 00:29:24
that the wave function would
have a value of zero.
389
00:29:24 --> 00:29:29
That is
how you solve for the value of r
390
00:29:29 --> 00:29:34
at which you have a node.
Now, this is really important
391
00:29:34 --> 00:29:37
here.
That is, at the radial node,
392
00:29:37 --> 00:29:42
the wave function changes sign.
The amplitude of the wave
393
00:29:42 --> 00:29:45
function goes from positive to
negative.
394
00:29:45 --> 00:29:50
That is important because at
all nodes, for all wave
395
00:29:50 --> 00:29:53
functions, the wave function
changes sign.
396
00:29:53 --> 00:29:59
And the reason the sign of the
wave function is so important is
397
00:29:59 --> 00:30:03
in chemical bonding.
But let me back up for a
398
00:30:03 --> 00:30:06
moment.
Many of you have talked about p
399
00:30:06 --> 00:30:09
orbitals or have seen p orbitals
before.
400
00:30:09 --> 00:30:14
Sometimes on a lobe of a p
orbital, you put a plus sign and
401
00:30:14 --> 00:30:16
sometimes you put a negative
sign.
402
00:30:16 --> 00:30:18
You have seen that,
right?
403
00:30:18 --> 00:30:20
Okay.
Well, what that is just
404
00:30:20 --> 00:30:25
referring to is the sign of the
amplitude of the wave function.
405
00:30:25 --> 00:30:29
It means that in that area the
amplitude is positive,
406
00:30:29 --> 00:30:34
and in the other area the
amplitude is negative.
407
00:30:34 --> 00:30:37
And the reason the amplitudes
are so important,
408
00:30:37 --> 00:30:42
or the sign of the amplitudes
are so important is because in a
409
00:30:42 --> 00:30:46
chemical reaction,
when you are bringing two atoms
410
00:30:46 --> 00:30:51
together and your electrons that
are represented by waves are
411
00:30:51 --> 00:30:56
overlapping, if you are bringing
in two wave functions that have
412
00:30:56 --> 00:30:59
the same sign,
well, then you are going to
413
00:30:59 --> 00:31:05
have constructive interference.
And you are going to have
414
00:31:05 --> 00:31:09
chemical bonding.
If you bring in two atoms,
415
00:31:09 --> 00:31:14
and the wave functions are
overlapping and they have
416
00:31:14 --> 00:31:17
opposite signs,
you have destructive
417
00:31:17 --> 00:31:22
interference and you are not
going to have any chemical
418
00:31:22 --> 00:31:25
bonding.
That is why the sign of those
419
00:31:25 --> 00:31:30
wave functions is so important.
So, that is Psi(2,
420
00:31:30 --> 00:31:34
0, 0).
What about Psi(3,
421
00:31:34 --> 00:31:38
0, 0)?
That is the last function,
422
00:31:38 --> 00:31:45
here, on the side walls.
And let me just write down the
423
00:31:45 --> 00:31:53
radial part, 27 minus 18(r over
a nought) plus 2 times (r over a
424
00:31:53 --> 00:32:01
nought) quantity squared times e
to the minus r over 3 a nought.
425
00:32:01 --> 00:32:09
426
00:32:09 --> 00:32:14
And now, if I plotted that
function, Psi(3,0,0),
427
00:32:14 --> 00:32:19
3s wave function,
I would find out that r equals
428
00:32:19 --> 00:32:26
zero, large value of psi finite,
it drops, it crosses zero,
429
00:32:26 --> 00:32:30
gets negative,
then gets less negative,
430
00:32:30 --> 00:32:36
crosses zero again,
and then drops off.
431
00:32:36 --> 00:32:40
In the case of the 3s wave
function, we have two radial
432
00:32:40 --> 00:32:43
nodes.
We have a radial node right in
433
00:32:43 --> 00:32:47
here, and we have a radial node
right in there.
434
00:32:47 --> 00:32:51
And, if you want to know what
those radial nodes are,
435
00:32:51 --> 00:32:56
you set the wave function equal
to zero and solve for the values
436
00:32:56 --> 00:33:01
of r that make that wave
function zero.
437
00:33:01 --> 00:33:04
And, if you do that,
you would find this would come
438
00:33:04 --> 00:33:08
out to be, in terms of units of
a nought,
439
00:33:08 --> 00:33:10
1.9 a nought.
And right here,
440
00:33:10 --> 00:33:13
it would be 7.1 a nought.
The wave function,
441
00:33:13 --> 00:33:17
in the case of the 3s,
has a positive value for r less
442
00:33:17 --> 00:33:20
than 1.9 a nought,
443
00:33:20 --> 00:33:25
has a negative value from 1.9 a
nought to 7.1 a nought,
444
00:33:25 --> 00:33:29
and then a positive value again
445
00:33:29 --> 00:33:33
from 7.1 a nought to infinity.
446
00:33:33 --> 00:33:37
So, those are the wave
447
00:33:37 --> 00:33:45
functions, the functional forms,
what they actually look like.
448
00:33:45 --> 00:33:53
Now, it is time to talk about
what the wave function actually
449
00:33:53 --> 00:34:00
means, and how does it represent
the electron.
450
00:34:00 --> 00:34:12
451
00:34:12 --> 00:34:16
Well, this was,
of course, a very puzzling
452
00:34:16 --> 00:34:20
question to the scientific
community.
453
00:34:20 --> 00:34:25
As soon as S wrote down is
Schrödinger equation,
454
00:34:25 --> 00:34:30
hmm, somehow these waves
represent the particles,
455
00:34:30 --> 00:34:35
but exactly how do they
represent where the particles
456
00:34:35 --> 00:34:40
are?
And the answer to that question
457
00:34:40 --> 00:34:46
is essentially there is no
answer to that question.
458
00:34:46 --> 00:34:50
Wave functions are wave
functions.
459
00:34:50 --> 00:34:57
It is one of these concepts
that you cannot draw a classical
460
00:34:57 --> 00:35:01
analogy to.
You want to say,
461
00:35:01 --> 00:35:05
well, a wave function does
this.
462
00:35:05 --> 00:35:10
But the only way you can
describe it is in terms of
463
00:35:10 --> 00:35:17
language that is something that
you experience everyday in your
464
00:35:17 --> 00:35:22
world, so you cannot.
A wave function is a wave
465
00:35:22 --> 00:35:26
function.
I cannot draw a correct analogy
466
00:35:26 --> 00:35:32
to a classical world.
Really, that is the case.
467
00:35:32 --> 00:35:39
However, it took a very smart
gentleman by the name of Max
468
00:35:39 --> 00:35:46
Born to look at this problem.
He said, "If I take the wave
469
00:35:46 --> 00:35:52
function and I square it,
if I interpret that as a
470
00:35:52 --> 00:35:57
probability density,
then I can understand all the
471
00:35:57 --> 00:36:04
predictions made by the
Schrödinger equation within that
472
00:36:04 --> 00:36:09
framework."
In other words,
473
00:36:09 --> 00:36:16
he said, let me take Psi and l
and m as a function r,
474
00:36:16 --> 00:36:23
theta, and phi and square it.
Let me interpret that as a
475
00:36:23 --> 00:36:27
probability density.
476
00:36:27 --> 00:36:32
477
00:36:32 --> 00:36:36
Probability density is not a
probability.
478
00:36:36 --> 00:36:41
It is a density.
Density is always per unit
479
00:36:41 --> 00:36:45
volume.
Probability density is a
480
00:36:45 --> 00:36:49
probability per unit volume.
481
00:36:49 --> 00:36:54
482
00:36:54 --> 00:36:57
It is a probability per unit
volume.
483
00:36:57 --> 00:37:00
Well, if I use that
interpretation,
484
00:37:00 --> 00:37:04
then I can understand all the
predictions made by the
485
00:37:04 --> 00:37:09
Schrödinger equation.
It makes sense.
486
00:37:09 --> 00:37:13
And, you know what,
that is it.
487
00:37:13 --> 00:37:19
Because that interpretation
does agree with our
488
00:37:19 --> 00:37:25
observations,
it is therefore believed to be
489
00:37:25 --> 00:37:32
correct.
But it is just an assumption.
490
00:37:32 --> 00:37:37
It is an interpretation.
There is no derivation for it.
491
00:37:37 --> 00:37:41
It is just that the
interpretation works.
492
00:37:41 --> 00:37:45
If it works,
we therefore believe it to be
493
00:37:45 --> 00:37:48
accurate.
There is no indication,
494
00:37:48 --> 00:37:54
there are no data that seem to
contradict that interpretation,
495
00:37:54 --> 00:38:00
so we think it is right.
That is what Max Born said.
496
00:38:00 --> 00:38:03
Now, Max Born was really
something in terms of his
497
00:38:03 --> 00:38:08
scientific accomplishments.
Not only did he recognize or
498
00:38:08 --> 00:38:12
have the insight to realize what
Psi squared was,
499
00:38:12 --> 00:38:16
but he is also the Born of the
Born-Oppenheimer Approximation
500
00:38:16 --> 00:38:20
that maybe some of you have
heard about before.
501
00:38:20 --> 00:38:24
He is also the Born in the
Distorted-Wave Born
502
00:38:24 --> 00:38:27
Approximation,
which probably none of you have
503
00:38:27 --> 00:38:32
heard before.
But, despite all of those
504
00:38:32 --> 00:38:37
accomplishments,
psi squared interpretation,
505
00:38:37 --> 00:38:42
Born-Oppenheimer Approximation,
Distorted-Wave Born
506
00:38:42 --> 00:38:46
Approximation,
he is best known for being the
507
00:38:46 --> 00:38:50
grandfather of Olivia
Newton-John.
508
00:38:50 --> 00:38:54
That's right.
Oliver Newton-John is a singer
509
00:38:54 --> 00:38:59
in Grease.
Two weeks ago in the Boston
510
00:38:59 --> 00:39:04
Globe Parade Magazine,
which I actually think is a
511
00:39:04 --> 00:39:09
magazine that goes throughout
the country in all the Sunday
512
00:39:09 --> 00:39:15
newspapers, there is a long
article on Olivia Newton-John
513
00:39:15 --> 00:39:19
and a short sentence about her
grandfather, Max Born.
514
00:39:19 --> 00:39:24
So, that is our interpretation,
thanks to Max Born.
515
00:39:24 --> 00:39:29
Now, how are we going to use
that?
516
00:39:29 --> 00:39:35
Well, first of all,
let's take our functional forms
517
00:39:35 --> 00:39:40
for Psi, here,
and square it and plot those
518
00:39:40 --> 00:39:47
probability densities for the
individual wave functions and
519
00:39:47 --> 00:39:50
see what we get.
520
00:39:50 --> 00:39:55
521
00:39:55 --> 00:40:00
The way I am going to plot the
probability density is by using
522
00:40:00 --> 00:40:04
some dots.
And the density of the dots is
523
00:40:04 --> 00:40:08
going to reflect the probability
density.
524
00:40:08 --> 00:40:13
The more dense the dots,
the larger the probability
525
00:40:13 --> 00:40:16
density.
If I take that functional form
526
00:40:16 --> 00:40:21
for the 1s wave function and
square it and then plot the
527
00:40:21 --> 00:40:27
value of that function squared
with this density dot diagram,
528
00:40:27 --> 00:40:33
then you can see that the dots
here are most dense right at the
529
00:40:33 --> 00:40:37
origin, and that they
exponentially decay in all
530
00:40:37 --> 00:40:41
directions.
The probability density here
531
00:40:41 --> 00:40:45
for 1s wave function is greatest
at the origin,
532
00:40:45 --> 00:40:48
r equals 0, and it decays
exponentially in all directions.
533
00:40:48 --> 00:40:52
It is spherically symmetric.
That is what you would expect
534
00:40:52 --> 00:40:56
because that is what the wave
function looks like.
535
00:40:56 --> 00:40:59
You square that,
you get another exponential,
536
00:40:59 --> 00:41:03
and it decays exponentially in
all directions.
537
00:41:03 --> 00:41:08
That is a probability density,
probability of finding the
538
00:41:08 --> 00:41:12
electron per unit volume at some
value r, theta,
539
00:41:12 --> 00:41:15
and phi.
And it turns out it doesn't
540
00:41:15 --> 00:41:20
matter what theta and phi are
because this is spherically
541
00:41:20 --> 00:41:23
symmetric.
What about the 2s wave
542
00:41:23 --> 00:41:26
function?
Well, here is the 2s
543
00:41:26 --> 00:41:30
probability density.
Again, you can see the
544
00:41:30 --> 00:41:34
probability density is a maximum
at the origin,
545
00:41:34 --> 00:41:37
at the nucleus.
That probability density decays
546
00:41:37 --> 00:41:41
uniformly in all directions.
And it decays so much that at
547
00:41:41 --> 00:41:45
some point, you have no
probability density.
548
00:41:45 --> 00:41:47
Why?
Because that is the node.
549
00:41:47 --> 00:41:50
If you square zero,
you still get zero.
550
00:41:50 --> 00:41:52
r equals 2 a nought.
551
00:41:52 --> 00:41:57
You can see that in the
probability density.
552
00:41:57 --> 00:42:00
But then again,
as you move up this way,
553
00:42:00 --> 00:42:03
as you increase r,
the probability density
554
00:42:03 --> 00:42:04
increases again.
Why?
555
00:42:04 --> 00:42:08
Remember the wave function?
It has changed sign.
556
00:42:08 --> 00:42:11
But in this area,
here, where it is negative,
557
00:42:11 --> 00:42:15
if I square it,
well, the probability density
558
00:42:15 --> 00:42:19
still is going to be large.
Square a negative number,
559
00:42:19 --> 00:42:22
you are going to have a large
positive number.
560
00:42:22 --> 00:42:27
That is why the probability
density increases right in here,
561
00:42:27 --> 00:42:32
and then, again,
it decays towards zero.
562
00:42:32 --> 00:42:37
You can see the radial node not
only in the wave function,
563
00:42:37 --> 00:42:41
but also in the probability
density.
564
00:42:41 --> 00:42:47
Here is the probability density
for the 3s wave function.
565
00:42:47 --> 00:42:52
Once again, probability density
is a maximum at r equals 0,
566
00:42:52 --> 00:42:58
and it decays uniformly in all
directions.
567
00:42:58 --> 00:43:01
It decays so much that at some
value of r, right here,
568
00:43:01 --> 00:43:03
the probability density is
zero.
569
00:43:03 --> 00:43:05
Why?
Because the wave function is
570
00:43:05 --> 00:43:07
zero.
You square it,
571
00:43:07 --> 00:43:11
and you are going to get a zero
for the probability density.
572
00:43:11 --> 00:43:14
And then the probability
density increases again.
573
00:43:14 --> 00:43:16
Why?
Because you are getting a more
574
00:43:16 --> 00:43:20
and more negative value for the
wave function right in this
575
00:43:20 --> 00:43:21
area.
Square that,
576
00:43:21 --> 00:43:25
and it is going to increase.
And then, as you continue to
577
00:43:25 --> 00:43:30
increase r, probability density
decreases.
578
00:43:30 --> 00:43:33
It decreases again,
so that you get a zero.
579
00:43:33 --> 00:43:37
You get a zero because the wave
function is zero right there.
580
00:43:37 --> 00:43:41
This is our second radial node.
But then, the probability
581
00:43:41 --> 00:43:45
density increases again.
It increases because the wave
582
00:43:45 --> 00:43:48
function increases.
Square that,
583
00:43:48 --> 00:43:51
we are going to get a high
probability density,
584
00:43:51 --> 00:43:55
and then it tapers off.
So, the important point here is
585
00:43:55 --> 00:44:00
the interpretation of the
probability density.
586
00:44:00 --> 00:44:05
Probability per unit volume.
The fact that the s wave
587
00:44:05 --> 00:44:08
functions are all spherically
symmetric.
588
00:44:08 --> 00:44:13
They do not have an angular
dependence to them.
589
00:44:13 --> 00:44:18
And what a radial node is.
If you want to get a radial
590
00:44:18 --> 00:44:23
node, you take the wave
function, set it equal to zero,
591
00:44:23 --> 00:44:30
solve for the value of r,
and that gives you a zero.
592
00:44:30 --> 00:44:34
Now, so far,
we have talked only about the
593
00:44:34 --> 00:44:38
probability density and this
interpretation.
594
00:44:38 --> 00:44:43
We have not talked about any
probabilities yet.
595
00:44:43 --> 00:44:48
And, to do so,
we are going to talk about this
596
00:44:48 --> 00:44:52
function, here.
It is called a radial
597
00:44:52 --> 00:44:58
probability distribution.
The radial probability
598
00:44:58 --> 00:45:04
distribution is the probability
of finding an electron in a
599
00:45:04 --> 00:45:08
spherical shell.
That spherical shell will be
600
00:45:08 --> 00:45:12
some distance r away from the
nucleus.
601
00:45:12 --> 00:45:18
That spherical shell will have
a radius r and will have a
602
00:45:18 --> 00:45:20
thickness.
And the thickness,
603
00:45:20 --> 00:45:26
we are going to call dr.
This is not a solid sphere.
604
00:45:26 --> 00:45:30
This is a shell.
This is a sphere,
605
00:45:30 --> 00:45:35
but the thickness of that
sphere is very small.
606
00:45:35 --> 00:45:41
The thickness of it is dr.
And, to try to represent that a
607
00:45:41 --> 00:45:45
little bit better,
I show you here a picture of
608
00:45:45 --> 00:45:50
the probability density for the
(1, 0, 0) state.
609
00:45:50 --> 00:45:54
This is kind of my density dot
diagram.
610
00:45:54 --> 00:46:00
And then, this blue thing is my
spherical shell.
611
00:46:00 --> 00:46:04
This blue thing,
here, has a radius r,
612
00:46:04 --> 00:46:08
and this blue thing has a
thickness dr.
613
00:46:08 --> 00:46:14
And so, I am saying that the
radial probability distribution
614
00:46:14 --> 00:46:21
is going to be the probability
of finding the electron in this
615
00:46:21 --> 00:46:26
spherical shell.
That spherical shell is a
616
00:46:26 --> 00:46:33
distance r from the nucleus and
has a thickness dr.
617
00:46:33 --> 00:46:37
Now, I want to point out that
the volume of the spherical
618
00:46:37 --> 00:46:41
shell is just the surface area,
here, 4 pi r squared,
619
00:46:41 --> 00:46:44
times the thickness,
which is dr.
620
00:46:44 --> 00:46:49
Not a very thick spherical
621
00:46:49 --> 00:46:51
shell.
The radial probability
622
00:46:51 --> 00:46:56
distribution is the probability
of finding that electron in that
623
00:46:56 --> 00:47:02
spherical shell.
It is like the probability of
624
00:47:02 --> 00:47:10
finding the electron a distance
r to r plus dr
625
00:47:10 --> 00:47:14
from the nucleus.
Why is that important?
626
00:47:14 --> 00:47:21
Well, because if I want to
calculate a probability,
627
00:47:21 --> 00:47:27
what I can do then is take the
probability density here,
628
00:47:27 --> 00:47:34
Psi squared for an s orbital,
which is probability per unit
629
00:47:34 --> 00:47:43
volume, and I can then multiply
it by that unit volume.
630
00:47:43 --> 00:47:48
In this case it was the 4pi r
squared dr.
631
00:47:48 --> 00:47:53
And the result will be a
probability, because I have
632
00:47:53 --> 00:47:57
probability density,
probability per unit volume
633
00:47:57 --> 00:48:02
times a volume,
and that is a probability.
634
00:48:02 --> 00:48:09
Now we are getting somewhere in
terms of figuring out what the
635
00:48:09 --> 00:48:15
probability is of finding the
electron some distance r to r
636
00:48:15 --> 00:48:22
plus dr from the nucleus.
In the case of the s orbitals,
637
00:48:22 --> 00:48:28
I said that the Psi was a
product of the radial part and
638
00:48:28 --> 00:48:34
the Y(lm) angular part.
Remember that the Y(lm) for the
639
00:48:34 --> 00:48:40
s orbitals was always one over
the square-root of one over 4pi.
640
00:48:40 --> 00:48:46
The Y(lm) squared is going to
641
00:48:46 --> 00:48:50
cancel with 4pi,
and you are just going to have
642
00:48:50 --> 00:48:54
r squared times the radial part
squared.
643
00:48:54 --> 00:48:57
For a 1s orbital,
if you want to actually
644
00:48:57 --> 00:49:01
calculate the probability at
some value r,
645
00:49:01 --> 00:49:07
you just have to take Psi
squared and multiply it by 4pi r
646
00:49:07 --> 00:49:12
squared dr.
647
00:49:12 --> 00:49:16
However, in the case of all
other orbitals,
648
00:49:16 --> 00:49:22
you cannot do that because they
are not spherically symmetric.
649
00:49:22 --> 00:49:28
And so, for all other orbitals,
you have to take the radial
650
00:49:28 --> 00:49:34
part and multiply it by r
squared dr.
651
00:49:34 --> 00:49:38
I will explain that a little
bit more next time.
652
00:49:38 --> 49:41
Okay.
See you on Monday.