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All right.
I am going to start with
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Friday's lecture notes because
there was a significant amount
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on them that I had not finished
up yet.
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00:00:26 --> 00:00:31
We had finally gotten to the
point where we were talking
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about what does a wave function
mean, what is the physical
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significance of it and how does
it actually represent the
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presence of an electron?
And what we saw was that the
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physically significant
representation of the wave
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function, if you have some wave
function Psi labeled by three
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quantum numbers,
n, l and m.
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And, of course,
it is a function of r,
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theta and phi.
The physically significant
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quantity was this wave function
squared.
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That wave function squared,
that was interpreted as a
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probability density.
The wave function squared has
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units.
It has units of inverse volume.
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It is a density.
It is a probability per unit
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volume.
Now, as an aside,
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because someone asked me,
I should tell you that the more
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comprehensive definition of the
probability density is Psi,
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not squared,
but Psi times Psi star,
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where Psi star is the
complex conjugate.
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Because it turns out that some
wave functions are imaginary
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functions.
And so, if you took an
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imaginary function and squared
it, then you would still get an
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imaginary function after it.
And then it is hard to
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interpret an imaginary function
as a probability density.
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And so the more comprehensive
definition is Psi times Psi
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star, where Psi star is the
complex conjugate of Psi.
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00:02:37 --> 00:02:42
And, when you multiply Psi by
Psi star, if Psi is a complex
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function, well,
then you get a real function.
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This is the more comprehensive
definition of the probability
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density, Psi times Psi star.
We won't use that.
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I just wanted to let you know
about it.
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So, probability density.
Not only do we want to know
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something about the probability
density.
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00:03:06 --> 00:03:12
We also want to know something
about the probability of finding
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00:03:12 --> 00:03:16
the electron some distance away
from the nucleus.
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And, to do that,
what we were talking about was
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this quantity,
this radial distribution,
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the radial probability
distribution.
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00:03:26 --> 00:03:30
And what that is,
is the probability of finding
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an electron in a spherical shell
of radius r and distance or
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thickness dr.
For example,
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if this gray portion here
represented the probability
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density of the 1s wave function
in our dot density diagram.
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Remember, we squared the wave
function, got the probability
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density and then represented it
with a dot density diagram,
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where the density of the dots
was proportional to the value of
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the wave function squared.
And, in the case of the 1s wave
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function, we saw that the
probability density was largest
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right at r equals 0,
and that is exponentially
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decayed in all directions
uniformly.
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That is what that gray part
represents.
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But now, this blue,
here, is my spherical shell.
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It has a radius r,
and it has a thickness,
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here, dr.
And the radial probability
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distribution is asking,
what is the probability of
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finding the electron in this
spherical shell?
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And that spherical shell has a
thickness dr.
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Another way to ask that is the
probability of finding the
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electron between r and r was dr.
That is what we wanted to know,
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and that is what the radial
probability distribution tells
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us.
Now, how do you get a value out
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of that?
How do you actually calculate
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the radial probability?
Well, to do that,
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what we have to know is this
volume, here,
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of the spherical shell.
The volume of this spherical
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shell is just the surface area
of that spherical shell,
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4 pi r squared,
and the volume is times this
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thickness, this thickness dr.
It is a very thin shell.
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It is an infinitesimally thin
shell of thickness dr.
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Well, if we know that volume,
then what we can do is take our
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probability density,
Psi squared,
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which has units of probability
per unit volume.
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And we are multiplying it,
here, by our unit volume.
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The unit volumes cancel,
and we are left with a
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probability.
So, that is our probability of
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finding that electron in a shell
of radius r and a thickness dr.
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Let's look at the result of
calculating the radial
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probability distribution for the
1s wave function.
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What did I do?
I took Psi squared for the 1s
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wave function at some value of
r, then I multiplied it by 4 pi
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r squared dr,
and I did that for many
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different values of r and
plotted the result here.
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That is what that radial
probability is as a function of
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r.
Well, the first thing you see
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is that the most probable value
of r, or the value of r where
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the electron has the highest
probability of being is at this
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value, a nought.
The most probable value of r is
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this value, a nought.
a nought is what we call
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the Bohr radius.
And today, in a moment or so,
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I will tell you why it is
called the Bohr radius.
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It has a numerical value of
0.529 angstroms.
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And so it is most likely that
the electron is about a half an
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angstrom away from the nucleus,
making, then,
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the diameter of the hydrogen
atom, on the average,
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a little bit over one angstrom.
That is how we think about the
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size of a hydrogen atom,
is to take this most probable
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value of r and double it to get
the diameter.
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The most probable value of r,
or the most probable distance
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of the electron from the
nucleus, is half an angstrom
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away.
The most probable distance of
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the electron from the nucleus is
not r equals 0 because the
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radial probability here is zero
at r equals 0.
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That seems a little strange
because the other day we plotted
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the probability density for the
1s wave function.
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And, when we did that,
here is Psi(1,
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0, 0) squared versus r,
what we saw was that the
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00:09:07 --> 00:09:14
probability density was some
maximum value at r equals 0 and
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that it exponentially decayed
with increasing r.
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And that is the case.
Probability density for the s
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wave functions is a maximum at r
equals 0.
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But the radial probability here
is actually zero at r equals 0.
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Why?
Look at how we defined that
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radial probability,
here.
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It is Psi squared times this
volume element.
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Our volume element is this
spherical shell.
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And, at r equals 0,
the spherical shell goes to a
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volume of zero.
So, our radial probability here
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is equal to zero at r equals 0.
That is really important,
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that you understand that this
radial probability here is
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always going to be zero at r
equals 0 for all of the wave
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functions that we are going to
look at.
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00:10:22 --> 00:10:26
And we will talk about this a
little bit more,
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the fact that the electron is
about a half an angstrom away
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from the nucleus.
But before I do that,
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I also just want to point out
that in your textbook,
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and sometimes in the notes,
that sometimes that radial
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probability is actually written
as the following.
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It is written as the r squared,
the distance variable,
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times the radial part of the
wave function.
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That is the radial part
squared.
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We talked about the radial and
the angular part last time.
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And the radial part is labeled
only by two quantum numbers,
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n and l.
And so, for the 1s,
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that is n equals 1,
l equals 0.
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Where does this come from?
Well, let me just emphasize or
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explain where this comes from.
This radial probability
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distribution here,
we said for the s wave
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functions, was Psi squared.
You could take Psi squared,
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00:11:34 --> 00:11:39
the probability density,
and multiply it by this unit
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00:11:39 --> 00:11:44
volume or the volume of the
shell, 4 pi r squared dr.
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00:11:44 --> 00:11:48
Let's write that out again,
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00:11:48 --> 00:11:54
but write it out now so that we
write out Psi squared in terms
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00:11:54 --> 00:11:59
of the radial part and the
angular part.
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00:11:59 --> 00:12:05
Remember, we said last time,
for the hydrogen atom wave
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functions, that Psi is always a
product of a factor only an r,
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00:12:11 --> 00:12:18
which was the radial part,
and a factor only in theta and
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00:12:18 --> 00:12:21
phi, which are the angular
parts.
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00:12:21 --> 00:12:28
Now, what you also have to
remember in looking at this is
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that the angular part for the 1s
wave functions,
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2s, 3s, all s wave functions,
was equal to 1 over 4 pi to the
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1/2.
If you square that,
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you are going to get 1 over 4
pi.
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Therefore, the 4pi's here are
going to cancel for the 1s wave
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functions.
And what you are going to have
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left is this r squared times
just the radial part dr.
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That is why the y-axis in your
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book is sometimes labeled this
way for the radial probability
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distribution.
But this is also important
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00:13:17 --> 00:13:23
because if you were calculating
the radial distribution function
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for something other than an s
wave function.
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The way you would do it is to
take just the radial part of
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that wave function times r
squared, or just the radial part
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00:13:40 --> 00:13:45
of that wave function and
evaluate it at that value of r
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times r squared dr.
You could not,
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for the other wave functions,
take psi squared times 4 pi r
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squared dr.
And that is because the angular
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part for the other wave
functions that are not
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spherically symmetric is not the
square root of 1 over 4 pi.
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This is a broader definition
for what the radial probability
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distribution function is.
It just works out,
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00:14:18 --> 00:14:22
in the case for the s wave
functions, these 4pi's cancel.
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00:14:22 --> 00:14:26
And so you can write the radial
probability for the s wave
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functions like that.
So, those are just some
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00:14:30 --> 00:14:35
definitions.
I want to talk some more about
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this radial probability
distribution function,
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00:14:40 --> 00:14:47
here, for the 1s wave function.
I want to talk about it and
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also explain why a nought
is called the Bohr radius.
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00:14:54 --> 00:15:00
The reason for that is the
following.
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The nucleus was discovered in
1911, the electron was known
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00:15:04 --> 00:15:09
before that, and Schrödinger did
not write down his wave equation
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00:15:09 --> 00:15:12
until 1926.
And, in between that,
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1911 to 1926,
the scientific community was
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00:15:15 --> 00:15:20
really working very hard to try
to understand the structure of
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00:15:20 --> 00:15:23
the atom.
And we saw how the classical
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ideas, as predicted,
would live a whopping 10^-10
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seconds.
And one of the people who were
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00:15:32 --> 00:15:35
working on that problem was
Niels Bohr.
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00:15:35 --> 00:15:40
And, in 1919,
Niels Bohr of course realized
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00:15:40 --> 00:15:46
that classical physics fails
this kind of planetary model for
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00:15:46 --> 00:15:52
the atom where you put the
nucleus in the center and the
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electron is going around that
nucleus with some fixed orbit.
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We will call it r.
Well, he knew that it was not
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00:16:03 --> 00:16:06
going to work,
that those classical ideas
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00:16:06 --> 00:16:11
predicted that this would
plummet into the nucleus in
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00:16:11 --> 00:16:13
10^-10 seconds.
But, he said,
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00:16:13 --> 00:16:18
obviously, that does not
happen, so let me just forget
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00:16:18 --> 00:16:23
classical physics at the moment.
Then, what he did was to impose
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00:16:23 --> 00:16:28
some quantization on this
classical model for the hydrogen
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00:16:28 --> 00:16:32
atom.
And the reason he got this idea
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00:16:32 --> 00:16:36
of quantization is because he
already knew the hydrogen atom
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emission spectrum.
He knew that in the hydrogen
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00:16:39 --> 00:16:43
atom emission spectrum that
light of only certain
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frequencies was emitted.
That is, there was some idea
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00:16:47 --> 00:16:51
that there was something about
this hydrogen atom that is
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quantized.
He said, well,
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00:16:52 --> 00:16:56
let me just ignore classical
physics for a moment.
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00:16:56 --> 00:17:00
Let me give this a circular
orbit.
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00:17:00 --> 00:17:05
But let me quantize something
about this hydrogen atom.
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00:17:05 --> 00:17:09
And, in particular,
what he went and did was
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00:17:09 --> 00:17:14
quantized the angular momentum
of that electron.
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00:17:14 --> 00:17:19
He kind of just pasted the
quantization onto a classical
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00:17:19 --> 00:17:24
model for the atom,
because he is trying to work
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00:17:24 --> 00:17:30
toward explaining what the
observations were.
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00:17:30 --> 00:17:35
When he pasted that
quantization onto this classic
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00:17:35 --> 00:17:40
model, he was able to calculate
a value of r.
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00:17:40 --> 00:17:46
And that value of r is what we
call the Bohr radius,
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00:17:46 --> 00:17:51
a nought, and has the value
0.529 angstroms.
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That came out of it.
And if you calculate for the
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radial probability distribution
function for this model,
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00:18:06 --> 00:18:15
which is called the Bohr atom,
would be one where that radial
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00:18:15 --> 00:18:23
probability is 1 right here at r
equals a nought.
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00:18:23 --> 00:18:28
In Bohr's model,
the electron had a
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well-defined,
precise orbit.
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00:18:34 --> 00:18:39
The value of r at which it went
around the nucleus was given by
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00:18:39 --> 00:18:43
a nought.
He knew exactly where the
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00:18:43 --> 00:18:47
electron was in his model.
This kind of model,
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00:18:47 --> 00:18:51
which is this classical model,
really, is what we call
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00:18:51 --> 00:18:55
deterministic.
It is deterministic because we
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00:18:55 --> 00:19:00
know exactly where the particle,
in this case the electron,
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00:19:00 --> 00:19:04
is.
I want you to contrast it with
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00:19:04 --> 00:19:09
the quantum mechanical result
from the Schrödinger equation.
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00:19:09 --> 00:19:13
What you see,
in the quantum mechanical
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00:19:13 --> 00:19:18
result, is that we don't really
know where the electron is,
241
00:19:18 --> 00:19:21
so to speak.
The best we can tell you is a
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00:19:21 --> 00:19:26
probability of finding the
electron at some value r to r
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00:19:26 --> 00:19:32
plus dr.
That is the best we can do
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00:19:32 --> 00:19:36
because quantum mechanics is
non-deterministic.
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00:19:36 --> 00:19:40
There is a limit to which we
can know the position of a
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00:19:40 --> 00:19:43
particle.
That limit is given by
247
00:19:43 --> 00:19:47
something called the uncertainty
principle.
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00:19:47 --> 00:19:52
The uncertainty principle is
not something we are going to
249
00:19:52 --> 00:19:57
discuss, but it tells us that
there is a limit to which we can
250
00:19:57 --> 00:20:03
know both the position and the
momentum of a particle.
251
00:20:03 --> 00:20:09
And that is the basis for why,
here, we have a probability
252
00:20:09 --> 00:20:15
distribution and knowing sort of
where the electron is.
253
00:20:15 --> 00:20:20
We don't exactly know where the
electron is, here.
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00:20:20 --> 00:20:27
This is the classical model on
which Bohr just kind of pasted
255
00:20:27 --> 00:20:33
the quantization of the angular
momentum of the electron onto
256
00:20:33 --> 00:20:37
it.
In the case of the Schrödinger
257
00:20:37 --> 00:20:41
equation, the quantization drops
out when you solve the
258
00:20:41 --> 00:20:44
differential equation.
It comes out of the equation
259
00:20:44 --> 00:20:47
just naturally.
We did not paste it onto it.
260
00:20:47 --> 00:20:51
We did not make an ad hoc kind
of representation.
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00:20:51 --> 00:20:55
That is the big difference here
between quantum mechanics and
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00:20:55 --> 00:21:00
classical mechanics.
In quantum mechanics,
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00:21:00 --> 00:21:04
it can only tell you about a
probability.
264
00:21:04 --> 00:21:09
It cannot tell you exactly
where the particle is going to
265
00:21:09 --> 00:21:11
be.
Questions on that?
266
00:21:11 --> 00:21:15
Okay.
Anyway, this value a nought,
267
00:21:15 --> 00:21:20
that is why it is called
the Bohr radius.
268
00:21:20 --> 00:21:24
And then it turns out,
quantum mechanically,
269
00:21:24 --> 00:21:29
that this value of r,
the most probable value of r
270
00:21:29 --> 00:21:35
is, in fact, exactly a nought.
271
00:21:35 --> 00:21:37
In a sense, Bohr was pretty
lucky.
272
00:21:37 --> 00:21:42
And this is kind of an accident
that he got a nought out
273
00:21:42 --> 00:21:46
of this, and it has to do with
the actual form of the Coulomb
274
00:21:46 --> 00:21:49
interaction.
But, of course,
275
00:21:49 --> 00:21:53
this doesn't work for anything
else, other than a hydrogen
276
00:21:53 --> 00:21:55
atom.
Whereas, the Schrödinger
277
00:21:55 --> 00:22:00
equation, as we are going to see
in a moment, is applicable to
278
00:22:00 --> 00:22:04
all the atoms that we know
about.
279
00:22:04 --> 00:22:08
So that is the radial
probability distribution
280
00:22:08 --> 00:22:11
function for the 1s atom,
for the 1s state.
281
00:22:11 --> 00:22:17
We want to take a look at the
radial probability distribution
282
00:22:17 --> 00:22:20
for 2s and for 3s.
Let me plot those.
283
00:22:20 --> 00:22:24
And you can actually put these
lights on here.
284
00:22:24 --> 00:22:28
That is okay.
I am going to use this board
285
00:22:28 --> 00:22:33
for a moment.
Here is the radial probability
286
00:22:33 --> 00:22:38
distribution function.
I can write it as little r
287
00:22:38 --> 00:22:42
times R(2,0) squared of r,
288
00:22:42 --> 00:22:45
or RPD.
This is for 2s versus r.
289
00:22:45 --> 00:22:50
And when I do that I get a
function that looks like this.
290
00:22:50 --> 00:22:55
And, if I evaluate it here,
what is this value of r at
291
00:22:55 --> 00:23:00
which the probability is a
maximum?
292
00:23:00 --> 00:23:04
Well, this most probable value
of r is 6 a nought.
293
00:23:04 --> 00:23:06
Look at that.
294
00:23:06 --> 00:23:10
The most probable value of r
for 1s was a nought.
295
00:23:10 --> 00:23:13
In the case of the 2s state
296
00:23:13 --> 00:23:17
here, the electron,
the most probable value is 6 a
297
00:23:17 --> 00:23:20
nought, six times as far from
the nucleus.
298
00:23:20 --> 00:23:24
If you have a hydrogen atom in
the first excited state,
299
00:23:24 --> 00:23:29
in a sense that hydrogen atom
is bigger.
300
00:23:29 --> 00:23:34
It is bigger in the sense that
the probability of you finding
301
00:23:34 --> 00:23:39
the electron at a larger
distance away from the nucleus
302
00:23:39 --> 00:23:42
is larger.
And that, in general,
303
00:23:42 --> 00:23:45
is the case.
The radial probability
304
00:23:45 --> 00:23:48
distribution,
here, also reflects the radial
305
00:23:48 --> 00:23:52
node that we talked about last
time.
306
00:23:52 --> 00:23:55
That radial node is r equals a
nought.
307
00:23:55 --> 00:24:00
Radial node is the value of r
that makes your wave function go
308
00:24:00 --> 00:24:04
to zero.
Notice, again,
309
00:24:04 --> 00:24:10
that this radial probability
distribution function right here
310
00:24:10 --> 00:24:14
is zero at r equals 0.
This is not a node.
311
00:24:14 --> 00:24:19
This is not a radial node.
This is a consequence,
312
00:24:19 --> 00:24:25
right here, of our definition
for the radial probability.
313
00:24:25 --> 00:24:30
Our volume element has gone to
zero.
314
00:24:30 --> 00:24:36
r equals 0 is never a radial
node in any wave function.
315
00:24:36 --> 00:24:40
What about 3s?
Well, let's plot 3s.
316
00:24:40 --> 00:24:44
Here is 3s.
This is the radial probability
317
00:24:44 --> 00:24:49
distribution.
I take Psi for 3s and square
318
00:24:49 --> 00:24:55
it, multiply by 4 pi r squared
dr,
319
00:24:55 --> 00:25:02
and do so for all the values of
r, and I am going to get
320
00:25:02 --> 00:25:10
something that looks like this.
Now this most probable value of
321
00:25:10 --> 00:25:16
r here, where the 3s wave
function is equal to 11.468 a
322
00:25:16 --> 00:25:22
nought.
For the second excited state of
323
00:25:22 --> 00:25:25
a hydrogen atom,
that electron,
324
00:25:25 --> 00:25:30
on the average,
is 11.5 times farther out from
325
00:25:30 --> 00:25:38
the nucleus than it is in the
case of the 1s state right here.
326
00:25:38 --> 00:25:43
Again, for that second excited
state, that hydrogen atom is
327
00:25:43 --> 00:25:49
bigger in the sense that the
probability of it being farther
328
00:25:49 --> 00:25:54
away from the nucleus is larger.
That radial probability
329
00:25:54 --> 00:26:00
distribution of the 3s also
reflects the two radial nodes in
330
00:26:00 --> 00:26:07
the 3s wave function.
The radial nodes are at 1.9 a
331
00:26:07 --> 00:26:11
nought, here,
and 7.1 a nought.
332
00:26:11 --> 00:26:19
Again, the value here at r
equals 0 is not a radial node.
333
00:26:19 --> 00:26:27
Now, as you look at this,
it is tempting to ask the
334
00:26:27 --> 00:26:33
following question.
You might want to ask,
335
00:26:33 --> 00:26:40
if the electron can be at these
values of r, and it can be at
336
00:26:40 --> 00:26:46
these values of r,
and it can be at these values
337
00:26:46 --> 00:26:53
of r, how does the electron
actually get from here to here
338
00:26:53 --> 00:27:00
to here if right at r equals 1.9
a nought and 7.1 a nought the
339
00:27:00 --> 00:27:08
probability is equal to zero?
Well, you might say maybe this
340
00:27:08 --> 00:27:14
probability isn't exactly zero.
It is something small.
341
00:27:14 --> 00:27:20
But I am telling you that it is
zero, goose egg,
342
00:27:20 --> 00:27:23
zilch, zippo,
nada, cipher,
343
00:27:23 --> 00:27:28
nix, nought.
Anybody else have another name?
344
00:27:28 --> 00:27:31
Nil.
It is nothing.
345
00:27:31 --> 00:27:34
It is zero.
How do you answer that
346
00:27:34 --> 00:27:37
question?
Well, it turns out,
347
00:27:37 --> 00:27:42
of course, that it isn't an
appropriate question.
348
00:27:42 --> 00:27:48
And the reason it is not is
because that question is asked
349
00:27:48 --> 00:27:52
in the framework of classical
mechanics.
350
00:27:52 --> 00:27:56
When you ask,
how does a particle get from
351
00:27:56 --> 00:28:01
one place to another,
you are asking about a
352
00:28:01 --> 00:28:06
trajectory.
You are asking about a path.
353
00:28:06 --> 00:28:10
Particles over here,
over here, over here,
354
00:28:10 --> 00:28:14
how does it get from one place
to another?
355
00:28:14 --> 00:28:18
And, in quantum mechanics,
we don't have the concept of
356
00:28:18 --> 00:28:22
trajectories.
Instead, what we have to think
357
00:28:22 --> 00:28:28
of is the electron as a wave.
And we already know that a wave
358
00:28:28 --> 00:28:32
can have amplitude
simultaneously at many different
359
00:28:32 --> 00:28:37
positions.
And so it has simultaneous
360
00:28:37 --> 00:28:42
amplitude or probability here,
here, and here,
361
00:28:42 --> 00:28:46
all at the same time.
We cannot talk about
362
00:28:46 --> 00:28:50
trajectories anymore.
And that, again,
363
00:28:50 --> 00:28:56
ties into the uncertainty
principle, our inability to know
364
00:28:56 --> 00:29:02
exactly the position and the
momentum of a particle at any
365
00:29:02 --> 00:29:07
given instance.
The best we can tell you is a
366
00:29:07 --> 00:29:11
probability.
We have to change the way we
367
00:29:11 --> 00:29:16
think about electrons.
You cannot cast them in the
368
00:29:16 --> 00:29:19
framework of your everyday
world.
369
00:29:19 --> 00:29:23
This is part of our world,
but you have to go do a
370
00:29:23 --> 00:29:30
specific type of experiment to
see this part of the world.
371
00:29:30 --> 00:29:35
That is why it seems so strange
to you, because it is not part
372
00:29:35 --> 00:29:40
of your everyday experience.
But this world works with
373
00:29:40 --> 00:29:45
different rules that you really
do have to accept that it just
374
00:29:45 --> 00:29:49
works differently.
Questions?
375
00:29:49 --> 00:30:00
376
00:30:00 --> 00:30:06
Now, I am going to stop talking
about the s wave functions and
377
00:30:06 --> 00:30:11
move on to talk about the p wave
functions.
378
00:30:11 --> 00:30:16
With the s wave functions,
we talked about the
379
00:30:16 --> 00:30:22
significance of the wave
function, probability density,
380
00:30:22 --> 00:30:26
radial probability
distribution.
381
00:30:26 --> 00:30:32
We talked about what a radial
node was.
382
00:30:32 --> 00:30:36
Now it is time to move onto the
p wave functions.
383
00:30:36 --> 00:30:41
And the p wave functions,
of course, are not spherically
384
00:30:41 --> 00:30:44
symmetric.
And to represent them,
385
00:30:44 --> 00:30:48
we are going to do our dot
density diagram again.
386
00:30:48 --> 00:30:54
We are going to take the wave
function and square it to get
387
00:30:54 --> 00:31:00
the probability density and then
plot that probability density as
388
00:31:00 --> 00:31:05
a density of dots.
We the dots are most dense,
389
00:31:05 --> 00:31:10
well, that means the highest
probability density.
390
00:31:10 --> 00:31:14
Here is the result for the pz
wave function.
391
00:31:14 --> 00:31:19
It is pz because you can see
the highest probability,
392
00:31:19 --> 00:31:23
here, is along the z-axis.
It is symmetric along the
393
00:31:23 --> 00:31:27
z-axis.
Here is the probability density
394
00:31:27 --> 00:31:32
for the px wave function.
You can see that the
395
00:31:32 --> 00:31:36
probability density is greatest
along the x-axis.
396
00:31:36 --> 00:31:39
It is symmetric along the
x-axis.
397
00:31:39 --> 00:31:43
And, if you look really
carefully, you can see that
398
00:31:43 --> 00:31:48
there is no probability density
in the y,z-plane for the px wave
399
00:31:48 --> 00:31:49
function.
And, over here,
400
00:31:49 --> 00:31:53
if you look carefully,
you can see that there is no
401
00:31:53 --> 00:31:58
probability density in the
x,y-plane for the pz wave
402
00:31:58 --> 00:32:03
function.
And here is a py wave function,
403
00:32:03 --> 00:32:08
the probability density of it.
The probability density is
404
00:32:08 --> 00:32:13
concentrated along the y-axis.
It is symmetric along the
405
00:32:13 --> 00:32:16
y-axis.
And, if you look very
406
00:32:16 --> 00:32:20
carefully, there is no
probability density,
407
00:32:20 --> 00:32:25
here, in the x,z-plane.
Well, the fact that there is no
408
00:32:25 --> 00:32:30
probability density,
here, in the x,y-plane,
409
00:32:30 --> 00:32:34
in the case of pz,
indicates that we have an
410
00:32:34 --> 00:32:39
angular node.
An angular node at theta equal
411
00:32:39 --> 00:32:42
90 degrees.
An angular node is the same
412
00:32:42 --> 00:32:46
thing as a radial node in the
sense that it is the value of
413
00:32:46 --> 00:32:51
the angle that makes the wave
function be equal to zero.
414
00:32:51 --> 00:32:53
Here is the wave function for
pz.
415
00:32:53 --> 00:32:57
You can see that when theta is
equal to zero,
416
00:32:57 --> 00:33:02
this wave function is going to
be equal to zero.
417
00:33:02 --> 00:33:08
An angular node is the value of
theta or phi that makes the wave
418
00:33:08 --> 00:33:12
function be zero.
And the consequence,
419
00:33:12 --> 00:33:19
then, is that we have a nodal
plane, because everywhere on the
420
00:33:19 --> 00:33:23
x,y-plane, theta is equal to 90
degrees.
421
00:33:23 --> 00:33:29
For the px wave function,
the value of the angle that
422
00:33:29 --> 00:33:35
gives you that nodal plane is
phi equals 90.
423
00:33:35 --> 00:33:41
That means everywhere in the
y,z-plane is phi equal to 90.
424
00:33:41 --> 00:33:46
In the case of py,
when phi is equal to zero,
425
00:33:46 --> 00:33:50
well, that is everywhere in the
x,z-plane.
426
00:33:50 --> 00:33:55
Everywhere in the x,z-plane,
phi is equal to zero.
427
00:33:55 --> 00:34:02
So, that is the angular nodes.
In general, and this is
428
00:34:02 --> 00:34:07
something you do have to know,
an orbital has n minus 1
429
00:34:07 --> 00:34:12
total nodes.
And what I mean by total nodes
430
00:34:12 --> 00:34:17
is angular plus radial nodes.
The number of angular nodes is
431
00:34:17 --> 00:34:20
given by this quantity,
l.
432
00:34:20 --> 00:34:25
The quantum number l that
labels your wave function always
433
00:34:25 --> 00:34:30
gives you the number of angular
nodes.
434
00:34:30 --> 00:34:34
Therefore, if n minus 1 is the
total and l is the number of
435
00:34:34 --> 00:34:38
angular, well then,
the number of radial nodes is n
436
00:34:38 --> 00:34:42
minus 1 minus l. This is
437
00:34:42 --> 00:34:44
something that you do have to
know.
438
00:34:44 --> 00:34:49
If I give you a wave function
and ask you how many radial and
439
00:34:49 --> 00:34:52
angular nodes it has,
you need to be able to
440
00:34:52 --> 00:34:56
calculate that,
and vice versa.
441
00:34:56 --> 00:35:02
Sometimes I will tell you a
function has three radial nodes
442
00:35:02 --> 00:35:07
and six or seven angular nodes
or something,
443
00:35:07 --> 00:35:12
what is the wave function?
So, we go both ways.
444
00:35:12 --> 00:35:19
Well, I also want to take a
look at the radial probability
445
00:35:19 --> 00:35:24
distribution functions for the p
wave functions.
446
00:35:24 --> 00:35:31
We looked at it for the s wave
functions already.
447
00:35:31 --> 00:35:36
I actually want to contrast the
radial probability distribution,
448
00:35:36 --> 00:35:40
say, for 2p,
here it is, with that of 2s
449
00:35:40 --> 00:35:45
that we looked at a moment ago.
Remember, how do you get the
450
00:35:45 --> 00:35:50
radial probability distribution
function here for 2p?
451
00:35:50 --> 00:35:55
It is the radial part of the 2p
wave function times r squared
452
00:35:55 --> 00:35:58
dr.
It gives me the probability of
453
00:35:58 --> 00:36:05
finding the electron a distance
between r and r plus dr.
454
00:36:05 --> 00:36:10
Again, what you see is that at
r equals 0, that is zero.
455
00:36:10 --> 00:36:15
That is not a radial node.
But what I really want to point
456
00:36:15 --> 00:36:20
out here is that the most
probable value of r,
457
00:36:20 --> 00:36:25
for the 2p wave function,
is actually smaller than it is
458
00:36:25 --> 00:36:31
for the 2s wave function.
That is, it is more likely for
459
00:36:31 --> 00:36:37
the electron in a 2p state to be
a little closer in to the
460
00:36:37 --> 00:36:40
nucleus than it is for the 2s
state.
461
00:36:40 --> 00:36:46
In general, as you increase the
angular momentum quantum number,
462
00:36:46 --> 00:36:51
the most probable value of r
gets smaller for the same value
463
00:36:51 --> 00:36:54
of n.
Similarly, here is the 3s
464
00:36:54 --> 00:37:01
radial probability distribution
function that we looked at.
465
00:37:01 --> 00:37:05
Here is a radial probability
distribution for 3p.
466
00:37:05 --> 00:37:08
Now, with the 3p,
you can see the value of the
467
00:37:08 --> 00:37:11
radial node.
You can see the radial
468
00:37:11 --> 00:37:15
probability distribution
reflects a radial node,
469
00:37:15 --> 00:37:17
here.
And here is the radial
470
00:37:17 --> 00:37:21
probability distribution
function for 3d.
471
00:37:21 --> 00:37:25
We did not look at the
probability density of 3d.
472
00:37:25 --> 00:37:30
You will do that with Professor
Cummins when you talk about
473
00:37:30 --> 00:37:35
transition metals.
But here, I just drew in the
474
00:37:35 --> 00:37:39
radial probability distribution
for 3d.
475
00:37:39 --> 00:37:44
But the point again that I want
to make is here is the most
476
00:37:44 --> 00:37:48
probable value of r for 3s,
here it is for 3p,
477
00:37:48 --> 00:37:53
here it is for 3d,
again, the most probable value
478
00:37:53 --> 00:37:59
for 3d is smaller than it is for
3p, than it is for 3s.
479
00:37:59 --> 00:38:05
Again, as you increase the
angular momentum quantum number,
480
00:38:05 --> 00:38:09
that most probable value gets
smaller.
481
00:38:09 --> 00:38:14
However, ironically,
if you actually look at the
482
00:38:14 --> 00:38:20
probability of the electron
being very, very close to the
483
00:38:20 --> 00:38:26
nucleus, that probability is
only significant for the s wave
484
00:38:26 --> 00:38:31
functions.
Look at the 3s wave function.
485
00:38:31 --> 00:38:36
Here, you see that you really
do have some probability very
486
00:38:36 --> 00:38:40
close to the nucleus.
You don't see that in the 3p
487
00:38:40 --> 00:38:43
wave function.
You certainly don't see that in
488
00:38:43 --> 00:38:48
the 3d wave function.
Again, in the 2s wave function,
489
00:38:48 --> 00:38:52
you have some significant
probability of the electron
490
00:38:52 --> 00:38:55
being really close to the
nucleus in 2s,
491
00:38:55 --> 00:39:00
but you don't in 2p.
That is important.
492
00:39:00 --> 00:39:05
And it seems in contradiction
to the fact that on the average,
493
00:39:05 --> 00:39:10
the most probable value of r
gets smaller as l gets larger.
494
00:39:10 --> 00:39:14
These two facts that look
contradictory are important.
495
00:39:14 --> 00:39:17
They dictate the behavior of
atoms.
496
00:39:17 --> 00:39:22
These two facts seem like kind
of loose threads at the moment
497
00:39:22 --> 00:39:27
in the sense that you are
probably wondering why I am
498
00:39:27 --> 00:39:31
telling you what I am telling
you.
499
00:39:31 --> 00:39:36
But we are going to use that
information in a few days,
500
00:39:36 --> 00:39:41
and you will see really the
significance of this plot.
501
00:39:41 --> 00:39:47
And this plot will be an
important one for you to refer
502
00:39:47 --> 00:39:49
back to.
Yes?
503
00:39:49 --> 00:40:07
504
00:40:07 --> 00:40:09
Probably.
I am not exactly sure of the
505
00:40:09 --> 00:40:12
picture you drew in high school,
but yes.
506
00:40:12 --> 00:40:17
If the electron in general is
further out from the nucleus,
507
00:40:17 --> 00:40:21
that is a higher energy state.
The electron is less strongly
508
00:40:21 --> 00:40:27
bound, as we are going to see in
the multi-electron atoms here.
509
00:40:27 --> 00:40:35
510
00:40:35 --> 00:40:38
Oh, no.
For the hydrogen no.
511
00:40:38 --> 00:40:42
Let me explain that.
For the hydrogen atom,
512
00:40:42 --> 00:40:48
the energies are only dictated
by the n quantum number,
513
00:40:48 --> 00:40:53
so 3s, 3p, 3d all have the same
energies.
514
00:40:53 --> 00:40:58
Where the energies become
degenerate is with a
515
00:40:58 --> 00:41:05
multi-electron atom.
And we are going to talk about
516
00:41:05 --> 00:41:11
that and how that reflects here,
these wave functions in the
517
00:41:11 --> 00:41:16
next day.
That is all I am going to say
518
00:41:16 --> 00:41:21
about the hydrogen atom.
Now it is time to move on,
519
00:41:21 --> 00:41:24
to helium.
And, of course,
520
00:41:24 --> 00:41:30
the Schrödinger equation
predicts the binding energies of
521
00:41:30 --> 00:41:39
the electrons to the nucleus in
a helium atom also very well.
522
00:41:39 --> 00:41:42
But, of course,
it is a much more complicated
523
00:41:42 --> 00:41:46
Schrödinger equation.
And I am not even going to
524
00:41:46 --> 00:41:50
write out the Hamiltonian in
this case, but I want to show
525
00:41:50 --> 00:41:54
you the wave function here.
See the wave function?
526
00:41:54 --> 00:41:59
The wave function is a function
of six variables.
527
00:41:59 --> 00:42:04
It is a function of two r's,
two distances from the nucleus,
528
00:42:04 --> 00:42:07
one for electron one,
one for electron two,
529
00:42:07 --> 00:42:12
two theta's and two phi's.
We have six variables for the
530
00:42:12 --> 00:42:16
wave function.
And the consequence of this is
531
00:42:16 --> 00:42:20
that our solutions for the
binding energies for the
532
00:42:20 --> 00:42:26
electrons in helium or any other
atoms are not going to be nice
533
00:42:26 --> 00:42:31
analytical forms.
We are no longer going to have
534
00:42:31 --> 00:42:35
e sub n equal minus the Rydberg
constant over n squared.
535
00:42:35 --> 00:42:39
If you
actually solve for those
536
00:42:39 --> 00:42:43
energies, and you have to do it
numerically, you are just going
537
00:42:43 --> 00:42:46
to get a list of numbers,
a table of numbers,
538
00:42:46 --> 00:42:50
but not a nice analytical form.
If you solve for the wave
539
00:42:50 --> 00:42:55
function, you are not going to
get a nice analytical form,
540
00:42:55 --> 00:42:59
like we got for hydrogen.
Instead, what you will get is a
541
00:42:59 --> 00:43:02
value for the amplitude of Psi
as a function of r,
542
00:43:02 --> 00:43:08
theta and phi.
But if you get actually much
543
00:43:08 --> 00:43:12
above three electrons,
it turns out that even
544
00:43:12 --> 00:43:18
numerically, you cannot solve
the Schrödinger equation,
545
00:43:18 --> 00:43:23
exactly.
You have to use approximations.
546
00:43:23 --> 00:43:30
And we are going to look at the
most basic approximation that is
547
00:43:30 --> 00:43:34
used that works,
amazingly.
548
00:43:34 --> 00:43:39
It works well enough for us to
have a framework in which to
549
00:43:39 --> 00:43:42
understand the reactions of
these atoms.
550
00:43:42 --> 00:43:47
And what is that approximation?
Well, that approximation is
551
00:43:47 --> 00:43:52
called the one-electron wave
approximation or the
552
00:43:52 --> 00:43:55
one-electron orbital
approximation.
553
00:43:55 --> 00:44:00
What does that mean?
Well, that means this.
554
00:44:00 --> 00:44:05
I am going to take my wave
function here for the helium
555
00:44:05 --> 00:44:10
atom, which strictly is a wave
function that is a function of
556
00:44:10 --> 00:44:15
six variables,
and I am going to separate it.
557
00:44:15 --> 00:44:20
I am going to let electron one
have its own wave function and
558
00:44:20 --> 00:44:24
electron two have its own wave
function.
559
00:44:24 --> 00:44:28
That is an approximation.
In addition,
560
00:44:28 --> 00:44:33
what I am going to do is let
the wave function for electron
561
00:44:33 --> 00:44:39
one have a hydrogen-like wave
function.
562
00:44:39 --> 00:44:43
I am going to say that it has
the 1s wave function,
563
00:44:43 --> 00:44:46
or the Psi(1,
0, 0) wave function of a
564
00:44:46 --> 00:44:49
hydrogen atom.
And I am going to let electron
565
00:44:49 --> 00:44:53
two have the Psi(1,
0, 0) wave function of a
566
00:44:53 --> 00:44:56
hydrogen atom.
Or, I am going to write it as
567
00:44:56 --> 00:45:00
1s of 1, for electron one,
times 1s of 2,
568
00:45:00 --> 00:45:04
for electron two.
569
00:45:04 --> 00:45:07
Or, another shorthand,
I am going to write it as 1s.
570
00:45:07 --> 00:45:09
squared.
And, if I continued on,
571
00:45:09 --> 00:45:13
here, it is for lithium.
Lithium, the wave function
572
00:45:13 --> 00:45:17
strictly has nine coordinates,
but I am going to let every one
573
00:45:17 --> 00:45:19
of those electrons,
in the one electron wave
574
00:45:19 --> 00:45:22
approximation,
have its own wave function.
575
00:45:22 --> 00:45:26
And I am going to let electron
one have a wave function that
576
00:45:26 --> 00:45:30
looks like a hydrogen atom
wavefunction.
577
00:45:30 --> 00:45:33
The 1s wave function.
The same thing with electron
578
00:45:33 --> 00:45:35
two.
And then I am going to let
579
00:45:35 --> 00:45:40
electron three have the 2s wave
function of the hydrogen atom.
580
00:45:40 --> 00:45:44
And in simplified notation,
that is just 1s squared 2s.
581
00:45:44 --> 00:45:47
And here is
beryllium, 16 variables,
582
00:45:47 --> 00:45:51
but I am going to let every
electron have its own wave
583
00:45:51 --> 00:45:54
function.
And I am going to give electron
584
00:45:54 --> 00:45:56
one the 1s wave function,
electron two,
585
00:45:56 --> 00:46:00
the 1s, electron three,
the 2s, electron four,
586
00:46:00 --> 00:46:03
the 2s.
I can also write that,
587
00:46:03 --> 00:46:07
as you have already done,
1s 2 2s 2.
588
00:46:07 --> 00:46:10
And I can keep going.
And these electron
589
00:46:10 --> 00:46:14
configurations that you have
been writing down in high
590
00:46:14 --> 00:46:18
school, that is what they are,
electron configurations,
591
00:46:18 --> 00:46:23
well, they are nothing more
than our shorthand notation for
592
00:46:23 --> 00:46:27
the electron wave functions
within this one-electron wave
593
00:46:27 --> 00:46:32
approximation.
That is what those were,
594
00:46:32 --> 00:46:37
that you were writing down.
Those were a shorthand notation
595
00:46:37 --> 00:46:42
for the wave functions in
Schrödinger's equation within
596
00:46:42 --> 00:46:46
this one-electron wave
approximation.
597
00:46:46 --> 00:46:50
Now, one thing you do notice is
that I did not,
598
00:46:50 --> 00:46:55
in the case of boron here,
let all five electrons be in
599
00:46:55 --> 00:46:59
the 1s state,
or let all five electrons be
600
00:46:59 --> 00:47:05
represented by a 1s hydrogen
atom wave function.
601
00:47:05 --> 00:47:10
I didn't because of a quantity
that you already know about,
602
00:47:10 --> 00:47:14
called spin.
You already know that if you
603
00:47:14 --> 00:47:19
are going to put electrons in
the 1s state here that one
604
00:47:19 --> 00:47:25
electron has to go in with spin
up and the other spin down.
605
00:47:25 --> 00:47:30
And the 2s, spin up and spin
down, etc.
606
00:47:30 --> 00:47:34
What is the phenomenon called
spin?
607
00:47:34 --> 00:47:42
Well, spin is entirely a
quantum mechanical phenomenon.
608
00:47:42 --> 00:47:48
There is no correct classical
analogy to spin.
609
00:47:48 --> 00:47:53
Spin is intrinsic angular
momentum.
610
00:47:53 --> 00:48:00
It is angular momentum that is
just part of a particle,
611
00:48:00 --> 00:48:07
such as an electron.
The spin quantum numbers
612
00:48:07 --> 00:48:11
actually come from solving the
relativistic Schrödinger
613
00:48:11 --> 00:48:15
equation, which we did not even
write down.
614
00:48:15 --> 00:48:20
When you solve the relativistic
Schrödinger equation,
615
00:48:20 --> 00:48:23
out drops a fourth quantum
number.
616
00:48:23 --> 00:48:28
That fourth quantum number we
are going to call m sub s.
617
00:48:28 --> 00:48:31
And we find that m sub s
618
00:48:31 --> 00:48:36
has two allowed values.
One of those values is one-half
619
00:48:36 --> 00:48:41
and the other is minus one-half.
Here, we have a case where the
620
00:48:41 --> 00:48:44
quantum number is not an
integer.
621
00:48:44 --> 00:48:47
It is one-half and it is minus
one-half.
622
00:48:47 --> 00:48:51
Now, if it helps you to think
about the electron spinning
623
00:48:51 --> 00:48:54
around its own axis,
like I depict here,
624
00:48:54 --> 00:48:58
well, if that is the case,
then the angular momentum
625
00:48:58 --> 00:49:03
quantum number is perpendicular,
here ,to this plane in which it
626
00:49:03 --> 00:49:08
is rotating.
And you might want to call that
627
00:49:08 --> 00:49:10
spin up.
And, of course,
628
00:49:10 --> 00:49:14
if it is spinning in the other
direction, well,
629
00:49:14 --> 00:49:17
then the angular momentum
vector is pointed in the
630
00:49:17 --> 00:49:21
opposite direction.
You might want to call this
631
00:49:21 --> 00:49:24
spin down.
If it helps for you to think
632
00:49:24 --> 00:49:28
about this, okay,
but remember that this is not
633
00:49:28 --> 00:49:32
correct.
This is a classical analogy
634
00:49:32 --> 00:49:37
that we are trying to draw here.
We are trying to say that this
635
00:49:37 --> 00:49:40
electron is rotating around its
own axis.
636
00:49:40 --> 00:49:44
That is not true.
This angular momentum is just
637
00:49:44 --> 00:49:47
an intrinsic part,
the intrinsic nature of a
638
00:49:47 --> 00:49:52
particular such as an electron.
Next time, I will tell you
639
00:49:52.099 --> 49:55
about Uhlenbeck and Goudsmith.
See you Wednesday.