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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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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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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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We also want to know something
about the probability of finding
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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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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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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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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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00:11:01,000 --> 00:11:06,000
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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the probability density,
and multiply it by this unit
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00:11:39,000 --> 00:11:44,000
volume or the volume of the
shell, 4 pi r squared dr.
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00:11:44,000 --> 00:11:48,000
Let's write that out again,
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00:11:48,000 --> 00:11:54,000
but write it out now so that we
write out Psi squared in terms
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of the radial part and the
angular part.
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00:11:59,000 --> 00:12:05,000
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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which was the radial part,
and a factor only in theta and
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phi, which are the angular
parts.
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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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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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00:13:34,000 --> 00:13:40,000
that wave function times r
squared, or just the radial part
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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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00:14:10,000 --> 00:14:14,000
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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in the case for the s wave
functions, these 4pi's cancel.
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00:14:22,000 --> 00:14:26,000
And so you can write the radial
probability for the s wave
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00:14:26,000 --> 00:14:30,000
functions like that.
So, those are just some
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definitions.
I want to talk some more about
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00:14:35,000 --> 00:14:40,000
this radial probability
distribution function,
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00:14:40,000 --> 00:14:47,000
here, for the 1s wave function.
I want to talk about it and
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00:14:47,000 --> 00:14:54,000
also explain why a nought
is called the Bohr radius.
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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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00:15:04,000 --> 00:15:09,000
before that, and Schrˆdinger did
not write down his wave equation
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until 1926.
And, in between that,
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00:15:12,000 --> 00:15:15,000
1911 to 1926,
the scientific community was
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00:15:15,000 --> 00:15:20,000
really working very hard to try
to understand the structure of
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00:15:20,000 --> 00:15:23,000
the atom.
And we saw how the classical
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00:15:23,000 --> 00:15:27,000
ideas, as predicted,
would live a whopping 10^-10
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00:15:27,000 --> 00:15:32,000
seconds.
And one of the people who were
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00:15:32,000 --> 00:15:35,000
working on that problem was
Niels Bohr.
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And, in 1919,
Niels Bohr of course realized
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00:15:40,000 --> 00:15:46,000
that classical physics fails
this kind of planetary model for
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00:15:46,000 --> 00:15:52,000
the atom where you put the
nucleus in the center and the
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00:15:52,000 --> 00:15:58,000
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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00:16:03,000 --> 00:16:06,000
going to work,
that those classical ideas
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00:16:06,000 --> 00:16:11,000
predicted that this would
plummet into the nucleus in
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00:16:11,000 --> 00:16:13,000
10^-10 seconds.
But, he said,
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00:16:13,000 --> 00:16:18,000
obviously, that does not
happen, so let me just forget
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00:16:18,000 --> 00:16:23,000
classical physics at the moment.
Then, what he did was to impose
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00:16:23,000 --> 00:16:28,000
some quantization on this
classical model for the hydrogen
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00:16:28,000 --> 00:16:32,000
atom.
And the reason he got this idea
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00:16:32,000 --> 00:16:36,000
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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00:16:39,000 --> 00:16:43,000
atom emission spectrum that
light of only certain
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00:16:43,000 --> 00:16:47,000
frequencies was emitted.
That is, there was some idea
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00:16:47,000 --> 00:16:51,000
that there was something about
this hydrogen atom that is
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00:16:51,000 --> 00:16:52,000
quantized.
He said, well,
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00:16:52,000 --> 00:16:56,000
let me just ignore classical
physics for a moment.
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00:16:56,000 --> 00:17:00,000
Let me give this a circular
orbit.
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00:17:00,000 --> 00:17:05,000
But let me quantize something
about this hydrogen atom.
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00:17:05,000 --> 00:17:09,000
And, in particular,
what he went and did was
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00:17:09,000 --> 00:17:14,000
quantized the angular momentum
of that electron.
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00:17:14,000 --> 00:17:19,000
He kind of just pasted the
quantization onto a classical
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00:17:19,000 --> 00:17:24,000
model for the atom,
because he is trying to work
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00:17:24,000 --> 00:17:30,000
toward explaining what the
observations were.
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00:17:30,000 --> 00:17:35,000
When he pasted that
quantization onto this classic
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00:17:35,000 --> 00:17:40,000
model, he was able to calculate
a value of r.
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00:17:40,000 --> 00:17:46,000
And that value of r is what we
call the Bohr radius,
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00:17:46,000 --> 00:17:51,000
a nought, and has the value
0.529 angstroms.
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00:17:51,000 --> 00:17:59,000
That came out of it.
And if you calculate for the
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00:17:59,000 --> 00:18:06,000
radial probability distribution
function for this model,
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00:18:06,000 --> 00:18:15,000
which is called the Bohr atom,
would be one where that radial
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00:18:15,000 --> 00:18:23,000
probability is 1 right here at r
equals a nought.
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00:18:23,000 --> 00:18:28,000
In Bohr's model,
the electron had a
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00:18:28,000 --> 00:18:34,000
well-defined,
precise orbit.
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00:18:34,000 --> 00:18:39,000
The value of r at which it went
around the nucleus was given by
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00:18:39,000 --> 00:18:43,000
a nought.
He knew exactly where the
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00:18:43,000 --> 00:18:47,000
electron was in his model.
This kind of model,
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00:18:47,000 --> 00:18:51,000
which is this classical model,
really, is what we call
235
00:18:51,000 --> 00:18:55,000
deterministic.
It is deterministic because we
236
00:18:55,000 --> 00:19:00,000
know exactly where the particle,
in this case the electron,
237
00:19:00,000 --> 00:19:04,000
is.
I want you to contrast it with
238
00:19:04,000 --> 00:19:09,000
the quantum mechanical result
from the Schrˆdinger equation.
239
00:19:09,000 --> 00:19:13,000
What you see,
in the quantum mechanical
240
00:19:13,000 --> 00:19:18,000
result, is that we don't really
know where the electron is,
241
00:19:18,000 --> 00:19:21,000
so to speak.
The best we can tell you is a
242
00:19:21,000 --> 00:19:26,000
probability of finding the
electron at some value r to r
243
00:19:26,000 --> 00:19:32,000
plus dr.
That is the best we can do
244
00:19:32,000 --> 00:19:36,000
because quantum mechanics is
non-deterministic.
245
00:19:36,000 --> 00:19:40,000
There is a limit to which we
can know the position of a
246
00:19:40,000 --> 00:19:43,000
particle.
That limit is given by
247
00:19:43,000 --> 00:19:47,000
something called the uncertainty
principle.
248
00:19:47,000 --> 00:19:52,000
The uncertainty principle is
not something we are going to
249
00:19:52,000 --> 00:19:57,000
discuss, but it tells us that
there is a limit to which we can
250
00:19:57,000 --> 00:20:03,000
know both the position and the
momentum of a particle.
251
00:20:03,000 --> 00:20:09,000
And that is the basis for why,
here, we have a probability
252
00:20:09,000 --> 00:20:15,000
distribution and knowing sort of
where the electron is.
253
00:20:15,000 --> 00:20:20,000
We don't exactly know where the
electron is, here.
254
00:20:20,000 --> 00:20:27,000
This is the classical model on
which Bohr just kind of pasted
255
00:20:27,000 --> 00:20:33,000
the quantization of the angular
momentum of the electron onto
256
00:20:33,000 --> 00:20:37,000
it.
In the case of the Schrˆdinger
257
00:20:37,000 --> 00:20:41,000
equation, the quantization drops
out when you solve the
258
00:20:41,000 --> 00:20:44,000
differential equation.
It comes out of the equation
259
00:20:44,000 --> 00:20:47,000
just naturally.
We did not paste it onto it.
260
00:20:47,000 --> 00:20:51,000
We did not make an ad hoc kind
of representation.
261
00:20:51,000 --> 00:20:55,000
That is the big difference here
between quantum mechanics and
262
00:20:55,000 --> 00:21:00,000
classical mechanics.
In quantum mechanics,
263
00:21:00,000 --> 00:21:04,000
it can only tell you about a
probability.
264
00:21:04,000 --> 00:21:09,000
It cannot tell you exactly
where the particle is going to
265
00:21:09,000 --> 00:21:11,000
be.
Questions on that?
266
00:21:11,000 --> 00:21:15,000
Okay.
Anyway, this value a nought,
267
00:21:15,000 --> 00:21:20,000
that is why it is called
the Bohr radius.
268
00:21:20,000 --> 00:21:24,000
And then it turns out,
quantum mechanically,
269
00:21:24,000 --> 00:21:29,000
that this value of r,
the most probable value of r
270
00:21:29,000 --> 00:21:35,000
is, in fact, exactly a nought.
271
00:21:35,000 --> 00:21:37,000
In a sense, Bohr was pretty
lucky.
272
00:21:37,000 --> 00:21:42,000
And this is kind of an accident
that he got a nought out
273
00:21:42,000 --> 00:21:46,000
of this, and it has to do with
the actual form of the Coulomb
274
00:21:46,000 --> 00:21:49,000
interaction.
But, of course,
275
00:21:49,000 --> 00:21:53,000
this doesn't work for anything
else, other than a hydrogen
276
00:21:53,000 --> 00:21:55,000
atom.
Whereas, the Schrˆdinger
277
00:21:55,000 --> 00:22:00,000
equation, as we are going to see
in a moment, is applicable to
278
00:22:00,000 --> 00:22:04,000
all the atoms that we know
about.
279
00:22:04,000 --> 00:22:08,000
So that is the radial
probability distribution
280
00:22:08,000 --> 00:22:11,000
function for the 1s atom,
for the 1s state.
281
00:22:11,000 --> 00:22:17,000
We want to take a look at the
radial probability distribution
282
00:22:17,000 --> 00:22:20,000
for 2s and for 3s.
Let me plot those.
283
00:22:20,000 --> 00:22:24,000
And you can actually put these
lights on here.
284
00:22:24,000 --> 00:22:28,000
That is okay.
I am going to use this board
285
00:22:28,000 --> 00:22:33,000
for a moment.
Here is the radial probability
286
00:22:33,000 --> 00:22:38,000
distribution function.
I can write it as little r
287
00:22:38,000 --> 00:22:42,000
times R(2,0) squared of r,
288
00:22:42,000 --> 00:22:45,000
or RPD.
This is for 2s versus r.
289
00:22:45,000 --> 00:22:50,000
And when I do that I get a
function that looks like this.
290
00:22:50,000 --> 00:22:55,000
And, if I evaluate it here,
what is this value of r at
291
00:22:55,000 --> 00:23:00,000
which the probability is a
maximum?
292
00:23:00,000 --> 00:23:04,000
Well, this most probable value
of r is 6 a nought.
293
00:23:04,000 --> 00:23:06,000
Look at that.
294
00:23:06,000 --> 00:23:10,000
The most probable value of r
for 1s was a nought.
295
00:23:10,000 --> 00:23:13,000
In the case of the 2s state
296
00:23:13,000 --> 00:23:17,000
here, the electron,
the most probable value is 6 a
297
00:23:17,000 --> 00:23:20,000
nought, six times as far from
the nucleus.
298
00:23:20,000 --> 00:23:24,000
If you have a hydrogen atom in
the first excited state,
299
00:23:24,000 --> 00:23:29,000
in a sense that hydrogen atom
is bigger.
300
00:23:29,000 --> 00:23:34,000
It is bigger in the sense that
the probability of you finding
301
00:23:34,000 --> 00:23:39,000
the electron at a larger
distance away from the nucleus
302
00:23:39,000 --> 00:23:42,000
is larger.
And that, in general,
303
00:23:42,000 --> 00:23:45,000
is the case.
The radial probability
304
00:23:45,000 --> 00:23:48,000
distribution,
here, also reflects the radial
305
00:23:48,000 --> 00:23:52,000
node that we talked about last
time.
306
00:23:52,000 --> 00:23:55,000
That radial node is r equals a
nought.
307
00:23:55,000 --> 00:24:00,000
Radial node is the value of r
that makes your wave function go
308
00:24:00,000 --> 00:24:04,000
to zero.
Notice, again,
309
00:24:04,000 --> 00:24:10,000
that this radial probability
distribution function right here
310
00:24:10,000 --> 00:24:14,000
is zero at r equals 0.
This is not a node.
311
00:24:14,000 --> 00:24:19,000
This is not a radial node.
This is a consequence,
312
00:24:19,000 --> 00:24:25,000
right here, of our definition
for the radial probability.
313
00:24:25,000 --> 00:24:30,000
Our volume element has gone to
zero.
314
00:24:30,000 --> 00:24:36,000
r equals 0 is never a radial
node in any wave function.
315
00:24:36,000 --> 00:24:40,000
What about 3s?
Well, let's plot 3s.
316
00:24:40,000 --> 00:24:44,000
Here is 3s.
This is the radial probability
317
00:24:44,000 --> 00:24:49,000
distribution.
I take Psi for 3s and square
318
00:24:49,000 --> 00:24:55,000
it, multiply by 4 pi r squared
dr,
319
00:24:55,000 --> 00:25:02,000
and do so for all the values of
r, and I am going to get
320
00:25:02,000 --> 00:25:10,000
something that looks like this.
Now this most probable value of
321
00:25:10,000 --> 00:25:16,000
r here, where the 3s wave
function is equal to 11.468 a
322
00:25:16,000 --> 00:25:22,000
nought.
For the second excited state of
323
00:25:22,000 --> 00:25:25,000
a hydrogen atom,
that electron,
324
00:25:25,000 --> 00:25:30,000
on the average,
is 11.5 times farther out from
325
00:25:30,000 --> 00:25:38,000
the nucleus than it is in the
case of the 1s state right here.
326
00:25:38,000 --> 00:25:43,000
Again, for that second excited
state, that hydrogen atom is
327
00:25:43,000 --> 00:25:49,000
bigger in the sense that the
probability of it being farther
328
00:25:49,000 --> 00:25:54,000
away from the nucleus is larger.
That radial probability
329
00:25:54,000 --> 00:26:00,000
distribution of the 3s also
reflects the two radial nodes in
330
00:26:00,000 --> 00:26:07,000
the 3s wave function.
The radial nodes are at 1.9 a
331
00:26:07,000 --> 00:26:11,000
nought, here,
and 7.1 a nought.
332
00:26:11,000 --> 00:26:19,000
Again, the value here at r
equals 0 is not a radial node.
333
00:26:19,000 --> 00:26:27,000
Now, as you look at this,
it is tempting to ask the
334
00:26:27,000 --> 00:26:33,000
following question.
You might want to ask,
335
00:26:33,000 --> 00:26:40,000
if the electron can be at these
values of r, and it can be at
336
00:26:40,000 --> 00:26:46,000
these values of r,
and it can be at these values
337
00:26:46,000 --> 00:26:53,000
of r, how does the electron
actually get from here to here
338
00:26:53,000 --> 00:27:00,000
to here if right at r equals 1.9
a nought and 7.1 a nought the
339
00:27:00,000 --> 00:27:08,000
probability is equal to zero?
Well, you might say maybe this
340
00:27:08,000 --> 00:27:14,000
probability isn't exactly zero.
It is something small.
341
00:27:14,000 --> 00:27:20,000
But I am telling you that it is
zero, goose egg,
342
00:27:20,000 --> 00:27:23,000
zilch, zippo,
nada, cipher,
343
00:27:23,000 --> 00:27:28,000
nix, nought.
Anybody else have another name?
344
00:27:28,000 --> 00:27:31,000
Nil.
It is nothing.
345
00:27:31,000 --> 00:27:34,000
It is zero.
How do you answer that
346
00:27:34,000 --> 00:27:37,000
question?
Well, it turns out,
347
00:27:37,000 --> 00:27:42,000
of course, that it isn't an
appropriate question.
348
00:27:42,000 --> 00:27:48,000
And the reason it is not is
because that question is asked
349
00:27:48,000 --> 00:27:52,000
in the framework of classical
mechanics.
350
00:27:52,000 --> 00:27:56,000
When you ask,
how does a particle get from
351
00:27:56,000 --> 00:28:01,000
one place to another,
you are asking about a
352
00:28:01,000 --> 00:28:06,000
trajectory.
You are asking about a path.
353
00:28:06,000 --> 00:28:10,000
Particles over here,
over here, over here,
354
00:28:10,000 --> 00:28:14,000
how does it get from one place
to another?
355
00:28:14,000 --> 00:28:18,000
And, in quantum mechanics,
we don't have the concept of
356
00:28:18,000 --> 00:28:22,000
trajectories.
Instead, what we have to think
357
00:28:22,000 --> 00:28:28,000
of is the electron as a wave.
And we already know that a wave
358
00:28:28,000 --> 00:28:32,000
can have amplitude
simultaneously at many different
359
00:28:32,000 --> 00:28:37,000
positions.
And so it has simultaneous
360
00:28:37,000 --> 00:28:42,000
amplitude or probability here,
here, and here,
361
00:28:42,000 --> 00:28:46,000
all at the same time.
We cannot talk about
362
00:28:46,000 --> 00:28:50,000
trajectories anymore.
And that, again,
363
00:28:50,000 --> 00:28:56,000
ties into the uncertainty
principle, our inability to know
364
00:28:56,000 --> 00:29:02,000
exactly the position and the
momentum of a particle at any
365
00:29:02,000 --> 00:29:07,000
given instance.
The best we can tell you is a
366
00:29:07,000 --> 00:29:11,000
probability.
We have to change the way we
367
00:29:11,000 --> 00:29:16,000
think about electrons.
You cannot cast them in the
368
00:29:16,000 --> 00:29:19,000
framework of your everyday
world.
369
00:29:19,000 --> 00:29:23,000
This is part of our world,
but you have to go do a
370
00:29:23,000 --> 00:29:30,000
specific type of experiment to
see this part of the world.
371
00:29:30,000 --> 00:29:35,000
That is why it seems so strange
to you, because it is not part
372
00:29:35,000 --> 00:29:40,000
of your everyday experience.
But this world works with
373
00:29:40,000 --> 00:29:45,000
different rules that you really
do have to accept that it just
374
00:29:45,000 --> 00:29:49,000
works differently.
Questions?
375
00:30:00,000 --> 00:30:06,000
Now, I am going to stop talking
about the s wave functions and
376
00:30:06,000 --> 00:30:11,000
move on to talk about the p wave
functions.
377
00:30:11,000 --> 00:30:16,000
With the s wave functions,
we talked about the
378
00:30:16,000 --> 00:30:22,000
significance of the wave
function, probability density,
379
00:30:22,000 --> 00:30:26,000
radial probability
distribution.
380
00:30:26,000 --> 00:30:32,000
We talked about what a radial
node was.
381
00:30:32,000 --> 00:30:36,000
Now it is time to move onto the
p wave functions.
382
00:30:36,000 --> 00:30:41,000
And the p wave functions,
of course, are not spherically
383
00:30:41,000 --> 00:30:44,000
symmetric.
And to represent them,
384
00:30:44,000 --> 00:30:48,000
we are going to do our dot
density diagram again.
385
00:30:48,000 --> 00:30:54,000
We are going to take the wave
function and square it to get
386
00:30:54,000 --> 00:31:00,000
the probability density and then
plot that probability density as
387
00:31:00,000 --> 00:31:05,000
a density of dots.
We the dots are most dense,
388
00:31:05,000 --> 00:31:10,000
well, that means the highest
probability density.
389
00:31:10,000 --> 00:31:14,000
Here is the result for the pz
wave function.
390
00:31:14,000 --> 00:31:19,000
It is pz because you can see
the highest probability,
391
00:31:19,000 --> 00:31:23,000
here, is along the z-axis.
It is symmetric along the
392
00:31:23,000 --> 00:31:27,000
z-axis.
Here is the probability density
393
00:31:27,000 --> 00:31:32,000
for the px wave function.
You can see that the
394
00:31:32,000 --> 00:31:36,000
probability density is greatest
along the x-axis.
395
00:31:36,000 --> 00:31:39,000
It is symmetric along the
x-axis.
396
00:31:39,000 --> 00:31:43,000
And, if you look really
carefully, you can see that
397
00:31:43,000 --> 00:31:48,000
there is no probability density
in the y,z-plane for the px wave
398
00:31:48,000 --> 00:31:49,000
function.
And, over here,
399
00:31:49,000 --> 00:31:53,000
if you look carefully,
you can see that there is no
400
00:31:53,000 --> 00:31:58,000
probability density in the
x,y-plane for the pz wave
401
00:31:58,000 --> 00:32:03,000
function.
And here is a py wave function,
402
00:32:03,000 --> 00:32:08,000
the probability density of it.
The probability density is
403
00:32:08,000 --> 00:32:13,000
concentrated along the y-axis.
It is symmetric along the
404
00:32:13,000 --> 00:32:16,000
y-axis.
And, if you look very
405
00:32:16,000 --> 00:32:20,000
carefully, there is no
probability density,
406
00:32:20,000 --> 00:32:25,000
here, in the x,z-plane.
Well, the fact that there is no
407
00:32:25,000 --> 00:32:30,000
probability density,
here, in the x,y-plane,
408
00:32:30,000 --> 00:32:34,000
in the case of pz,
indicates that we have an
409
00:32:34,000 --> 00:32:39,000
angular node.
An angular node at theta equal
410
00:32:39,000 --> 00:32:42,000
90 degrees.
An angular node is the same
411
00:32:42,000 --> 00:32:46,000
thing as a radial node in the
sense that it is the value of
412
00:32:46,000 --> 00:32:51,000
the angle that makes the wave
function be equal to zero.
413
00:32:51,000 --> 00:32:53,000
Here is the wave function for
pz.
414
00:32:53,000 --> 00:32:57,000
You can see that when theta is
equal to zero,
415
00:32:57,000 --> 00:33:02,000
this wave function is going to
be equal to zero.
416
00:33:02,000 --> 00:33:08,000
An angular node is the value of
theta or phi that makes the wave
417
00:33:08,000 --> 00:33:12,000
function be zero.
And the consequence,
418
00:33:12,000 --> 00:33:19,000
then, is that we have a nodal
plane, because everywhere on the
419
00:33:19,000 --> 00:33:23,000
x,y-plane, theta is equal to 90
degrees.
420
00:33:23,000 --> 00:33:29,000
For the px wave function,
the value of the angle that
421
00:33:29,000 --> 00:33:35,000
gives you that nodal plane is
phi equals 90.
422
00:33:35,000 --> 00:33:41,000
That means everywhere in the
y,z-plane is phi equal to 90.
423
00:33:41,000 --> 00:33:46,000
In the case of py,
when phi is equal to zero,
424
00:33:46,000 --> 00:33:50,000
well, that is everywhere in the
x,z-plane.
425
00:33:50,000 --> 00:33:55,000
Everywhere in the x,z-plane,
phi is equal to zero.
426
00:33:55,000 --> 00:34:02,000
So, that is the angular nodes.
In general, and this is
427
00:34:02,000 --> 00:34:07,000
something you do have to know,
an orbital has n minus 1
428
00:34:07,000 --> 00:34:12,000
total nodes.
And what I mean by total nodes
429
00:34:12,000 --> 00:34:17,000
is angular plus radial nodes.
The number of angular nodes is
430
00:34:17,000 --> 00:34:20,000
given by this quantity,
l.
431
00:34:20,000 --> 00:34:25,000
The quantum number l that
labels your wave function always
432
00:34:25,000 --> 00:34:30,000
gives you the number of angular
nodes.
433
00:34:30,000 --> 00:34:34,000
Therefore, if n minus 1 is the
total and l is the number of
434
00:34:34,000 --> 00:34:38,000
angular, well then,
the number of radial nodes is n
435
00:34:38,000 --> 00:34:42,000
minus 1 minus l. This is
436
00:34:42,000 --> 00:34:44,000
something that you do have to
know.
437
00:34:44,000 --> 00:34:49,000
If I give you a wave function
and ask you how many radial and
438
00:34:49,000 --> 00:34:52,000
angular nodes it has,
you need to be able to
439
00:34:52,000 --> 00:34:56,000
calculate that,
and vice versa.
440
00:34:56,000 --> 00:35:02,000
Sometimes I will tell you a
function has three radial nodes
441
00:35:02,000 --> 00:35:07,000
and six or seven angular nodes
or something,
442
00:35:07,000 --> 00:35:12,000
what is the wave function?
So, we go both ways.
443
00:35:12,000 --> 00:35:19,000
Well, I also want to take a
look at the radial probability
444
00:35:19,000 --> 00:35:24,000
distribution functions for the p
wave functions.
445
00:35:24,000 --> 00:35:31,000
We looked at it for the s wave
functions already.
446
00:35:31,000 --> 00:35:36,000
I actually want to contrast the
radial probability distribution,
447
00:35:36,000 --> 00:35:40,000
say, for 2p,
here it is, with that of 2s
448
00:35:40,000 --> 00:35:45,000
that we looked at a moment ago.
Remember, how do you get the
449
00:35:45,000 --> 00:35:50,000
radial probability distribution
function here for 2p?
450
00:35:50,000 --> 00:35:55,000
It is the radial part of the 2p
wave function times r squared
451
00:35:55,000 --> 00:35:58,000
dr.
It gives me the probability of
452
00:35:58,000 --> 00:36:05,000
finding the electron a distance
between r and r plus dr.
453
00:36:05,000 --> 00:36:10,000
Again, what you see is that at
r equals 0, that is zero.
454
00:36:10,000 --> 00:36:15,000
That is not a radial node.
But what I really want to point
455
00:36:15,000 --> 00:36:20,000
out here is that the most
probable value of r,
456
00:36:20,000 --> 00:36:25,000
for the 2p wave function,
is actually smaller than it is
457
00:36:25,000 --> 00:36:31,000
for the 2s wave function.
That is, it is more likely for
458
00:36:31,000 --> 00:36:37,000
the electron in a 2p state to be
a little closer in to the
459
00:36:37,000 --> 00:36:40,000
nucleus than it is for the 2s
state.
460
00:36:40,000 --> 00:36:46,000
In general, as you increase the
angular momentum quantum number,
461
00:36:46,000 --> 00:36:51,000
the most probable value of r
gets smaller for the same value
462
00:36:51,000 --> 00:36:54,000
of n.
Similarly, here is the 3s
463
00:36:54,000 --> 00:37:01,000
radial probability distribution
function that we looked at.
464
00:37:01,000 --> 00:37:05,000
Here is a radial probability
distribution for 3p.
465
00:37:05,000 --> 00:37:08,000
Now, with the 3p,
you can see the value of the
466
00:37:08,000 --> 00:37:11,000
radial node.
You can see the radial
467
00:37:11,000 --> 00:37:15,000
probability distribution
reflects a radial node,
468
00:37:15,000 --> 00:37:17,000
here.
And here is the radial
469
00:37:17,000 --> 00:37:21,000
probability distribution
function for 3d.
470
00:37:21,000 --> 00:37:25,000
We did not look at the
probability density of 3d.
471
00:37:25,000 --> 00:37:30,000
You will do that with Professor
Cummins when you talk about
472
00:37:30,000 --> 00:37:35,000
transition metals.
But here, I just drew in the
473
00:37:35,000 --> 00:37:39,000
radial probability distribution
for 3d.
474
00:37:39,000 --> 00:37:44,000
But the point again that I want
to make is here is the most
475
00:37:44,000 --> 00:37:48,000
probable value of r for 3s,
here it is for 3p,
476
00:37:48,000 --> 00:37:53,000
here it is for 3d,
again, the most probable value
477
00:37:53,000 --> 00:37:59,000
for 3d is smaller than it is for
3p, than it is for 3s.
478
00:37:59,000 --> 00:38:05,000
Again, as you increase the
angular momentum quantum number,
479
00:38:05,000 --> 00:38:09,000
that most probable value gets
smaller.
480
00:38:09,000 --> 00:38:14,000
However, ironically,
if you actually look at the
481
00:38:14,000 --> 00:38:20,000
probability of the electron
being very, very close to the
482
00:38:20,000 --> 00:38:26,000
nucleus, that probability is
only significant for the s wave
483
00:38:26,000 --> 00:38:31,000
functions.
Look at the 3s wave function.
484
00:38:31,000 --> 00:38:36,000
Here, you see that you really
do have some probability very
485
00:38:36,000 --> 00:38:40,000
close to the nucleus.
You don't see that in the 3p
486
00:38:40,000 --> 00:38:43,000
wave function.
You certainly don't see that in
487
00:38:43,000 --> 00:38:48,000
the 3d wave function.
Again, in the 2s wave function,
488
00:38:48,000 --> 00:38:52,000
you have some significant
probability of the electron
489
00:38:52,000 --> 00:38:55,000
being really close to the
nucleus in 2s,
490
00:38:55,000 --> 00:39:00,000
but you don't in 2p.
That is important.
491
00:39:00,000 --> 00:39:05,000
And it seems in contradiction
to the fact that on the average,
492
00:39:05,000 --> 00:39:10,000
the most probable value of r
gets smaller as l gets larger.
493
00:39:10,000 --> 00:39:14,000
These two facts that look
contradictory are important.
494
00:39:14,000 --> 00:39:17,000
They dictate the behavior of
atoms.
495
00:39:17,000 --> 00:39:22,000
These two facts seem like kind
of loose threads at the moment
496
00:39:22,000 --> 00:39:27,000
in the sense that you are
probably wondering why I am
497
00:39:27,000 --> 00:39:31,000
telling you what I am telling
you.
498
00:39:31,000 --> 00:39:36,000
But we are going to use that
information in a few days,
499
00:39:36,000 --> 00:39:41,000
and you will see really the
significance of this plot.
500
00:39:41,000 --> 00:39:47,000
And this plot will be an
important one for you to refer
501
00:39:47,000 --> 00:39:49,000
back to.
Yes?
502
00:40:07,000 --> 00:40:09,000
Probably.
I am not exactly sure of the
503
00:40:09,000 --> 00:40:12,000
picture you drew in high school,
but yes.
504
00:40:12,000 --> 00:40:17,000
If the electron in general is
further out from the nucleus,
505
00:40:17,000 --> 00:40:21,000
that is a higher energy state.
The electron is less strongly
506
00:40:21,000 --> 00:40:27,000
bound, as we are going to see in
the multi-electron atoms here.
507
00:40:35,000 --> 00:40:38,000
Oh, no.
For the hydrogen no.
508
00:40:38,000 --> 00:40:42,000
Let me explain that.
For the hydrogen atom,
509
00:40:42,000 --> 00:40:48,000
the energies are only dictated
by the n quantum number,
510
00:40:48,000 --> 00:40:53,000
so 3s, 3p, 3d all have the same
energies.
511
00:40:53,000 --> 00:40:58,000
Where the energies become
degenerate is with a
512
00:40:58,000 --> 00:41:05,000
multi-electron atom.
And we are going to talk about
513
00:41:05,000 --> 00:41:11,000
that and how that reflects here,
these wave functions in the
514
00:41:11,000 --> 00:41:16,000
next day.
That is all I am going to say
515
00:41:16,000 --> 00:41:21,000
about the hydrogen atom.
Now it is time to move on,
516
00:41:21,000 --> 00:41:24,000
to helium.
And, of course,
517
00:41:24,000 --> 00:41:30,000
the Schrˆdinger equation
predicts the binding energies of
518
00:41:30,000 --> 00:41:39,000
the electrons to the nucleus in
a helium atom also very well.
519
00:41:39,000 --> 00:41:42,000
But, of course,
it is a much more complicated
520
00:41:42,000 --> 00:41:46,000
Schrˆdinger equation.
And I am not even going to
521
00:41:46,000 --> 00:41:50,000
write out the Hamiltonian in
this case, but I want to show
522
00:41:50,000 --> 00:41:54,000
you the wave function here.
See the wave function?
523
00:41:54,000 --> 00:41:59,000
The wave function is a function
of six variables.
524
00:41:59,000 --> 00:42:04,000
It is a function of two r's,
two distances from the nucleus,
525
00:42:04,000 --> 00:42:07,000
one for electron one,
one for electron two,
526
00:42:07,000 --> 00:42:12,000
two theta's and two phi's.
We have six variables for the
527
00:42:12,000 --> 00:42:16,000
wave function.
And the consequence of this is
528
00:42:16,000 --> 00:42:20,000
that our solutions for the
binding energies for the
529
00:42:20,000 --> 00:42:26,000
electrons in helium or any other
atoms are not going to be nice
530
00:42:26,000 --> 00:42:31,000
analytical forms.
We are no longer going to have
531
00:42:31,000 --> 00:42:35,000
e sub n equal minus the Rydberg
constant over n squared.
532
00:42:35,000 --> 00:42:39,000
If you
actually solve for those
533
00:42:39,000 --> 00:42:43,000
energies, and you have to do it
numerically, you are just going
534
00:42:43,000 --> 00:42:46,000
to get a list of numbers,
a table of numbers,
535
00:42:46,000 --> 00:42:50,000
but not a nice analytical form.
If you solve for the wave
536
00:42:50,000 --> 00:42:55,000
function, you are not going to
get a nice analytical form,
537
00:42:55,000 --> 00:42:59,000
like we got for hydrogen.
Instead, what you will get is a
538
00:42:59,000 --> 00:43:02,000
value for the amplitude of Psi
as a function of r,
539
00:43:02,000 --> 00:43:08,000
theta and phi.
But if you get actually much
540
00:43:08,000 --> 00:43:12,000
above three electrons,
it turns out that even
541
00:43:12,000 --> 00:43:18,000
numerically, you cannot solve
the Schrˆdinger equation,
542
00:43:18,000 --> 00:43:23,000
exactly.
You have to use approximations.
543
00:43:23,000 --> 00:43:30,000
And we are going to look at the
most basic approximation that is
544
00:43:30,000 --> 00:43:34,000
used that works,
amazingly.
545
00:43:34,000 --> 00:43:39,000
It works well enough for us to
have a framework in which to
546
00:43:39,000 --> 00:43:42,000
understand the reactions of
these atoms.
547
00:43:42,000 --> 00:43:47,000
And what is that approximation?
Well, that approximation is
548
00:43:47,000 --> 00:43:52,000
called the one-electron wave
approximation or the
549
00:43:52,000 --> 00:43:55,000
one-electron orbital
approximation.
550
00:43:55,000 --> 00:44:00,000
What does that mean?
Well, that means this.
551
00:44:00,000 --> 00:44:05,000
I am going to take my wave
function here for the helium
552
00:44:05,000 --> 00:44:10,000
atom, which strictly is a wave
function that is a function of
553
00:44:10,000 --> 00:44:15,000
six variables,
and I am going to separate it.
554
00:44:15,000 --> 00:44:20,000
I am going to let electron one
have its own wave function and
555
00:44:20,000 --> 00:44:24,000
electron two have its own wave
function.
556
00:44:24,000 --> 00:44:28,000
That is an approximation.
In addition,
557
00:44:28,000 --> 00:44:33,000
what I am going to do is let
the wave function for electron
558
00:44:33,000 --> 00:44:39,000
one have a hydrogen-like wave
function.
559
00:44:39,000 --> 00:44:43,000
I am going to say that it has
the 1s wave function,
560
00:44:43,000 --> 00:44:46,000
or the Psi(1,
0, 0) wave function of a
561
00:44:46,000 --> 00:44:49,000
hydrogen atom.
And I am going to let electron
562
00:44:49,000 --> 00:44:53,000
two have the Psi(1,
0, 0) wave function of a
563
00:44:53,000 --> 00:44:56,000
hydrogen atom.
Or, I am going to write it as
564
00:44:56,000 --> 00:45:00,000
1s of 1, for electron one,
times 1s of 2,
565
00:45:00,000 --> 00:45:04,000
for electron two.
566
00:45:04,000 --> 00:45:07,000
Or, another shorthand,
I am going to write it as 1s.
567
00:45:07,000 --> 00:45:09,000
squared.
And, if I continued on,
568
00:45:09,000 --> 00:45:13,000
here, it is for lithium.
Lithium, the wave function
569
00:45:13,000 --> 00:45:17,000
strictly has nine coordinates,
but I am going to let every one
570
00:45:17,000 --> 00:45:19,000
of those electrons,
in the one electron wave
571
00:45:19,000 --> 00:45:22,000
approximation,
have its own wave function.
572
00:45:22,000 --> 00:45:26,000
And I am going to let electron
one have a wave function that
573
00:45:26,000 --> 00:45:30,000
looks like a hydrogen atom
wavefunction.
574
00:45:30,000 --> 00:45:33,000
The 1s wave function.
The same thing with electron
575
00:45:33,000 --> 00:45:35,000
two.
And then I am going to let
576
00:45:35,000 --> 00:45:40,000
electron three have the 2s wave
function of the hydrogen atom.
577
00:45:40,000 --> 00:45:44,000
And in simplified notation,
that is just 1s squared 2s.
578
00:45:44,000 --> 00:45:47,000
And here is
beryllium, 16 variables,
579
00:45:47,000 --> 00:45:51,000
but I am going to let every
electron have its own wave
580
00:45:51,000 --> 00:45:54,000
function.
And I am going to give electron
581
00:45:54,000 --> 00:45:56,000
one the 1s wave function,
electron two,
582
00:45:56,000 --> 00:46:00,000
the 1s, electron three,
the 2s, electron four,
583
00:46:00,000 --> 00:46:03,000
the 2s.
I can also write that,
584
00:46:03,000 --> 00:46:07,000
as you have already done,
1s 2 2s 2.
585
00:46:07,000 --> 00:46:10,000
And I can keep going.
And these electron
586
00:46:10,000 --> 00:46:14,000
configurations that you have
been writing down in high
587
00:46:14,000 --> 00:46:18,000
school, that is what they are,
electron configurations,
588
00:46:18,000 --> 00:46:23,000
well, they are nothing more
than our shorthand notation for
589
00:46:23,000 --> 00:46:27,000
the electron wave functions
within this one-electron wave
590
00:46:27,000 --> 00:46:32,000
approximation.
That is what those were,
591
00:46:32,000 --> 00:46:37,000
that you were writing down.
Those were a shorthand notation
592
00:46:37,000 --> 00:46:42,000
for the wave functions in
Schrˆdinger's equation within
593
00:46:42,000 --> 00:46:46,000
this one-electron wave
approximation.
594
00:46:46,000 --> 00:46:50,000
Now, one thing you do notice is
that I did not,
595
00:46:50,000 --> 00:46:55,000
in the case of boron here,
let all five electrons be in
596
00:46:55,000 --> 00:46:59,000
the 1s state,
or let all five electrons be
597
00:46:59,000 --> 00:47:05,000
represented by a 1s hydrogen
atom wave function.
598
00:47:05,000 --> 00:47:10,000
I didn't because of a quantity
that you already know about,
599
00:47:10,000 --> 00:47:14,000
called spin.
You already know that if you
600
00:47:14,000 --> 00:47:19,000
are going to put electrons in
the 1s state here that one
601
00:47:19,000 --> 00:47:25,000
electron has to go in with spin
up and the other spin down.
602
00:47:25,000 --> 00:47:30,000
And the 2s, spin up and spin
down, etc.
603
00:47:30,000 --> 00:47:34,000
What is the phenomenon called
spin?
604
00:47:34,000 --> 00:47:42,000
Well, spin is entirely a
quantum mechanical phenomenon.
605
00:47:42,000 --> 00:47:48,000
There is no correct classical
analogy to spin.
606
00:47:48,000 --> 00:47:53,000
Spin is intrinsic angular
momentum.
607
00:47:53,000 --> 00:48:00,000
It is angular momentum that is
just part of a particle,
608
00:48:00,000 --> 00:48:07,000
such as an electron.
The spin quantum numbers
609
00:48:07,000 --> 00:48:11,000
actually come from solving the
relativistic Schrˆdinger
610
00:48:11,000 --> 00:48:15,000
equation, which we did not even
write down.
611
00:48:15,000 --> 00:48:20,000
When you solve the relativistic
Schrˆdinger equation,
612
00:48:20,000 --> 00:48:23,000
out drops a fourth quantum
number.
613
00:48:23,000 --> 00:48:28,000
That fourth quantum number we
are going to call m sub s.
614
00:48:28,000 --> 00:48:31,000
And we find that m sub s
615
00:48:31,000 --> 00:48:36,000
has two allowed values.
One of those values is one-half
616
00:48:36,000 --> 00:48:41,000
and the other is minus one-half.
Here, we have a case where the
617
00:48:41,000 --> 00:48:44,000
quantum number is not an
integer.
618
00:48:44,000 --> 00:48:47,000
It is one-half and it is minus
one-half.
619
00:48:47,000 --> 00:48:51,000
Now, if it helps you to think
about the electron spinning
620
00:48:51,000 --> 00:48:54,000
around its own axis,
like I depict here,
621
00:48:54,000 --> 00:48:58,000
well, if that is the case,
then the angular momentum
622
00:48:58,000 --> 00:49:03,000
quantum number is perpendicular,
here ,to this plane in which it
623
00:49:03,000 --> 00:49:08,000
is rotating.
And you might want to call that
624
00:49:08,000 --> 00:49:10,000
spin up.
And, of course,
625
00:49:10,000 --> 00:49:14,000
if it is spinning in the other
direction, well,
626
00:49:14,000 --> 00:49:17,000
then the angular momentum
vector is pointed in the
627
00:49:17,000 --> 00:49:21,000
opposite direction.
You might want to call this
628
00:49:21,000 --> 00:49:24,000
spin down.
If it helps for you to think
629
00:49:24,000 --> 00:49:28,000
about this, okay,
but remember that this is not
630
00:49:28,000 --> 00:49:32,000
correct.
This is a classical analogy
631
00:49:32,000 --> 00:49:37,000
that we are trying to draw here.
We are trying to say that this
632
00:49:37,000 --> 00:49:40,000
electron is rotating around its
own axis.
633
00:49:40,000 --> 00:49:44,000
That is not true.
This angular momentum is just
634
00:49:44,000 --> 00:49:47,000
an intrinsic part,
the intrinsic nature of a
635
00:49:47,000 --> 00:49:52,000
particular such as an electron.
Next time, I will tell you
636
00:49:52,099 --> 00:49:55,000
about Uhlenbeck and Goudsmith.
See you Wednesday.