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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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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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volume or the volume of the
shell, 4 pi r squared dr.

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Let's write that out again,

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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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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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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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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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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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definitions.
I want to talk some more about

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this radial probability
distribution function,

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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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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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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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1911 to 1926,
the scientific community was

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really working very hard to try
to understand the structure of

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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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working on that problem was
Niels Bohr.

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And, in 1919,
Niels Bohr of course realized

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that classical physics fails
this kind of planetary model for

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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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going to work,
that those classical ideas

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predicted that this would
plummet into the nucleus in

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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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classical physics at the moment.
Then, what he did was to impose

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some quantization on this
classical model for the hydrogen

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atom.
And the reason he got this idea

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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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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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00:16:47,000 --> 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.