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All right. I am going to start with
Friday's lecture notes because there was a significant amount
on them that I had not finished up yet.
We had finally gotten to the point where we were talking
about what does a wave function mean, what is the physical
significance of it and how does it actually represent the
presence of an electron? And what we saw was that the
physically significant representation of the wave
function, if you have some wave function Psi labeled by three
quantum numbers, n, l and m.
And, of course, it is a function of r,
theta and phi. The physically significant
quantity was this wave function squared.
That wave function squared, that was interpreted as a
probability density. The wave function squared has
units. It has units of inverse volume.
It is a density. It is a probability per unit
volume. Now, as an aside,
because someone asked me, I should tell you that the more
comprehensive definition of the probability density is Psi,
not squared, but Psi times Psi star,
where Psi star is the complex conjugate.
Because it turns out that some wave functions are imaginary
functions. And so, if you took an
imaginary function and squared it, then you would still get an
imaginary function after it. And then it is hard to
interpret an imaginary function as a probability density.
And so the more comprehensive definition is Psi times Psi
star, where Psi star is the complex conjugate of Psi.
And, when you multiply Psi by Psi star, if Psi is a complex
function, well, then you get a real function.
This is the more comprehensive definition of the probability
density, Psi times Psi star. We won't use that.
I just wanted to let you know about it.
So, probability density. Not only do we want to know
something about the probability density.
We also want to know something about the probability of finding
the electron some distance away from the nucleus.
And, to do that, what we were talking about was
this quantity, this radial distribution,
the radial probability distribution.
And what that is, is the probability of finding
an electron in a spherical shell of radius r and distance or
thickness dr. For example,
if this gray portion here represented the probability
density of the 1s wave function in our dot density diagram.
Remember, we squared the wave function, got the probability
density and then represented it with a dot density diagram,
where the density of the dots was proportional to the value of
the wave function squared. And, in the case of the 1s wave
function, we saw that the probability density was largest
right at r equals 0, and that is exponentially
decayed in all directions uniformly.
That is what that gray part represents.
But now, this blue, here, is my spherical shell.
It has a radius r, and it has a thickness,
here, dr. And the radial probability
distribution is asking, what is the probability of
finding the electron in this spherical shell?
And that spherical shell has a thickness dr.
Another way to ask that is the probability of finding the
electron between r and r was dr. That is what we wanted to know,
and that is what the radial probability distribution tells
us. Now, how do you get a value out
of that? How do you actually calculate
the radial probability? Well, to do that,
what we have to know is this volume, here,
of the spherical shell. The volume of this spherical
shell is just the surface area of that spherical shell,
4 pi r squared, and the volume is times this
thickness, this thickness dr. It is a very thin shell.
It is an infinitesimally thin shell of thickness dr.
Well, if we know that volume, then what we can do is take our
probability density, Psi squared,
which has units of probability per unit volume.
And we are multiplying it, here, by our unit volume.
The unit volumes cancel, and we are left with a
probability. So, that is our probability of
finding that electron in a shell of radius r and a thickness dr.
Let's look at the result of calculating the radial
probability distribution for the 1s wave function.
What did I do? I took Psi squared for the 1s
wave function at some value of r, then I multiplied it by 4 pi
r squared dr, and I did that for many
different values of r and plotted the result here.
That is what that radial probability is as a function of
r. Well, the first thing you see
is that the most probable value of r, or the value of r where
the electron has the highest probability of being is at this
value, a nought. The most probable value of r is
this value, a nought. a nought is what we call
the Bohr radius. And today, in a moment or so,
I will tell you why it is called the Bohr radius.
It has a numerical value of 0.529 angstroms.
And so it is most likely that the electron is about a half an
angstrom away from the nucleus, making, then,
the diameter of the hydrogen atom, on the average,
a little bit over one angstrom. That is how we think about the
size of a hydrogen atom, is to take this most probable
value of r and double it to get the diameter.
The most probable value of r, or the most probable distance
of the electron from the nucleus, is half an angstrom
away. The most probable distance of
the electron from the nucleus is not r equals 0 because the
radial probability here is zero at r equals 0.
That seems a little strange because the other day we plotted
the probability density for the 1s wave function.
And, when we did that, here is Psi(1,
0, 0) squared versus r, what we saw was that the
probability density was some maximum value at r equals 0 and
that it exponentially decayed with increasing r.
And that is the case. Probability density for the s
wave functions is a maximum at r equals 0.
But the radial probability here is actually zero at r equals 0.
Why? Look at how we defined that
radial probability, here.
It is Psi squared times this volume element.
Our volume element is this spherical shell.
And, at r equals 0, the spherical shell goes to a
volume of zero. So, our radial probability here
is equal to zero at r equals 0. That is really important,
that you understand that this radial probability here is
always going to be zero at r equals 0 for all of the wave
functions that we are going to look at.
And we will talk about this a little bit more,
the fact that the electron is about a half an angstrom away
from the nucleus. But before I do that,
I also just want to point out that in your textbook,
and sometimes in the notes, that sometimes that radial
probability is actually written as the following.
It is written as the r squared, the distance variable,
times the radial part of the wave function.
That is the radial part squared.
We talked about the radial and the angular part last time.
And the radial part is labeled only by two quantum numbers,
n and l. And so, for the 1s,
that is n equals 1, l equals 0.
Where does this come from? Well, let me just emphasize or
explain where this comes from. This radial probability
distribution here, we said for the s wave
functions, was Psi squared. You could take Psi squared,
the probability density, and multiply it by this unit
volume or the volume of the shell, 4 pi r squared dr.
Let's write that out again,
but write it out now so that we write out Psi squared in terms
of the radial part and the angular part.
Remember, we said last time, for the hydrogen atom wave
functions, that Psi is always a product of a factor only an r,
which was the radial part, and a factor only in theta and
phi, which are the angular parts.
Now, what you also have to remember in looking at this is
that the angular part for the 1s wave functions,
2s, 3s, all s wave functions, was equal to 1 over 4 pi to the
1/2. If you square that,
you are going to get 1 over 4 pi.
Therefore, the 4pi's here are going to cancel for the 1s wave
functions. And what you are going to have
left is this r squared times just the radial part dr.
That is why the y-axis in your
book is sometimes labeled this way for the radial probability
distribution. But this is also important
because if you were calculating the radial distribution function
for something other than an s wave function.
The way you would do it is to take just the radial part of
that wave function times r squared, or just the radial part
of that wave function and evaluate it at that value of r
times r squared dr. You could not,
for the other wave functions, take psi squared times 4 pi r
squared dr. And that is because the angular
part for the other wave functions that are not
spherically symmetric is not the square root of 1 over 4 pi.
This is a broader definition for what the radial probability
distribution function is. It just works out,
in the case for the s wave functions, these 4pi's cancel.
And so you can write the radial probability for the s wave
functions like that. So, those are just some
definitions. I want to talk some more about
this radial probability distribution function,
here, for the 1s wave function. I want to talk about it and
also explain why a nought is called the Bohr radius.
The reason for that is the following.
The nucleus was discovered in 1911, the electron was known
before that, and Schrˆdinger did not write down his wave equation
until 1926. And, in between that,
1911 to 1926, the scientific community was
really working very hard to try to understand the structure of
the atom. And we saw how the classical
ideas, as predicted, would live a whopping 10^-10
seconds. And one of the people who were
working on that problem was Niels Bohr.
And, in 1919, Niels Bohr of course realized
that classical physics fails this kind of planetary model for
the atom where you put the nucleus in the center and the
electron is going around that nucleus with some fixed orbit.
We will call it r. Well, he knew that it was not
going to work, that those classical ideas
predicted that this would plummet into the nucleus in
10^-10 seconds. But, he said,
obviously, that does not happen, so let me just forget
classical physics at the moment. Then, what he did was to impose
some quantization on this classical model for the hydrogen
atom. And the reason he got this idea
of quantization is because he already knew the hydrogen atom
emission spectrum. He knew that in the hydrogen
atom emission spectrum that light of only certain
frequencies was emitted. That is, there was some idea
that there was something about this hydrogen atom that is
quantized. He said, well,
let me just ignore classical physics for a moment.
Let me give this a circular orbit.
But let me quantize something about this hydrogen atom.
And, in particular, what he went and did was
quantized the angular momentum of that electron.
He kind of just pasted the quantization onto a classical
model for the atom, because he is trying to work
toward explaining what the observations were.
When he pasted that quantization onto this classic
model, he was able to calculate a value of r.
And that value of r is what we call the Bohr radius,
a nought, and has the value 0.529 angstroms.
That came out of it. And if you calculate for the
radial probability distribution function for this model,
which is called the Bohr atom, would be one where that radial
probability is 1 right here at r equals a nought.
In Bohr's model, the electron had a
well-defined, precise orbit.
The value of r at which it went around the nucleus was given by
a nought. He knew exactly where the
electron was in his model. This kind of model,
which is this classical model, really, is what we call
deterministic. It is deterministic because we
know exactly where the particle, in this case the electron,
is. I want you to contrast it with
the quantum mechanical result from the Schrˆdinger equation.
What you see, in the quantum mechanical
result, is that we don't really know where the electron is,
so to speak. The best we can tell you is a
probability of finding the electron at some value r to r
plus dr. That is the best we can do
because quantum mechanics is non-deterministic.
There is a limit to which we can know the position of a
particle. That limit is given by
something called the uncertainty principle.
The uncertainty principle is not something we are going to
discuss, but it tells us that there is a limit to which we can
know both the position and the momentum of a particle.
And that is the basis for why, here, we have a probability
distribution and knowing sort of where the electron is.
We don't exactly know where the electron is, here.
This is the classical model on which Bohr just kind of pasted
the quantization of the angular momentum of the electron onto
it. In the case of the Schrˆdinger
equation, the quantization drops out when you solve the
differential equation. It comes out of the equation
just naturally. We did not paste it onto it.
We did not make an ad hoc kind of representation.
That is the big difference here between quantum mechanics and
classical mechanics. In quantum mechanics,
it can only tell you about a probability.
It cannot tell you exactly where the particle is going to
be. Questions on that?
Okay. Anyway, this value a nought,
that is why it is called the Bohr radius.
And then it turns out, quantum mechanically,
that this value of r, the most probable value of r
is, in fact, exactly a nought.
In a sense, Bohr was pretty lucky.
And this is kind of an accident that he got a nought out
of this, and it has to do with the actual form of the Coulomb
interaction. But, of course,
this doesn't work for anything else, other than a hydrogen
atom. Whereas, the Schrˆdinger
equation, as we are going to see in a moment, is applicable to
all the atoms that we know about.
So that is the radial probability distribution
function for the 1s atom, for the 1s state.
We want to take a look at the radial probability distribution
for 2s and for 3s. Let me plot those.
And you can actually put these lights on here.
That is okay. I am going to use this board
for a moment. Here is the radial probability
distribution function. I can write it as little r
times R(2,0) squared of r,
or RPD. This is for 2s versus r.
And when I do that I get a function that looks like this.
And, if I evaluate it here, what is this value of r at
which the probability is a maximum?
Well, this most probable value of r is 6 a nought.
Look at that.
The most probable value of r for 1s was a nought.
In the case of the 2s state
here, the electron, the most probable value is 6 a
nought, six times as far from the nucleus.
If you have a hydrogen atom in the first excited state,
in a sense that hydrogen atom is bigger.
It is bigger in the sense that the probability of you finding
the electron at a larger distance away from the nucleus
is larger. And that, in general,
is the case. The radial probability
distribution, here, also reflects the radial
node that we talked about last time.
That radial node is r equals a nought.
Radial node is the value of r that makes your wave function go
to zero. Notice, again,
that this radial probability distribution function right here
is zero at r equals 0. This is not a node.
This is not a radial node. This is a consequence,
right here, of our definition for the radial probability.
Our volume element has gone to zero.
r equals 0 is never a radial node in any wave function.
What about 3s? Well, let's plot 3s.
Here is 3s. This is the radial probability
distribution. I take Psi for 3s and square
it, multiply by 4 pi r squared dr,
and do so for all the values of r, and I am going to get
something that looks like this. Now this most probable value of
r here, where the 3s wave function is equal to 11.468 a
nought. For the second excited state of
a hydrogen atom, that electron,
on the average, is 11.5 times farther out from
the nucleus than it is in the case of the 1s state right here.
Again, for that second excited state, that hydrogen atom is
bigger in the sense that the probability of it being farther
away from the nucleus is larger. That radial probability
distribution of the 3s also reflects the two radial nodes in
the 3s wave function. The radial nodes are at 1.9 a
nought, here, and 7.1 a nought.
Again, the value here at r equals 0 is not a radial node.
Now, as you look at this, it is tempting to ask the
following question. You might want to ask,
if the electron can be at these values of r, and it can be at
these values of r, and it can be at these values
of r, how does the electron actually get from here to here
to here if right at r equals 1.9 a nought and 7.1 a nought the
probability is equal to zero? Well, you might say maybe this
probability isn't exactly zero. It is something small.
But I am telling you that it is zero, goose egg,
zilch, zippo, nada, cipher,
nix, nought. Anybody else have another name?
Nil. It is nothing.
It is zero. How do you answer that
question? Well, it turns out,
of course, that it isn't an appropriate question.
And the reason it is not is because that question is asked
in the framework of classical mechanics.
When you ask, how does a particle get from
one place to another, you are asking about a
trajectory. You are asking about a path.
Particles over here, over here, over here,
how does it get from one place to another?
And, in quantum mechanics, we don't have the concept of
trajectories. Instead, what we have to think
of is the electron as a wave. And we already know that a wave
can have amplitude simultaneously at many different
positions. And so it has simultaneous
amplitude or probability here, here, and here,
all at the same time. We cannot talk about
trajectories anymore. And that, again,
ties into the uncertainty principle, our inability to know
exactly the position and the momentum of a particle at any
given instance. The best we can tell you is a
probability. We have to change the way we
think about electrons. You cannot cast them in the
framework of your everyday world.
This is part of our world, but you have to go do a
specific type of experiment to see this part of the world.
That is why it seems so strange to you, because it is not part
of your everyday experience. But this world works with
different rules that you really do have to accept that it just
works differently. Questions?
Now, I am going to stop talking about the s wave functions and
move on to talk about the p wave functions.
With the s wave functions, we talked about the
significance of the wave function, probability density,
radial probability distribution.
We talked about what a radial node was.
Now it is time to move onto the p wave functions.
And the p wave functions, of course, are not spherically
symmetric. And to represent them,
we are going to do our dot density diagram again.
We are going to take the wave function and square it to get
the probability density and then plot that probability density as
a density of dots. We the dots are most dense,
well, that means the highest probability density.
Here is the result for the pz wave function.
It is pz because you can see the highest probability,
here, is along the z-axis. It is symmetric along the
z-axis. Here is the probability density
for the px wave function. You can see that the
probability density is greatest along the x-axis.
It is symmetric along the x-axis.
And, if you look really carefully, you can see that
there is no probability density in the y,z-plane for the px wave
function. And, over here,
if you look carefully, you can see that there is no
probability density in the x,y-plane for the pz wave
function. And here is a py wave function,
the probability density of it. The probability density is
concentrated along the y-axis. It is symmetric along the
y-axis. And, if you look very
carefully, there is no probability density,
here, in the x,z-plane. Well, the fact that there is no
probability density, here, in the x,y-plane,
in the case of pz, indicates that we have an
angular node. An angular node at theta equal
90 degrees. An angular node is the same
thing as a radial node in the sense that it is the value of
the angle that makes the wave function be equal to zero.
Here is the wave function for pz.
You can see that when theta is equal to zero,
this wave function is going to be equal to zero.
An angular node is the value of theta or phi that makes the wave
function be zero. And the consequence,
then, is that we have a nodal plane, because everywhere on the
x,y-plane, theta is equal to 90 degrees.
For the px wave function, the value of the angle that
gives you that nodal plane is phi equals 90.
That means everywhere in the y,z-plane is phi equal to 90.
In the case of py, when phi is equal to zero,
well, that is everywhere in the x,z-plane.
Everywhere in the x,z-plane, phi is equal to zero.
So, that is the angular nodes. In general, and this is
something you do have to know, an orbital has n minus 1
total nodes. And what I mean by total nodes
is angular plus radial nodes. The number of angular nodes is
given by this quantity, l.
The quantum number l that labels your wave function always
gives you the number of angular nodes.
Therefore, if n minus 1 is the total and l is the number of
angular, well then, the number of radial nodes is n
minus 1 minus l. This is
something that you do have to know.
If I give you a wave function and ask you how many radial and
angular nodes it has, you need to be able to
calculate that, and vice versa.
Sometimes I will tell you a function has three radial nodes
and six or seven angular nodes or something,
what is the wave function? So, we go both ways.
Well, I also want to take a look at the radial probability
distribution functions for the p wave functions.
We looked at it for the s wave functions already.
I actually want to contrast the radial probability distribution,
say, for 2p, here it is, with that of 2s
that we looked at a moment ago. Remember, how do you get the
radial probability distribution function here for 2p?
It is the radial part of the 2p wave function times r squared
dr. It gives me the probability of
finding the electron a distance between r and r plus dr.
Again, what you see is that at r equals 0, that is zero.
That is not a radial node. But what I really want to point
out here is that the most probable value of r,
for the 2p wave function, is actually smaller than it is
for the 2s wave function. That is, it is more likely for
the electron in a 2p state to be a little closer in to the
nucleus than it is for the 2s state.
In general, as you increase the angular momentum quantum number,
the most probable value of r gets smaller for the same value
of n. Similarly, here is the 3s
radial probability distribution function that we looked at.
Here is a radial probability distribution for 3p.
Now, with the 3p, you can see the value of the
radial node. You can see the radial
probability distribution reflects a radial node,
here. And here is the radial
probability distribution function for 3d.
We did not look at the probability density of 3d.
You will do that with Professor Cummins when you talk about
transition metals. But here, I just drew in the
radial probability distribution for 3d.
But the point again that I want to make is here is the most
probable value of r for 3s, here it is for 3p,
here it is for 3d, again, the most probable value
for 3d is smaller than it is for 3p, than it is for 3s.
Again, as you increase the angular momentum quantum number,
that most probable value gets smaller.
However, ironically, if you actually look at the
probability of the electron being very, very close to the
nucleus, that probability is only significant for the s wave
functions. Look at the 3s wave function.
Here, you see that you really do have some probability very
close to the nucleus. You don't see that in the 3p
wave function. You certainly don't see that in
the 3d wave function. Again, in the 2s wave function,
you have some significant probability of the electron
being really close to the nucleus in 2s,
but you don't in 2p. That is important.
And it seems in contradiction to the fact that on the average,
the most probable value of r gets smaller as l gets larger.
These two facts that look contradictory are important.
They dictate the behavior of atoms.
These two facts seem like kind of loose threads at the moment
in the sense that you are probably wondering why I am
telling you what I am telling you.
But we are going to use that information in a few days,
and you will see really the significance of this plot.
And this plot will be an important one for you to refer
back to. Yes?
Probably. I am not exactly sure of the
picture you drew in high school, but yes.
If the electron in general is further out from the nucleus,
that is a higher energy state. The electron is less strongly
bound, as we are going to see in the multi-electron atoms here.
Oh, no. For the hydrogen no.
Let me explain that. For the hydrogen atom,
the energies are only dictated by the n quantum number,
so 3s, 3p, 3d all have the same energies.
Where the energies become degenerate is with a
multi-electron atom. And we are going to talk about
that and how that reflects here, these wave functions in the
next day. That is all I am going to say
about the hydrogen atom. Now it is time to move on,
to helium. And, of course,
the Schrˆdinger equation predicts the binding energies of
the electrons to the nucleus in a helium atom also very well.
But, of course, it is a much more complicated
Schrˆdinger equation. And I am not even going to
write out the Hamiltonian in this case, but I want to show
you the wave function here. See the wave function?
The wave function is a function of six variables.
It is a function of two r's, two distances from the nucleus,
one for electron one, one for electron two,
two theta's and two phi's. We have six variables for the
wave function. And the consequence of this is
that our solutions for the binding energies for the
electrons in helium or any other atoms are not going to be nice
analytical forms. We are no longer going to have
e sub n equal minus the Rydberg constant over n squared.
If you actually solve for those
energies, and you have to do it numerically, you are just going
to get a list of numbers, a table of numbers,
but not a nice analytical form. If you solve for the wave
function, you are not going to get a nice analytical form,
like we got for hydrogen. Instead, what you will get is a
value for the amplitude of Psi as a function of r,
theta and phi. But if you get actually much
above three electrons, it turns out that even
numerically, you cannot solve the Schrˆdinger equation,
exactly. You have to use approximations.
And we are going to look at the most basic approximation that is
used that works, amazingly.
It works well enough for us to have a framework in which to
understand the reactions of these atoms.
And what is that approximation? Well, that approximation is
called the one-electron wave approximation or the
one-electron orbital approximation.
What does that mean? Well, that means this.
I am going to take my wave function here for the helium
atom, which strictly is a wave function that is a function of
six variables, and I am going to separate it.
I am going to let electron one have its own wave function and
electron two have its own wave function.
That is an approximation. In addition,
what I am going to do is let the wave function for electron
one have a hydrogen-like wave function.
I am going to say that it has the 1s wave function,
or the Psi(1, 0, 0) wave function of a
hydrogen atom. And I am going to let electron
two have the Psi(1, 0, 0) wave function of a
hydrogen atom. Or, I am going to write it as
1s of 1, for electron one, times 1s of 2,
for electron two.
Or, another shorthand, I am going to write it as 1s.
squared. And, if I continued on,
here, it is for lithium. Lithium, the wave function
strictly has nine coordinates, but I am going to let every one
of those electrons, in the one electron wave
approximation, have its own wave function.
And I am going to let electron one have a wave function that
looks like a hydrogen atom wavefunction.
The 1s wave function. The same thing with electron
two. And then I am going to let
electron three have the 2s wave function of the hydrogen atom.
And in simplified notation, that is just 1s squared 2s.
And here is beryllium, 16 variables,
but I am going to let every electron have its own wave
function. And I am going to give electron
one the 1s wave function, electron two,
the 1s, electron three, the 2s, electron four,
the 2s. I can also write that,
as you have already done, 1s 2 2s 2.
And I can keep going. And these electron
configurations that you have been writing down in high
school, that is what they are, electron configurations,
well, they are nothing more than our shorthand notation for
the electron wave functions within this one-electron wave
approximation. That is what those were,
that you were writing down. Those were a shorthand notation
for the wave functions in Schrˆdinger's equation within
this one-electron wave approximation.
Now, one thing you do notice is that I did not,
in the case of boron here, let all five electrons be in
the 1s state, or let all five electrons be
represented by a 1s hydrogen atom wave function.
I didn't because of a quantity that you already know about,
called spin. You already know that if you
are going to put electrons in the 1s state here that one
electron has to go in with spin up and the other spin down.
And the 2s, spin up and spin down, etc.
What is the phenomenon called spin?
Well, spin is entirely a quantum mechanical phenomenon.
There is no correct classical analogy to spin.
Spin is intrinsic angular momentum.
It is angular momentum that is just part of a particle,
such as an electron. The spin quantum numbers
actually come from solving the relativistic Schrˆdinger
equation, which we did not even write down.
When you solve the relativistic Schrˆdinger equation,
out drops a fourth quantum number.
That fourth quantum number we are going to call m sub s.
And we find that m sub s
has two allowed values. One of those values is one-half
and the other is minus one-half. Here, we have a case where the
quantum number is not an integer.
It is one-half and it is minus one-half.
Now, if it helps you to think about the electron spinning
around its own axis, like I depict here,
well, if that is the case, then the angular momentum
quantum number is perpendicular, here ,to this plane in which it
is rotating. And you might want to call that
spin up. And, of course,
if it is spinning in the other direction, well,
then the angular momentum vector is pointed in the
opposite direction. You might want to call this
spin down. If it helps for you to think
about this, okay, but remember that this is not
correct. This is a classical analogy
that we are trying to draw here. We are trying to say that this
electron is rotating around its own axis.
That is not true. This angular momentum is just
an intrinsic part, the intrinsic nature of a
particular such as an electron. Next time, I will tell you
about Uhlenbeck and Goudsmith. See you Wednesday.