Let's look at the allowable values of the four quantum
numbers.
Allowable values.
There are the values that they can take on.
And sometimes quantum numbers get a little bit confusing
because we use the term numbers, yet sometimes,
as you'll see in a moment, that we used letters
to refer to the numbers.
So it can be a little confusing.
Well, it can be quite confusing actually.
And I hope by structuring this is as best I
can in an organised fashion, it's
not too intimidating for you.
So first off, the principle quantum number
can take on integer values starting at 1.
n equals 1, 2, 3, 4, and so on.
The angular momentum quantum number
then can take on values from 0 up to n minus 1.
So for example--
For example, if we had--
Oh, let me see.
We have n equals 2.
Well, if n equals 2, we can l equals 0, or l equals 1.
And we'll do some more examples of this as we proceed.
And then the magnetic quantum number
can take on values from negative l-- OK, negative
l-- to positive l.
So that's all this means, anywhere
from positive to negative l.
So and example, if we continue with this n equals 2,
l equals 0 or 1.
Well, if had l equals 1, we could proceed and have values
for l equals 1, or for m sub l equal to 1.
Sorry.
For values of l equals 1-- that is the angular momentum quantum
number equal to 1-- m sub l, the magnetic quantum number,
could take on values of negative 1, 0, and 1.
And in fact, that in the previous example
refers to those different spatial orientations--
--about x, y, and z as you often see.
I'm going to come back in just a and give you
some more concrete examples.
The spin quantum number is fairly straightforward,
plus 1/2 or minus 1/2.
But I warned you that we're going
to use some letters and things as well to refer synonymously
to these numbers.
So some disciplines, some spectroscopists
prefer instead of n equals 1, 2, 3, 4, they use capital letters.
And so K refers to n equals 1.
L, M, N. In fact, there's certain series of energy
transitions, these spectral lines that correspond
to electrons dropping down to the K shell or the L,
thinks like K alpha, for example,
if you've ever seen that, refers to a transition down to n
equals 1.
So then the angular momentum quantum number,
which you remember I told you describes
the shape-- describes the shape.
So if it describes the shape, the shape that corresponds to l
equals 0 is a sphere.
And the letter that's often used is the letter s.
And that actually refers to a historical reason
from spectroscopy.
There's a sharp line.
But not sphere.
So don't get confused there.
It's not that it starts with an s for sphere.
l equals 1 describes sort of a dumbbell shape.
And that's what I had sketched previously.
Let's see if I can sketch that again for you.
This is a rough shape if you can imagine
that in three dimensions-- OK, shade that for you-- of the p
orbital.
And that corresponds to l equals 1.
And those three states of it here,
of the spatial orientations, then you
could say well then this is p sub x.
This is p sub y.
And this is p sub z.
These different values of m sub l.
Sorry, I gave you s I forgot to give you p.
And that was what I just showed you here. p is l equals 1.
That's p.
And then d is l equals 2, and so on.
So f would be next.
But we'll stick around s, p, and d for this course.
Let's see.
So we've described the principal quantum number.
Could use either uppercase letters or the numbers.
So the angular momentum quantum number, most frequently,
these letters s, p , d, f are used.
I'd also like to show you another way that you've
potentially seen this previously.
Lot of people maybe have done some of this in high school,
or you've seen it in a textbook or something.
You've perhaps seen a few things.
You've seen these electron box diagrams perhaps.
That's the s in that little box.
You might see p going for sort of a more of a rectangular box.
d, and then even longer rectangular box.
And these boxes are broken down.
This one would have three boxes within it for p.
This one would have-- I wasn't very clear-- would have 5.
So this is depicting the values of the angular momentum quantum
number.
OK.
So for l equals 0, s, that shows you
that there's one possible spatial orientation.
That is if l equals 0, well then spatial orientations possible
range only from negative 0 to positives 0.
So that's just only one value.
So if we have the s subshell, you can only have spin up
or spin down it in, There's only going to be one orientation.
But then for l equals 1, we've got p,
and three possible spatial orientations
of it-- negative 1, 0, and 1, or p
sub x, p sub y, p sub z, or negative 1, 0, or 1.
Now be a little careful depending
on your discipline as to whether you
assign px, py, and pz to negative 1, 0,
and 1 as I've done.
I believe in organic chemistry, there's
usually a convention to assign a bond along the z axis.
And I could be wrong, but I believe that that often
is attributed to a value of 0 for m sub l.
But for our purposes, it doesn't really
matter what the mapping is.
It's really just kind of a cute memory aid
anyway with these spatial orientations.
But hopefully you can see this is starting to piece together.
And with the spin values, we have spin up, spin down,
going to each.
So there's three possible states we say within p.
And d has five.
And again, you can electron spin up and spin down
into each of them.
OK?
So those are the allowable values for the quantum numbers.
The final thing that I will show you
is again, one more thing that you might have seen
dealing with quantum numbers.
And it's just, OK, what's the relative energy
level for all these possible combinations
of quantum numbers?
And there's a little memory aid that
may have seen where you take-- you write 1s.
And then below that, you write 2s and then 2p. and then 3s
and 3p, and then 3d.
So we're going up to increasing values of angular momentum
quantum number.
And in fact you can see the same.
This little pyramid or half pyramid is describing, again,
these same allowable values that I just
mentioned to you up there.
So then you get to n equals 4, well we'd have 4s, 4p.
4d, 4f.
And then the neat thing though with this
is after you've done that, the next trick is you draw
this little diagonal arrow.
And it shows you, OK, you go through 1s.
That means 1s has the lowest energy.
Then 2s, then 2p, and 3s.
So no surprise.
You're going in increasing order of principal quantum number
as you would really just imagine from the Bohr model.
The Bohr model, you've get these different shelves.
It makes sense that the lowest energy
would be the one closest to the nucleus and so on.
But then an interesting thing happens here.
After you go through 3p, you notice
that 4s has the arrows go through it before 3d.
And what that tells us is in fact that 4s
is a slightly lower energy upon filling to make
a neutral atom than 3d is.
So this is, again, another way that you can often
refer to the energy levels.
And it's using these same quantum numbers
that we saw up here and those same allowable values.
So I really hope that this is starting to come together
and is not too intimidating.
But give it some time.
It does take some time to digest.