Quantum Numbers
These are conventionally known as




The principal quantum number (n = 1, 2, 3, 4 ...) denotes the eigenvalue of H with
the J2 part removed. This number therefore has a dependence only on the distance
between the electron and the nucleus (ie, the radial coordinate, r). The average
distance increases with n, and hence quantum states with different principal
quantum numbers are said to belong to different shells.
The azimuthal quantum number (l = 0, 1 ... n−1) (also known as the angular
quantum number or orbital quantum number) gives the angular momentum
through the relation J2 = l(l+1) h/2π, where h is the universal constant known as
the Planck's constant. In chemistry, this quantum number is very important, since
it specifies the shape of an atomic orbital and strongly influences chemical bonds
and bond angles. In some contexts, l=0 is called an s orbital, l=1, a p orbital, l=2,
a d orbital and l=3, an f orbital.
The magnetic quantum number (ml = −l, −l+1 ... 0 ... l−1, l) is the eigenvalue,
Jz=mlh/2π.
The spin quantum number (ms = −1/2 or +1/2) was found experimentally from
spectroscopy.
To summarize, the quantum state of an electron is determined by the quantum numbers:
name
symbol
orbital
meaning
principal quantum
number
shell
azimuthal quantum
number
subshell
magnetic quantum
energy shift
range of values
value example
for
:
for
:
number
spin quantum
number
spin
always only:
Quantum Numbers
Each electron has a set of four numbers, called
quantum numbers, that specify it completely;
no two electrons in the same atom can have the
same four. That's a more precise statement of the
Pauli exclusion principle Bob was discussing. (He also
mentioned still another way of expressing this important
idea.)
Is there a special reason why there are four, and
not three or six or fifty-nine?
Good question. There are certainly reasons, but I
won't be able to explain them to you here, any
more than Bob could explain where his rules
were coming from. What I can offer you is a
mathematical expression of those rules, which I hope
will make them easier to work with and perhaps provide
some insight into the underlying patterns.
Okay, I can live with that. Tell me about the four
numbers.
First, the "primary quantum number," which is
given the symbol n, corresponds to those colored
rows you saw in the chart. The lowest row, the
pink one, has electrons with n=1; the yellow row
is n=2, and they go up from there.
All right, so n tells you which of the "main" energy
levels you're in. I suppose there's another quantum
number that goes with the sublevels--s, p, d, and all
that.
Very good. The second quantum number is
known as l. A value of l=0 corresponds to s, l=1
is p, l=2 is d, and so forth.
This all seems very abstract to me. What does l
really mean? Can you give me some concrete way
to think about it?
I have two answers for that. First, l,
unlike n, does have an association with
angular momentum. If you'd like to
know more about this, click on the
"advanced" button at right.
If "angular momentum" means nothing to you, don't
despair. You can also picture its significance this way: l,
along with n and the third quantum number, m, is
responsible for determining the shape of an electron's
probability cloud. Here are a few examples:
n=1, l=0, m=0
n=3, l=2, m=1
n=3, l=2, m=2
n=4, l=2, m=2
Quantum Numbers II
What does the third quantum number mean?
The number m also has a connection with
angular momentum, but it's not necessary to
know the details of that in order to make some
sense of m's significance. The key point about m is that it
does not affect the electron's energy, although, as you've
seen, it does change the shape of the electron cloud.
So when Bob said before that there could be
different kinds of clouds at the same energy, what
he meant was that there could be different values of
m for the same n and l.
That's absolutely right. For example, here are the
quantum numbers for the two different p states
Bob showed you:
n=2, l=1, m=0
n=2, l=1, m=1
That reminds me of some other questions I had.
First, Bob said that the number of sublevels keeps
going up with each primary level. Can you explain
that, in terms of quantum numbers?
What's happening is that there are restrictions on
the possible values of each quantum number. n is
allowed to be any positive integer. Within the level given
by a particular n, l can take on only integer values from 0
to n-1.
So when n is 1, l can only be 0, and that's why the
first row has only s states. Then when n=2, l can be
either 0 or 1, and that gives you s and p--I get it!
I also remember Bob saying that there's only one kind of
s state, three kinds of p states, five kinds of d states, and
so on. Does that mean there are also restrictions on what
m can be?
Very good. Given a particular l, m is entitled to be
any integer from minus l up to l. For example, when
l=1, m can be -1, 0, or 1; those are your three p
states. If you work it out, you'll see that for a given l,
there are 2l+1 different values of m.
Quantum Numbers and Spin
Well, three quantum numbers down, one to go. You
haven't mentioned anything about spin yet...
How perceptive of you. The fourth quantum number,
s, does indeed pertain to an electron's spin.
I know that each electron has either "spin up" or "spin
down"...
Yes, and, correspondingly, s only has two possible
values. For good reasons that I shall not explain
here, s= +1/2 means "spin up" and s= -1/2 means
"spin down."
So that's it; those four quantum numbers completely
describe an electron. I see how this fits in with Bob's
visual picture of a "quantum state":
+
n=3, l=2, m=0
s= -1/2