Physics 2000 Science Trek Elements as Atoms

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.

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