Physics 2000 Science Trek Elements as Atoms

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



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