Penumbral Lore; the Shadows of Science

The first time I was exposed to Quantum Theory the word ‘quantum’ wasn’t even mentioned. It was in a High School chemistry class, and we were discussing electron shells, and the lecture covered how the various sub shells explained the periodic table of the elements. I think it was the first time a chemistry lecture actually made sense to me (probably an ADD thing - harder concepts become much easier if I can find ‘teh shiny’).

I’m going to try to reproduce the explanation here, because an understanding of electron shells is a necessary part of getting the experimental proof of the Heisenberg uncertainty principle I mentioned before. As before, I draw on Wikipedia as a source, with common sense checks. This time I also found a reference at Georgia State University, which explains the Pauli Exclusion Principle with much more precise language. Please let me know when I go around the bend…

In 1925, Wolfgang Pauli came up with the idea that certain particles (called fermions) interact in a weird way. No two of them can exist ‘in the same state’ at the same time. For example, each electron in an atom will have its own set of ‘quantum numbers’ describing it. Quantum numbers are whatever set of metrics are needed to describe the quantum state of the system in question - <s> apparently physicists can use a bunch of them at will</s> (see Edwards explanation in comments - there is logic to how they are chosen). There are typically four quantum numbers which describe the electrons in an atom:

n

the “principle quantum number” or the number of the ‘electron shell’.  Roughly corresponds to the distance between the electron and the atomic nucleus, <i>and is a good indicator of the energy held by the electron</i>. Can be any positive integer (1,2,3…)

l

the “orbital quantum number”. It gives the ‘orbital angular momentum’ of the electron. It’s in integer ranging from 0 to n-1, so an electron in electron shell 1 (n=1) l can only equal 0. If n=2, l can equal 0 or 1, etc… For n=3 l can be 0, 1 or 2 (which are labeled s,p,d just to make it more confusing all around).

m₁

the “magnetic quantum number”, sometimes called ‘eigenvalue’ - it’s the projection of the orbital angular momentum along a specified axis. Values can range from -l through l.

m_s

the “spin” can be either -½ or +½ - so for any n l m1 combination there can be a ’spin up’ or ’spin down’. I usually picture a tiny, spherical electron spinning clockwise or counterclockwise, but that’s not actually what’s happening.

So, any electron in an atom will have four numbers describing it, a unique address, as it were.

For any given atom, there can be two, and no more than two, electrons with a principle quantum number of 1.
n=1 therefore l=0 therefore m₁=0 and m_s=½ or -½. If there are two electrons with n=1, we’d say that electron shell is “closed”. Atoms with closed electron shells are very stable (like helium).

I sometimes indulge in anthropomorphism. That’s a fancy way of saying I think of them as though they were people, with personalities, likes and dislikes. Atoms ‘want’ closed electron shells. Two hydrogen atoms have two electrons between them. If they share the electrons then each has a full n=1 shell, and I often say to myself this ‘makes them happy’. It actually makes them stable. Whether stability is the same thing as happiness in people is possibly a question for another post altogether.

So, each principle electron number can hold more electrons than the number before it. Atoms which have all the electrons which their principle electron number can hold are very stable, and don’t share electrons easily (if at all).

As n increases, the possible number of electrons in the orbital increase. There are also various ’suborbitals’ which describe the different l and m₁ levels. Some of them are more stable than others, etc and so on. As you compare elements with different full and empty suborbitals and orbitals you find that their chemical properties are similar in those with similar electron shells. You can predict an elements chemical properties from it’s atomic number (the number of protons it has), if you understand what suborbitals are full or empty in its outermost orbital.

The periodic table shows this, more or less easily. The really cool thing is that Dmitri Mendeleev came up with the first version of the periodic table in 1865, well before Pauli. He understood it well enough to put elements in the right order, even though their atomic weights didn’t match up with the atomic numbers (that whole neutron thing), and he predicted the existence and properties of a number of elements which hadn’t been discovered yet.

It really is a beautiful thing.

Edited on 9 November to reflect Edward’s clarifications.