Two Fermions Walk Into a Bar...

So, in the previous post about symmetry and the difference between bosons and fermions, I threw in a bunch of teasing comments about how the requirement that quantum particles be indistinguishable has surprising and interesting consequences. Of course, I never quite explained what all that was about. Which, I suppose, means I'm obliged to pull out something pretty big to hold up as an example of an interesting consequence of the symmetry requirements.

Well, how about chemistry? Not some sub-part of it-- the whole field. If it weren't for the requirement that quantum particles be indistinguishable, we wouldn't have chemistry.

(Explanation after the cut.)

If you took chemistry in high school, you spent a lot of time drawing diagrams that look something like this:

↑ ↓ ↑ ↓ ↑ ↓ ↑_

The arrows in those diagrams represent electrons, which you put into available energy shells in pairs, one up, and one down. You're only allowed to put two electrons in each state, and there are different numbers of states available in the different shells.

For any given element, you fill these shells up until you run out of electrons, and the chemical properties of that element are determined by the state of the last shell. An alkali atom like sodium ends up with one lonely electron in the outermost shell, and is highly reactive as a result. A noble gas like krypton ends up with a full complement of electrons in its outermost shell, and as a result doesn't react with anything.

These diagrams are a mnemonic device for applying what's known as the Pauli Exclusion Principle, which is just a consequence of the fact that electrons are fermions, and must be in anti-symmetric wavefunctions.

The little arrows represent a property of the electrons called "spin," which can take on one of two values, conventionally called "up" and "down." Each electron has its own spin, which can be either up or down, depending on the physics of the system.

When you set out to write the wavefunction of the electron, then, there are two parts you need to keep track of. There's the normal spatial wavefunction, which tells you what the probability of finding an electron in a given position is, and there's the spin part of the wavefunction. If you have two electrons, the total state of the system needs to be anti-symmetric (it needs to pick up a negative sign when you change the labels on the electrons), which means that one of the two parts of the wavefunction has to be anti-symmetric.

It's easy to understand the symmetry properties of the spin wavefunction. If the two spins are both up, or both down, then the state is symmetric. Swapping the "A" and "B" on a state like ↑_A↑_B doesn't change anything. If you want the state to be anti-symmetric, you need to have the two spins be different-- one up, and the other down. Actually, you really need them to be in a singlet state like (↑_A↓_B - ↓_A↑_B), but that's a detail. The key thing is that the spins are different.

It's a little harder to see how to assign symmetry to spatial wavefunctions, but I hope it's obvious that having the two electrons exactly on top of one another is symmetric. If you want to get an anti-symmetric state, you need to have a state with a high probability of finding the two electrons in different places.

If you put these together, you get the Pauli Exclusion Principle. If you have two electrons, there's no way to put them in exactly the same state, meaning the same spatial and spin wavefunctions. If the two have the same spin, that's a symmetric spin state, and if they have the same position, that's a symmetric spatial state. One or the other needs to be different.

The "energy shells" that you fill up in chemistry classes represent different spatial wavefunctions for an electron orbiting the nucleus of an atom. Each energy level represents a particular probability distribution for finding the electron at a given position, and putting two electrons in the same energy level is equivalent to forcing them to have the same position. If they have the same position, the spatial wavefunction is symmetric, which forces the spin wavefunction to be anti-symmetric-- that is, the spins have to be different. There's no way to have the two have the same spin state, and there's no way to jam a third electron in there and still have the wavefunction be antisymmetric.

That's Pauli Exclusion, which is commonly expressed as something like "No two fermions are allowed to occupy the same quantum state." It has a number of important consequences, the biggest being the field of chemistry. If not for Pauli Exclusion, all the electrons in an atoms would occupy the same state, typically the lowest energy state, and there would be no particular reason for them to bind together to form molecules. In a very deep sense, the only reason we're here at all is because electrons are fermions, and have to occupy anti-symmetric wavefunctions.

This also affects protons and neutrons, which has important consequences in nuclear physics. You can get a basic understanding of why certain nuclei are unstable just by thinking of protons and neutrons are up and down arrows that fill up the available nuclear states in the same way that you fill electron energy levels in chemistry. It's also the key to determining the electrical behavior of solids. Electrons in a solid fill up energy bands due to Pauli exclusion, and the energy of the final electrons to go in to a given material is what determines whether you have an insulator, a conductor, or a semiconductor.

So, the consequences of the symmetry requirements of quantum mechanics are pretty far-reaching, at least when it comes to fermions. Bosons do different things, which are pretty interesting in their own right, but they're a little more exotic than fermions, and will have to wait for another post...

(The title, by the way, is the opening to an incredibly dorky joke:

Two fermions walk into a bar.

The bartender asks, "So, what'll it be."

The first one says, "I'll have a gin and tonic."

The second one replies, "Dammit, that's what I wanted!"

(Thank you, I'll be here all week. Try the fish.)