Last time on Sunday Function we talked about two types of symmetries that a real function might have: odd and even symmetry under reflection about the y-axis. Much more than I expected even as an undergraduate student, these types of symmetries turn out to be of amazingly fundamental importance in fundamental physics. One of these fundamental symmetry results in the Pauli exclusion principle, which says no two electrons (or any Fermions for that matter) can be in the same quantum state.

To do a little review, there’s a property of subatomic particles called *spin*. An electron isn’t actually spinning about a physical axis, but nonetheless this spin property is a real live angular momentum that’s just intrinsic to what an electron is quantum mechanically. Along with charge and mass and a few other basic quantum numbers, spin is simply a fundamental part of electron-ness.

An electron has a particular amount of spin. Without going into a lot of quantum mechanical background, it suffices to say that an electron has a spin of 1/2. Other particles of this class could have a spin of 3/2 or 5/2 or 7/2 and so on. These half-integer spin particles are known as Fermions.

Other particles (photons perhaps most obviously) have an integer spin. Their spins are 0 or 1 or 2 and so on. These are Bosons.

And so far as we know, there aren’t any other kinds of particles. No one has yet seen a particle with a spin of 3/5 or pi or anything that’s not n/2 for n integer. I think most theorists would be surprised if there turned out to be any such thing, but weirder things have happened.

Now in quantum mechanics particles are described by wavefunctions which specify everything you can know about a system. There’s lots of ways you can write them, but the easiest way to think of them conceptually is to imagine them as mostly describing the probability for a particle or system of particles to be found in a certain state. While there’s no *a prior*i reason for this to be true in nonrelativistic quantum mechanics, it so happens that the wavefunction of a system will be even under interchange of the particles if the particles are Bosons, and the wavefunction will be odd under the interchange of particles if the particles are Fermions. (This turns out to be provable in relativistic quantum mechanics. It’s rather complicated and to be honest I’ve never studied it in depth, but if you’re curious just Google the spin-statistics theorem and have fun.)

In other words, if you measure one particle’s position with the coordinate r1 and the other particle with coordinate r2, the overall wave function of the two-particle system for Bosons will satisfy this relationship:

So if you grab both particles and swap them you’ll have the same wavefunction for the system. Don’t take the “swapping” analogy too literally though, formally speaking that’s not really a well defined thing to say.

Now if the particles are Fermions they satisfy an antisymmetric relationship:

Swap the positions of the particles and you get the same wavefunction back, but with a minus sign in front of it.

This minor difference is the root of the Pauli exclusion principle. We’ll take a brief intermission here and come back tomorrow to actually derive the principle from the antisymmetry of Fermion wavefunctions.