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.

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In two dimensions you can have particles with any spin, suitably called anyons.

http://en.wikipedia.org/wiki/Anyons

yeah, true, you can have anyons. they are collective excitations of particles (quasi particles) rather than fundamental particles.

fundamental particles have quantized half integer or integer spins.

Matter is fundamentally fermionic, f(x) = -f(-x). Physical theory is fundamentally bosonic, f(x) = f(-x). Observation versus fundamental theory has three resolutions:

Begin with beautiful maximal symmetries and then add exceptions, or write theory with no testable predictions, or write theory ab initio consistent with empirical reality. The "beautiful" cases are degeneracies, as the formula for paralellepiped volume

(abc)sqrt[(1 - cos^2(alpha) - cos^2(beta) - cos^2(gamma) - 2cos(alpha)cos(beta)cos(gamma)] = a^3 for a cube.

Chirality requires N+1 non-coplanar points in N dimensions. It is an emergent phenomenon and an extrinsic property. It empirically exists as a nuisance not as an origin. Metric gravitation is beautiful but untested in the manner of Yang and Lee. Quantum gravitation theories require supplementing Einstein-Hilbert action with an odd-parity Chern-Simons term, f(x) = -f(-x). Do chemically identical, self-similar, opposite parity atomic mass distributions violate the Equivalence Principle? Somebody should look.

"if you measure one particle's position with the coordinate r1 and the other particle with coordinate r2"

I thought you couldn't measure the coordinate position of any particle due to the uncertainty principle?

@Krist: The uncertainty principle limits the precision to which you can simultaneously measure position and momentum. You can choose to measure one or the other (but not both) with as much precision as your equipment allows.

Right, a wavefunction can't pin down the location of a particle exactly. The two variables r1 and r2 are continuous variables which serve as labels for the two particles. For a 1-d wavefunction of one particle you'd have the wavefunction psi(x), which measures the probability density for a particle to be near x, where x is whatever location you want - a variable, in other words. It doesn't mean the particle is fixed at some particular x constant.

So roughly speaking the two-particle wavefunction is a measure of the probability density for having particle #1 near r1 and particle #2 near r2, for whatever values of the variables r1 and r2 you want.

In Griffith's Intro to Particle Physics, he has you prove that the angular momentum operator doesn't commute with the hamiltonian (actually the 0 component of the 4-momentum operator). Hence, angular momentum is not conserved.

But add in the spin angular momentum to get the total ang. mom. and voila, the commutator is zero.

But in Nonrelativistic QM, [H,L] = 0 (for a central force) without adding spin. This suggests to me that spin is some kind of relativistic artifact.

This is supported by the fact that H = cp_0 = c\gamma^0( \vec{\gamma} \cdot \vec{p} + mc).

(here the \vec{\gamma} is the vector (\gamma^1,\gamma^2,\gamma^3) )

When v<

It would probably make more sense if that second to last sentence were to read:

When v is small compared to c, H ~= mc^2, and that commutes with L.

Thank you for the clarification. it makes sense.