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What exactly is spin? Why does Pauli's exclusion pirnciple hold true? What is a Lepton number and why does it have to be conserved?
- Bob (age 19)
Seattle, WA, USA
Wow, it doesn't take many words to ask some very tough questions.
On spin: many particles have some internal state which is often described as having two possible states. These two states have angular momentum of +/- h_bar/2 along some chosen axis. To be more careful, what we really mean is that any of the possible internal states can be made up out of a combination of these two states. Formally, it's a vector space of dimension 2. Usually the two states also have different (and opposite) magnetic moments. Saying that the spin is 1/2 (in units of h_bar) means that rotation of the state by 360° multiplies by -1, not by +1 as you might expect for a complete rotation. I think I've told you what spin is, but know that's not a very intuitively satisfying answer.
On Pauli's exclusion principle: There's something called the spin-statistics theorem. It says among other things that for spin-1/2 particles, only 0 or 1 particle can go in any quantum state. The argument is not trivial. You start by representing the possible states for some type of particle in terms of things called creation and annihilation operators, which either add a particle of some type or remove it. You then ask how to represent the change in the state as you go to a moving reference frame in terms of those operators. You then make a loop of those changes by adding little motions to your reference frame until you come back to the starting reference frame's motion. You can show that this is equivalent to rotating the starting frame a little. So now you have an expression for how the state changes as you rotate it a little in terms of the creation and annihilation operators. You manipulate this a bit and find that for s=1/2 the product of two creation operators for the same state is zero. So you can't have two of these in the same state. You call that the Pauli exclusion principle. See? Easy!
That strains my mind enough for today. Let me steer you to another site for lepton number: http://en.wikipedia.org/wiki/Lepton_number
(published on 06/09/2011)
Follow-Up #1: spin statistics
Particles with a 1/2 spin are fermions and they all obey Pauli's exclusion principle. Particles with a 1 spin are bosons and they do not obey Pauli's exclusion principle thus allowing them to be in the same quantum state at the same place at the same time making them very different from fermions in nature. However, supersymmetry predicts subatomic particles with a 0 spin and a 3/2 spin. What would be the properties of particles of such spins? Would they obey Pauli's or would they obey other principles? And how can they be candidates for dark matter?
This is close to another question, so I've marked it as a follow up.
As my extremely sketchy outline of the spin-statistics theorem indicates, whether a particle is a boson or a fermion depends only on whether its spin is an integer or a half-integer. So 0 is boson, 3/2 is fermion.
There are plenty of composite S=0 and S=3/2 particles around, so we aren't just relying on theory here, although the theory is extremely solid. There are numerous indications that S=0 fundamental particles exist quite aside from supersymmetry.
I don't know much about which particles are good candidates for dark matter. The key traits needed are to have some rest mass (to allow clumping with galaxies) and to lack electric charge and QCD "charge" so as not to interact much with ordinary matter.
(published on 07/08/11)
Follow-up on this answer.