Yesterday we talked about how fermions and bosons had different values of spin and thus their wavefunctions had different symmetry properties. In particular, fermions are antisymmetric under exchange of particles. We’d like to write the overall two-particle wavefunction in terms of the individual wavefunctions for each of the two particles. The result will look like this:
You can see that if you switch around r1 and r2, you get the same function back with a negative sign in front of it. If these particles were bosons, we’d need a plus sign in the middle instead of a minus sign so that we’d get the same function back without a negative sign. Now at this point I’m a little torn. Calculating the so-called exchange interaction is a somewhat involved operation that does require a no-kidding understanding of the mathematical methods of quantum mechanics, and as much as I’d like to do this in full formal detail I’d rather not glaze over too many eyeballs. So for the moment we’ll do this the easy way, and at some point in the future we’ll do it the hard way.
Imagine that the two particles did have the same wavefunctions: phi1 = phi2. Looking at the overall two-particle wavefunction, we see that the second term would be the same as the first term. Since they’re being subtracted, the result is a flat 0. Thus two fermions can’t be in the same state, or there wouldn’t be a wavefunction in the first place. This restriction for fermions is called the Pauli exclusion principle. Bosons have no such restriction. Since they have a plus sign instead of a minus sign, you just get a sum of the two terms – not zero. Thus they can be in the same state if they feel like it.
And that’s it. How could you have a hard way? The answer is that interesting things happen when you have two particles in close to the same state. Doing the math, it’s possible to show that as phi1 and phi2 get closer and closer to being identical, this is equivalent to increasing the energy of the system. A change in energy with respect to volume (as the particles are forced closer) is the definition of pressure. It’s this pressure that helps keep solid objects solid, most dramatically in situations like neutron stars.
Fundamental as this is, it’s not something physics has known forever. The principle itself is less than a century old, and a good understanding of the consequences of Pauli’s principle with respect to the behavior of solids was only well-developed starting in the 60s.