Bose-Einstein Condensates, pt. 1

It’s been a while since we’ve worked on some slightly more heavy-duty physics, so I’m going to spend the next couple of days going through the basics of the theory of Bose-Einstein condensation. Let me set the stage:

Particles can be divided into two classes based on a quantum mechanical property called spin. Spin in this context isn’t a physical rotation of a solid object, but an intrinsic quantum mechanical property that happens to be analogous in many ways to the physical rotations we’re used to seeing in the real world. For some reason the world is constructed in such a way so that a particle can’t have whatever spin it wants. Spin comes in quantized units. In terms of the multiples of this fundamental unit, a particle can have spin 0, 1, 2, 3… and so on. Or it can have spin 1/2, 3/2, 5/2… and so on. Integers and half-integers, that’s it. Nothing in between. The particles with integer spin are called bosons and the particles with half-integer spin are called fermions.

You may also know that spin isn’t the only quantity in quantum mechanics that’s restricted to particular values. Other properties – energy in particular – also comes in discrete quantities, the size of each quanta being determined by the particular properties of the system in question. Each of these quantized properties is described by a quantum number, and the complete set of quantum numbers for a particle is called a state of that particle.

There is a law of nature called the Pauli exclusion principle that says no two fermions can share the same state. It’s just not possible. Toss a bunch of electrons into a box (electrons are fermions) and only one of them* can occupy the lowest energy state no matter how much you refrigerate the box. Bosons have no such restriction. Pull enough energy out of the box and you can pack as many bosons into that lowest quantum state (also called the ground state) as you want.

But it turns out there’s a trick. Not only is there no limit to how many particles the ground state can hold, under certain low-temperature conditions there is a finite limit on how many particles all of the excited states can hold. This will force all the remainder of the particles into the ground state en masse, which is the heart of Bose-Einstein condensation.

That’s the wordy version. Over the next few days we’ll go through the detail. I’m going to try to keep the explanations between the math accessible for everyone, but the math itself assumes a fair fluency with calculus and the methods of advanced undergrad / intro graduate statistical mechanics. It’ll be fun, I promise!

*Two, if the ground state energy is degenerate with respect to the spin of the electron itself.

1. #1 rob
April 22, 2009

you forgot spin 0 particles.

the pion has spin 0. so do a bunch of un-discovered particles, such as the higgs boson.

2. #2 rob
April 22, 2009

oh, and i think paris hilton is obviously a boson, since she hangs around with anyone.

heh.

3. #3 Matt Springer
April 22, 2009

Indeed. Fixed!

4. #4 magista
April 22, 2009

Heh. There should be some sort of celebrity-QM corrolary that says that they need to be seen a certain minimum number of times in public to continue to exist…

Thanks, I’m looking forward to seeing if I can still sling the necessary math. High school teaching tends to atrophy the advanced math brain.

5. #5 andyb
April 23, 2009

…and photons are bosons, which permits laser-beams.

6. #6 Mark
May 5, 2009

Great series, I just read the final post. But I either missed this, or you didn’t explain:

How big is the box that the Pauli exclusion principle applies to? What defines it? Can no two fermions in the entire universe share the same state, or just those near each other?