Spin, Rotation, and a Plate Trick

In my introduction to the quantum number spin, I mentioned that particles can have half-integer or integer spin, and that which they have deeply affects their behavior. This is not an easy statement to understand, especially without seeing the math. The allowed values for spin come from solutions to quantum mechanical energy equations. But what do differences in these values mean? How does a spin-1/2 particle behave differently than a spin-0 particle?

One major difference is in the behavior under rotation. When we try to calculate how rotation affects a particle with spin-0, we find that it doesn’t matter: the particle is indistinguishable before and after any rotation. However, a spin-1 particle requires a 360° rotation to return to its initial state, and a spin-2 particle requires a 180° rotation to return to its initial state. This may seem strange, but what it means is that the spin value describes the symmetry of the particle. If you imagine a deck of cards, the spin-2 particles are like face cards that look the same when rotated 180°. Spin-1 particles are like number cards which must be rotated 360° to look the same as they did when they started. Particles with integer spin are called bosons, after the Indian physicist Satyendra Bose.

There are no playing cards which must be rotated 720° in order to look the same, and yet this is the case with spin-1/2 particles. There are few macroscopic objects that can demonstrate this property, but one of them is your hand! Place any object on your hand, palm up, and rotate it without dropping your palm. After 360° you will find your arm to be pretty contorted, but after 720° of rotation your arm has regained its initial position! Another way to think of it is that, instead of a 360° rotation bringing the object back to its initial state, which would be like multiplying by 1, the 360° brings the object to another state like multiplying by -1, and then an additional 360° rotation multiplies by (-1)*(-1) which equals 1. Every spin-1/2 particle shares this behavior, such as quarks (the constituents of protons and neutrons) and electrons. We call these particles fermions, after the physicist Enrico Fermi.

That factor of -1 becomes important because of the idea in quantum mechanics that particles are interchangeable or identical. That is, we cannot tell one specific electron from another. Mathematically, you can state this by writing a function that describes the positions of two particles, and seeing what happens to that function when you exchange the particles. If you do this, what you find is that bosons are symmetric under particle interchange and the function stays the same, but fermions are antisymmetric under particle exchange, and the function is multiplied by -1.

This idea, that bosons are symmetric and fermions are antisymmetric under exchange of identical particles, is called the spin-statistics theorem. A thorough proof requires relativity and quantum field theory, but the fundamental cause is the differing rotational behavior due to spin as a measure of symmetry. One very important consequence of all of this is that if you have two fermions occupying the same state, and you exchange them, you find that the function describing their position cancels out to zero. This is a mathematical statement of the Pauli exclusion principle forbidding two fermions from being in the same quantum mechanical state!

On the other hand, we find that bosons are perfectly happy to all pile into the same quantum mechanical state, at least at low temperatures. This is the concept behind the Bose-Einstein condensate, the state of matter experimentally realized only 30 years ago in which bosons can be cooled into occupying the same state.

I hope this makes the connections between spin, the Pauli exclusion principle, and particle types clearer. But if nothing else, the rotational exercise with an object on your hand, better known as Feynman’s plate trick, is fun at parties.