This idea, that bosons are symmetric and fermions are antisymmetric under exchange of identical particles, is called the spin-statistics theorem. 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. 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. That is, we cannot tell one specific electron from another. That factor of -1 becomes important because of the idea in quantum mechanics that particles are interchangeable or identical. We call these particles fermions, after the physicist Enrico Fermi. Every spin-1/2 particle shares this behavior, such as quarks (the constituents of protons and neutrons) and electrons. 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. 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. 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. Particles with integer spin are called bosons, after the Indian physicist Satyendra Bose. Spin-1 particles are like number cards which must be rotated 360° to look the same as they did when they started. If you imagine a deck of cards, the spin-2 particles are like face cards that look the same when rotated 180°. This may seem strange, but what it means is that the spin value describes the symmetry of the particle. 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. 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. One major difference is in the behavior under rotation. But what do differences in these values mean? How does a spin-1/2 particle behave differently than a spin-0 particle? The allowed values for spin come from solutions to quantum mechanical energy equations. This is not an easy statement to understand, especially without seeing the math. 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.
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