# Question regarding antisymmetry of a system of fermions

1. Apr 22, 2004

### karma345

I'm having a little difficulty grasping this concept of antisymmetry in a system of particles with half integer spins... well, let me put it this way. I can see what antisymmetry means in that - if we take one of the particles and interchange it with another - because of Pauli's exclusion principle we can't have any of them with the same eigenstate - we'll see a difference. This as opposed to a group of Bosons which can share eigenstates, naturally if they're noninteracting indistinguishable particles, say a beam of light, we won't see any difference in the total state of the beam. So, say we have a He atom - and one of the electrons goes into a higher state, it absorbs a photon and rises to a higher shell - you can see it in the state of the He atom. So from that point of view, I get the concept - it's antisymmetric. But I'm trying to relate it to the mathematical concept of antisymmetric relations, and maybe I'm going off on some tangent I needn't be. The term antisymmetric relation means that if I have a binary operator relating two pieces of the puzzle - for example R is our binary relator less than or equal to, and we have two variables, a and b - if aRb AND bRa, then a=b is the definition (mathematically) of antisymmetry. I'm not seeing that here. Let's say we have two particles in our system X and Y both elements of the set F, the set F being our system. X and Y are both half integer spin particles - let's say they're electrons for simplicity's sake. (Note that I am now using X and Y to mean the eigenstates of these electrons) By Pauli, X cannot equal Y, and therefore Y cannot equal X - we can say that X is less than Y, and that Y is greater than X - but that doesn't give us our binary relation anymore, we'd have to have X less than Y and Y less than X which will never work out in a physical system. In other words, it seems to me that because of the binary relation needed we are implying that X does not equal Y which would be the exact opposite of symmetry, not antisymmetry. Is the concept of antisymmetry in the case of many noninteracting particles different, similar or precisely the same as antisymmetrical relation in mathematics? What am I not seeing here?