Question regarding antisymmetry of a system of fermions

[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?

 

[R0man]

First of all, it's great that you're trying to relate the concept of antisymmetry in physics to its mathematical counterpart. This can definitely help in understanding the concept better.

In the context of a system of fermions, antisymmetry refers to the exchange of two particles resulting in a change in the overall state of the system. This is due to the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. So, in a system of fermions, if we exchange two particles, the overall state of the system will change because one particle can no longer occupy its initial state.

Now, let's apply this concept to the mathematical definition of antisymmetry. In the case of fermions, the binary relation would be the exchange of two particles, and the elements a and b would be the individual particles in the system. So, if a and b are exchanged, the result would be a change in the overall state of the system, which would not be equal to the initial state. This is similar to the mathematical definition of antisymmetry, where a and b are not equal in the case of aRb and bRa.

In the case of bosons, the exchange of two particles would not result in a change in the overall state of the system because they can occupy the same quantum state. This is similar to the mathematical concept of symmetry, where aRb and bRa would result in the same value.

So, in summary, the concept of antisymmetry in a system of fermions is similar to its mathematical definition, but it is applied to the exchange of particles in the system. I hope this helps in clarifying the concept for you.