Question regarding antisymmetry of a system of fermions

In summary: In contrast, a group of bosons (e.g. a beam of light) can share eigenstates, so there is no change in the total state of the system.In summary, the concept of antisymmetry in a system of particles with half integer spins is related to Pauli's exclusion principle and the mathematical definition of antisymmetry. It involves an antisymmetrical relation between particles and their respective eigenstates, where two particles cannot have the same eigenstate. This is different from the concept of antisymmetrical relation in mathematics, which relates to binary operators and elements.
  • #1
karma345
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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?
 
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  • #2
It sounds like you have a good grasp on the physical concept of antisymmetry, but it can be helpful to look at the mathematics of it as well. The mathematical definition of antisymmetry is that if a binary relation (e.g. less than or equal to) holds between two elements (a and b), then if aRb and bRa then a=b. In the case of particles with half integer spins, this can be thought of in terms of Pauli's exclusion principle - two particles with the same quantum state cannot occupy the same space. So in this case, the antisymmetrical relation is between two particles, X and Y, and their respective eigenstates. If X is less than Y, then Y must be greater than X (and vice versa). This means that X and Y cannot have the same eigenstate, and thus the antisymmetrical relation holds.

To further illustrate this, consider the example of a helium atom. If one of the electrons absorbs a photon and rises to a higher energy level, then the total state of the helium atom will be different from its original state. This is because the electron has shifted to a different eigenstate, thus breaking the antisymmetrical relation between the two electrons.
 
  • #3


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.
 

1. What is antisymmetry in a system of fermions?

Antisymmetry in a system of fermions refers to the property that the wave function describing the state of the system must be antisymmetric under the interchange of any two fermions. This means that the wave function changes sign when the positions of any two fermions are swapped.

2. Why is antisymmetry important in systems of fermions?

Antisymmetry is important in systems of fermions because it is a fundamental property of fermions, which are particles with half-integer spin. This property ensures that no two fermions can occupy the same quantum state, which is known as the Pauli exclusion principle. Without antisymmetry, the behavior of fermions would be drastically different and many fundamental laws and phenomena in physics would be violated.

3. How is antisymmetry mathematically represented in a system of fermions?

In a system of fermions, antisymmetry is mathematically represented through the use of the Slater determinant. This is a mathematical expression that ensures the correct antisymmetric behavior of the wave function by taking into account the interchange of fermions.

4. Can you give an example of a system of fermions that exhibits antisymmetry?

An example of a system of fermions that exhibits antisymmetry is the electron configuration in atoms. The electrons in an atom are fermions and their wave function must be antisymmetric, which leads to the observed electron configurations and their distinct energy levels.

5. How does antisymmetry affect the behavior and properties of a system of fermions?

Antisymmetry has a significant impact on the behavior and properties of a system of fermions. It determines the allowed energy levels and configurations of fermions, as well as their interactions with each other and with other particles. Additionally, antisymmetry plays a crucial role in understanding and predicting phenomena such as superconductivity and magnetism.

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