# Three Plus Anti-symmetric Particles

• I
• JohnH
In summary, fermions are anti-symmetric under exchange, and this applies even when there are three or more particles. The wave function for three electrons must reverse sign under the exchange of any two particles, and the same applies for any pair of particles in a multi-particle system.

#### JohnH

So I understand that fermions are anti-symmetric under exchange, but in the contexts I've seen this explained they were always talking about two particles, or at least two wavefunctions. I'm curious how this works when there are three or more particles. Is any two given pairs of those 3+ particles anti-symmetric under exchange or is it more systematic? Or is it that there's essentially one wavefunction for all the particles (quanta of energy) in one spin and another wavefunction for all the quanta of energy in the other spin such that it's just those two wavefunctions that are anti-symmetric under exchange?

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JohnH said:
So I understand that fermions are anti-symmetric under exchange, but in the contexts I've seen this explained they were always talking about two particles, or at least two wavefunctions. I'm curious how this works when there are three or more particles.
The wave function for three electrons is covered here:

https://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm

The total wavefunction must reverse sign under the exchange of any two particles.
JohnH said:
Is any two given pairs of those 3+ particles anti-symmetric under exchange or is it more systematic? Or is it that there's essentially one wavefunction for all the particles (quanta of energy) in one spin and another wavefunction for all the quanta of energy in the other spin such that it's just those two wavefunctions that are anti-symmetric under exchange?
I can't make any sense of this part of your question.

JohnH and topsquark
PeroK said:
I can't make any sense of this part of your question.
Yeah, just trying to answer my own question last minute. Probably better off waiting for an answer. Anyway, thank you for it.

The wave functions of indistinguishable fermions must be antisymmetric under exchange of any pair of arguments, ##(\vec{x}_j,\sigma_j##, where ##\vec{x}_j## is the postition and ##\sigma_j## the spin-##z##-component (##\sigma_j\in \{-s,-s+1,\ldots,s-1,s \}##, where ##s## is a half-integer positive number, ##s \in \{1/2,3/2,\ldots \}##), i.e., if ##\psi(t,\vec{x}_1,\sigma_1;\vec{x}_2,\sigma_2;\ldots; \vec{x}_N,\sigma_N)## is an ##N##-particle fermionic wave function, then it's antisymmetric by exchanging any pair ##(\vec{x}_i,\sigma_i)## and ##(\vec{x}_j,\sigma_j)##.

JohnH