This came up during a Particle Physics discussion :

Here is a proof that the wavefunction of n identical particles has to be either completely symmetric or anti-symmetric.

Wavefunction ψ=ψ(a_{1},a_{2},……..,a_{n})

Let **P**_{ij} be the operator which flips particles i and j

Now indistinguishability requires

**P**_{ij}ψ=c ψ

Since flipping twice results in the same wavefunction

**P**_{ij} **P**_{ij} ψ = ψ

Implies c=1 or -1

Now consider this relation

**P**_{ij}=**P**_{ik} **P**_{kj} **P**_{ki}

One can easily veryify that this true by looking at the action of **P**_{ij} on the ordrerd set (i,j,k).

Since **P**_{ik} **P**_{ik}=1

**P**_{ij}=**P**_{kj}

Extending this simlarly one gets

**P**_{ij} = **P**_{ik} =** P**_{kl} for any k,l

Thus **P**_{fg} = c for all f,g

Thus the wavefunction has to be **completely **symmetric or antisymmetric.

Venkateshan

PS: I took the liberty of tidying up the notation, hope I didn't make any mistake/ change your argument. –Shanth

5 Responses »