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[tex]$\Psi = \psi_1(r_1)\psi_2(r_2) ... \psi_n(r_n)$[/tex]

If I then know that the parity of [tex]$ \Psi $[/tex] is even. Can I then show that the wavefunction i symmetric under switching any two particles with each other. That is

[tex]$\psi_1(r_1)\psi_2(r_2) ...\psi_i(r_i) ... \psi_j(r_j) ... \psi_n(r_n) = =\psi_1(r_1)\psi_2(r_2) ...\psi_j(r_j) ... \psi_i(r_i) ... \psi_n(r_n)$[/tex]

for any i and j between 1 and n?

It may be used that the parity operator commutes with the hamilton of the system if nessesary, and that the interaction between the particles only depends on the distance between any two particles.

It is clear that if the system only consist of two particles, and we use one of the particles as the 0-point of our coordinatesystem, the parity operator does the same as changing the particles, and then even parity means even space function, but when n is greater than 2, I can't see it.

Hope someone understand what i'm asking, because the result is used frequently in my course of particle physics.