## Even parity => symmetric space wave function?

If I have af wavefunction that is a product of many particle wavefunctions

$$\Psi = \psi_1(r_1)\psi_2(r_2) ... \psi_n(r_n)$$

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

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

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.
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 Recognitions: Homework Help Science Advisor For two particles, parity changes r_12 into -r_12, which has the same effect as interchanging r_1 and r_2. For your many body wave function this isn't true, so parity is unrelated to particle interchange. Parity changes all r_i to -r_i. You have to look at the detailed wave function to see what this does. It depends on how your WF relates all the coordinates.
 ok that was my thought too. Let me give an example how it is used, because I would like to know how th author solves the excercise. It's from nuclear and particle physics by B.R. Martin. Consider a scenario where overall hadronic wavefunction $$\Psi$$ consist of just spin and space part, i.e. $$\Psi = \psi_{space} \psi_{spin}$$. What would be the resulting multplet structure of the lowest-lying baryon states composed of u,d and s quarks? The autors own solution: 'Low lying' implies that the internal orbital momentum between the quarks is zero. Hence the parity is P = +1 and space is symmetric. Since the Pauli principle requires the overall wavefunction to be antisymmetric under the interchange of any pair of like quarks, it follows that $$\psi_{spin}$$ is antisymmetric... The rest of the excercise makes sense, but how does he conclude that the space part i symmetric, to me it seems like he use parity symmetric and symmetric under switching of particles.

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