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Could anyone please explain why the system of two identical particles should have anti-symmetric wave function when their relative orbital angular momentum is $l=1$?

I tried hard to find the reason from Griffiths' and Shankar's Quantum Mechanics textbooks. But even I spent two days, I still have no idea how you can predict the wave function of two particle systems with different relative angular momentum $l$.

It would be very appreciated, when you could give me an explanation.

The page where I found the above statement is here. The first answer says that the two neutral pions have anti-symmetric wave function when $l=1$.

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This is because of the parity properties of spherical harmonics which describe the states of definite angular momentum. Namely, $Y_l \rightarrow (-1)^l Y_l$. In the case of identical particles, to exchange their positions is the same as to make the parity transformation, thus the conclusion.

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