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I have a question about two particles in a ring.

Okay, so as far as I know the wavefunction of a particle in a ring is cos(kθ) with k=0,1,2,3...

So, what is the wavefunction (total one) for the two particles? Im guessing it must be the multiplication of the two:

totalphi= cos(kθ1)*cos(kθ2) Yes?

I've been asked what special care is needed when writing down the total wavefunction and i don't know what to say. Im guessing that you just have to consider them together yes? Or maybe repulsions?

help!