# When does Pauli's exclusion principle kick in?

Imagine that I prepare a fermion in the $\left|\uparrow \right\rangle$ state and a second one far away in the $\left|\downarrow \right\rangle$ state and set them in a path for collision.

According to Pauli's exclusion principle, the composite wave function must be anti-symmetric. Does the wave function become anti-symmetric as they collide or was it like this from the start? Can one predict if the composite wave function will correspond to a singlet or a triplet state from the moment we prepare the separate fermions?

This can be done because don't make sense to have a far away systems in the same state! If the systems are far away one of another then exist some subspace in Hilbert space of the whole system that permits the definition of "far away". And is precisely the complementar of the subspace generated by $(\left|\uparrow \right\rangle\left|\downarrow \right\rangle,\left|\downarrow \right\rangle\left|\uparrow \right\rangle,\left|\downarrow \right\rangle\left|\downarrow \right\rangle,\left|\uparrow \right\rangle\left|\uparrow \right\rangle)$ that accounted the Pauli principle. Is the complementar space that has position-momentum observables and other degrees of freedom.
For proceed with this issues we can substitute the pauli principle for some exchange energy-interaction. $$H=g(l_{tp})S_1.S_2$$
where the $g(l)$ is the coupling in function of the typical lenght $l_{tp}$. Locality may tell us that if $l_{tp}\rightarrow \infty$ then $g(l_{tp})\rightarrow 0$ .