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?


To say you have put them into states, you have implicitly considered them as part of a single system (independent of how "far away" they are from each other). The wavefunction is anti-symmetric by definition, so they will behave corresponding to the Pauli Exclusion Principle "from the start".


Let's ignore the position-momentum observables. Your definition of far away has a precisely definition. We can treat one electron and ignore the another. So we don't have to worry about pauli principle. When we want to collide this two electrons, and no more work in far away paradigm, we need to define some length when pauli principle is applied. We can think in the length when one electron can reach the another in space by quantum fluctuations.If this electrons are confined by cells for example, is the tunneling length.

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.

Here for more

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 $g(l_{tp})\rightarrow 0$ .


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