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The exclusion principle makes a good sense to me mathematically but not so much physically intuitive.

Suppose we have a quantum system of two electrons.

Now if they are on a metal thin film they cannot have the same spin and the same momentum. That's ok because they are somehow related by being on the same metal film.

But what if we put these two electrons on two separate metal thin films with some distance from each other?

These two electrons can still be thought of as one quantum system and therefore have to obey antisymmetric wavefunction statistics.

So the Pauli exclusion principle should apply again.

But I cannot have a good physical intuitive sense of it when electrons are on two different plates with arbitrary distance from each other. Why they should not have the same spin and momentum? How can they relate to each other when they are not on the same metal thin film?

Put it in other way how we determine the system boundary in the physical space before making sense of the exclusion principle?

Are we assuming some interaction here to set the system boundary?

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Remember the relationship between position and momentum in quantum mechanics: knowing the momentum of an electron means the position is completely undetermined, and vice versa.

If your two electrons have well-defined momentum, they have undefined position - they exist as plane waves distributed equally throughout the universe. Thus they are not in one thin film or another, but everywhere at once. In this sense, no two electrons in the universe can have the same well-defined momenta at any point in time.

More realistically, electrons exist in superpositions of different momenta and positions, with both quantities only being narrowly determined when a measurement is made. Therefore we usually talk about electrons existing "within a thin film" with a narrowly defined range of momenta, approximately a plane wave state, but not exactly.

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