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Is the Pauli Exclusion Principle a force?

If an electron has the conditions to occupy a state that is already being occupied by another electron, wouldn't it have to be stopped somehow to not occupy that space? Should there be a force to prevent that, almost like a 'wall' saying 'no, you can't go there'?
gpt
Pauli exclusion itself is not a force, however it changes the way that forces “act” between identical fermions.

If you have two particles which are not identical to each other and not correlated, you can write the total state as a tensor product of their individual state vectors: |1>|2>.

Then if they interact via some potential operator V, you can calculate the interaction matrix element in this state: <1|<2|V|1>|2>.

This is what you might colloquially refer to as the “force” between the particles (really it’s an expectation value of a potential energy).

However if you consider identical fermions, you can’t write the state as a product of the form |1>|2>, because the total state vector must be antisymmetric under particle exchange. Instead, you have to write it as: (|1>|2> - |2>|1>)/sqrt(2).

Now if you take the matrix element of the interaction operator in this state, you have more than one term:

<V> = [<1|<2|V|1>|2> - <1|<2|V|2>|1> - <2|<1|V|1>|2> + <2|<1|V|2>|1>]/2.

The terms with positive signs are called *direct terms* and the terms with negative signs are called *exchange terms*, because in the exchange terms, you’re calculating matrix elements between terms when the two particles are switched.

The operator which encodes the “force” between the particles, V, hasn’t necessarily changed at all. But fact that we have identical fermions has changed the form of the total matrix element.

So while the Pauli principle is not itself a force, it changes the way that the particles interact, compared to the way they would if they weren’t identical.

If you go to more complicated systems, like gases of many fermions, you find that the energy stored in the gas is affected by the fact that identical fermions can’t occupy the same state. When you go to calculate the pressure of the gas, the standard way you’d do so for any system in statistical mechanics, you find that there’s degeneracy pressure. The physical interpretation is that the Pauli principle is not allowing fermions to occupy the same state.