Undergrad How does the Pauli exclusion principle work?

[BillKet]

Hello! I am a bit confused about the mechanism behind the Pauli exclusion principle. From what I read, it is motivated based on QFT arguments (for example if you don't impose antisymmetry of the fermionic wavefunction you get non-locality, or infinitely negative energies etc.) so mathematically it makes sense. However, if you try to place 2 identical fermions in the same quantum state, what exactly is the thing that exerts the force (and hence the pressure) that prevents you to do so. Mathematically if you do that, you get a zero wavefunction, so the state simply doesn't exists, but in an experiment what is actually going on? Thank you!

 

[PeterDonis]

if you try to place 2 identical fermions in the same quantum state, what exactly is the thing that exerts the force (and hence the pressure) that prevents you to do so

In order to even formulate this question consistently, you would have to specify how you are trying to place 2 identical fermions in the same quantum state. And right there you come up against a problem: if the two fermions are truly identical, i.e., indistinguishable, then by definition you can't manipulate them separately; you can only manipulate them together, as a single 2-fermion system. If you could manipulate the two fermions separately, they wouldn't be indistinguishable, because your ability to manipulate just one and not the other would count as distinguishing them.

And if you can only manipulate the two fermions as a single 2-fermion system, then it doesn't even make sense to talk about "trying to place them in the same state", because you don't have two systems, you only have one. And you can only put that one system into a state that exists. And questions like, for example, why do 2-fermion systems confined to smaller spaces exert higher pressure, are answerable simply by looking at the properties of the states of the 2-fermion system and how the observables of that system behave as the system's state changes.

 

[zonde]

However, if you try to place 2 identical fermions in the same quantum state, what exactly is the thing that exerts the force (and hence the pressure) that prevents you to do so.

Let's say you take a bunch of helium atoms and try to push them together so that they s1 orbitals overlap. Then all electrons won't have available quantum states near helium nucleus and some of them will occupy next available quantum states using up some energy. So you will end up trying to squeeze positively charged helium ions. If you try to take away that extra energy that you put into electrons QM says you can't get it. Electrons won't give up their energy if there are no lower energy levels that they can occupy.
Now let's say you try to squeeze bunch of neutral neutrons. You can't use electromagnetic interactions to squeeze them but of course you can use gravity for that. And again if there are no available quantum states at the lower energy levels of potential well some neutrons will occupy higher energy levels i.e. the system will expand. Well, you can make potential well deeper by adding some more neutrons but you can add them only in next available quantum states which will make the system even larger. And adding particle to the system means that you are putting it in a potential well and you have to take energy away from it in order to do it. So whenever particle can't occupy lower energy state you can't take away any energy because particle simply won't give it up i.e. you can't squeeze the system tighter.