My understanding is that when it comes to forming a white dwarf, it is the electron degeneracy pressure, due to the Pauli Exclusion Principle, preventing collapse in of the white dwarf. If the gravitational force is sufficiently large, then the electrons in the white dwarf will be forced to fuse with the protons to form neutrons, and the neutron star resists collapse in by neutron degeneracy pressure. If the gravitational force is even greater, then black hole will form.
How does the Pauli Exclusion Principle actually create a force? It seems to me from various things I have read that the force due to the Pauli Exclusion Principle increases as the fermions are squeezed closer together, although I am not sure why there is an increasing force and it is not simply the case that the fermions cannot be pushed into exactly the same position. It is as if the fermions know when they are approaching each other?
Answer
How does the Pauli Exclusion Principle actually create a force?
The Pauli exclusion principle doesn't really say that two fermions can't be in the same place. It's both stronger and weaker than that. It says that they can't be in the same state, i.e., if they're standing waves, two of them can't have the same standing wave pattern. But for bulk matter, for our purposes, it becomes a decent approximation to treat the exclusion principle as saying that if $n$ particles are confined to a volume $V$, they must each be confined to a space of about $V/n$. Since volume goes like length cubed, this means that their wavelengths must be $\lesssim (V/n)^{1/3}$. As $V$ shrinks, this maximum wavelength shrinks as well, and the de Broglie relation then tells us that the momentum goes up. The increased momentum shows up as a pressure, just as it would if you increased the momenta of all the molecules in a sample of air. A degenerate body like a neutron star or white dwarf is in a state where this pressure is in equilibrium with gravity.
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