quantum mechanics - Violation of Pauli exclusion principle

From hyperphysics (emphasis mine):

Neutron degeneracy is a stellar application of the Pauli Exclusion Principle, as is electron degeneracy. No two neutrons can occupy identical states, even under the pressure of a collapsing star of several solar masses. For stellar masses less than about 1.44 solar masses (the Chandrasekhar limit), the energy from the gravitational collapse is not sufficient to produce the neutrons of a neutron star, so the collapse is halted by electron degeneracy to form white dwarfs. Above 1.44 solar masses, enough energy is available from the gravitational collapse to force the combination of electrons and protons to form neutrons. As the star contracts further, all the lowest neutron energy levels are filled and the neutrons are forced into higher and higher energy levels, filling the lowest unoccupied energy levels. This creates an effective pressure which prevents further gravitational collapse, forming a neutron star. However, for masses greater than 2 to 3 solar masses, even neutron degeneracy can't prevent further collapse and it continues toward the black hole state.

How then, can they collapse, without violating the Pauli Exclusion Principle? At a certain point does it no longer apply?

Answer

The Pauli exclusion principle is being applied here to FREE neutrons. There are always free energy/momentum states for the neutrons to fill, even if they are compressed to ultra-high densities; these free states just have higher and higher energies (and momenta).

One way of thinking about this is in terms of the uncertainty principle. Each quantum state occupies approximately $h^3$ of position-momentum phase-space. i.e. $(\Delta p)^3 \times (\Delta x)^3 \geq h^3$. (Actually each momentum state can accomodate 2 fermions - spin up and spin down).

If you increase the density, $\Delta x$ becomes small, so $\Delta p$ has to become large. i.e. the neutrons will occupy higher and higher momentum states as the Fermi energy is increased. You can make the density as high as you like and the PEP is not violated because the particles gain higher momenta.

The increasing momenta of the neutrons supplies an increasing degeneracy pressure. However, there is a "saturation", because eventually all the neutrons become ultrarelativistic and so an increase in density does not lead to such a big increase in pressure. Technically - $P \propto \rho$ at extremely high densities.

It is then a bit of standard textbook astrophysics to show that a star supported by such an equation of state is not stable and will collapse given the slightest perturbation.

In reality neutron stars are not supported by ideal degeneracy pressure - there is a strong repulsive force when they are compressed beyond the nuclear saturation density, with something like $P \propto \rho^2$. Yet even here, an instability is reached at finite density because in General Relativity, the increasing pressure contributes (in addition to the density) to the extreme curvature of space and ultimately means that the star collapses at a finite density and pressure.