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Re: Whence Degeneracy Pressure?



Regarding Mike Monce's comment:

Been lurking on this topic and been thinking about neutron stars
also. I think "allowed" is not the right word. We use it when stating a
naive form of the Pauli Principle, but fermions aren't forbidden from
being in the same state; the total wavefunction goes to zero in that case.
the superposition yields a 'node'.

Those are two related but different effects. The nodes in the many
body wave function are for its value when multiple fermion coordinate
arguments (i.e. position & spin or momentum & spin) are set equal to
each other. The prohibition on multiple state occupancy pertains to
the many body wave function basis set made of superpostions of Slater
determinants. Each such basis determinant is an antisymmetrized
product of *single particle* states, and no two factors of those
*single particle* states in the antisymmetrized product can be the
same single particle wave function in each such Slater determinant.
Otherwise the antisymmetrization process causes the resulting basis
function identically vanish to the zero function which is *never* a
legitimate basis function.

You can also think of this occupancy restriction as a consequence of
the nilpotency (coming from the anti-commutation relations) of each
of the creation operators for single particle fermion states, if you
want to look at the situation from a 2nd quantized point of view.

I can see where John's view of
pressure comes in with regard to a neutron star. I assume here that the
confinement 'box' he speaks of would be the gravitational potential well
the neutrons are confined in.

Actually a real box is not necessary because the precise box boundary
conditions become irrelevant for the bulk macroscopic size and
densities of interest. All that is necessary is an appropriate
density of fermions in a sufficiently large region to qualify as an
asymptotically macroscopic volume. Such a macroscopic volume is
still infinitesimal in size compared to the size of the whole neutron
star and its gravitational potential well.

But couldn't we still write this as a force
with a vector nature in the opposite direction of the gravity?

Not any more than is done when we discuss ordinary static fluids in
equilibrium in any gravitational field. For each parcel of the fluid
the downward gravitational force on the parcel is opposed by a net
upward bouyant force that is the result of the difference in the
hydrostatic pressure on the parcel at its bottom and at its top. So
(except for a couple of tricky but conceptually relatively
unimportant general relativistic factors mostly applicable deep
inside neutron stars close to the Oppenheimer-Volkhoff limit) for all
practical purposes the static pressure follows the local bulk
equation: dP(r)/dr = - g(r)*rho(r) where P(r) is the local pressure,
g(r) is the local gravitational field strength, and rho(r) is the
local density (all at radius r from the center). Of course this
equation needs to be supplemented with an expression that determines
g(r) in terms of all the matter inside a sphere of radius r, and
another one which is effectively the equation of state for the fluid
that determines the fluid's local pressure in terms of its local
density.

The main difference between the degenerate neutron case for a neutron
star and the increase in water pressure in the ocean with increasing
depth is that the two cases have different equations of state
relating the pressure to the density (reflecting the effectively
incompressible nature of sea water in one case and the degeneracy
condition on the neutrons in the other). Also, the expression for
g(r) in the ocean case coming from the mass of the earth below the
height r is essentially purely Newtonian in character, whereas the
necessary analogous expression in the neutron star case needs some
subtle GR corrections built in to it.

Any neutron star theorists here? How do they model the entire star?

Since I'm not such a neutron star theorist I don't know for certain.
But I believe they model it as a static bulk fluid with a complicated
equation of state reflecting not only the neutrons' degeneracy but
height dependent differences in composition and, to some extent,
temperature. Neutron stars are not just a collection of neutrons;
they have a minority admixture of electrons and protons whose
relative concentrations depend on the appropriate Fermi levels (i.e.
chemical potentials) and they depend on the local density, etc., and
hence height. As I recall, sometimes calcuations have been done
where deep in a massive neutron star close to the O-V limit is a
region of a quark-matter phase with a nonzero equilibrium
concentration of strange quarks besides the usual u & d quarks &
pions.

David Bowman
David_Bowman@georgetowncollege.edu