Baseem T. answered • 04/12/19
Experienced UC Berkeley Graduate (Maths, Chemistry, Physics)
Degeneracy pressure is the term we use for the macroscopic manifestation of the exchange interaction or Pauli force. Loosely speaking, the Pauli exclusion principle states that the probability of finding two fermions in the exact same quantum state rapidly drops to zero as their wavefunctions begin to overlap, causing the energy of the system to increase— no two fermions can occupy the same quantum state. (A simple derivation of this using the exchange operator can be found; I would provide it, but it makes me exceed my character limit)
Degeneracy Pressure
Suppose we take two atoms and let them approach one another in an attempt to form a molecule. If you examine a graph of the potential energy (preferably the Morse potential), you may feel tempted to think that the dominant force at shorter distances is the Coulombic repulsion between the similarly charged nuclei and electrons. The interesting fact, however, is that the huge spike in energy as they get even closer is not due to this repulsive Coulombic force. Instead, it is due to the exchange force or exchange interaction, which occurs when you try to swap the wavefunctions of two fermions; every fermionic wavefunction begins to overlap, and the electrons become less and less distinct as they're shared between the atoms (it becomes impossible to tell which electron "belongs" to which nucleus at this point). Pauli's exclusion principle states that the probability of this occurring must quickly drop to zero, and this interaction due to a simple particle exchange (ergo "exchange" interaction) contributes an extremely high energy penalty, increasing the energy of the system much, much faster than the Coulombic repulsion.
How is this related to pressure? And why do we sometimes call this the exchange force? We know it's not really a force, because this isn't the result of any of the fundamental forces you alluded to in your question— it's a purely quantum mechanical effect. The reason we sometimes refer to it as the Pauli force is because it serves a mechanically similar purpose. Pressure (p) is more generally defined as the negative partial derivative of the energy (E) with respect to the physical volume (V):
p = - (∂E/∂V)S,N
at constant entropy (S) and particle number (N). [The energy here is specifically the Helmholtz free energy] This should agree with your classical intuition— for a system with positive pressure, like a balloon, if the volume increases spontaneously, it performs work on its surroundings and the energy decreases, or if the volume decreases, it's because work was applied to it by an external force, and the energy increases. In the case of our atoms, as we brought them closer together (decreasing the physical volume), the energy began to increase due to the exchange interaction. This corresponds to a type of quantum mechanical pressure, and when it occurs in degenerate materials like room-temperature metals or neutron stars, we call it degeneracy pressure.
The electrons in a metal/white dwarf or the neutrons in a neutron star all try to occupy the lowest energy state available. However, because they're fermions, they have to neatly fill up the energy states one at a time (although two can collapse into the same position-space wavefunction if we introduce the spin component of the wavefunction, which is what lets electrons of different spin pair up). This results in a "Fermi sea", where the occupied energy levels are all stacked up on top of one another, and we call collection of particles a Fermi gas— it's the quantum mechanical analogue of an ideal gas. The term "degenerate" specifically refers to whether or not the temperature is below the "Fermi temperature", which serves as a cutoff point for whether or not the material behaves like a Fermi gas.
As the white dwarf or neutron star tries to collapse and decrease in volume, the fermions are forced closer and closer to one another, and they begin to occupy the same quantum state. The energy increases, and just like a balloon, it feels like it's pushing back against you. This positive pressure is the electron/neutron degeneracy pressure.
