Quantum Degeneracy Pressure Calculator

Quantum degeneracy pressure is the outward pressure exerted by a gas of fermions (like electrons) purely because the Pauli exclusion principle forbids two identical fermions from occupying the same quantum state. Even at absolute zero, where thermal motion vanishes, fermions must still fill up a range of momentum states, and this forced motion produces real, measurable pressure.

For a non-relativistic, zero-temperature free electron gas, P = (2/5) n E_F, where n is the electron number density and E_F is the Fermi energy. Substituting the Fermi energy formula gives the equivalent form P = (ħ²/5m)(3π²)^(2/3) n^(5/3), so pressure grows with the five-thirds power of density.

Ordinary (thermal) gas pressure comes from random thermal motion and vanishes as temperature approaches absolute zero, following the classical ideal gas law P = nkT. Degeneracy pressure has nothing to do with temperature, it exists purely because identical fermions are forced into different quantum states, and survives even at T = 0.

A white dwarf has exhausted its nuclear fuel and has no fusion left to resist gravitational collapse. Instead, electron degeneracy pressure, arising from the extremely high density of free electrons packed into the stellar core, holds the star up. This balance between gravity and degeneracy pressure is what makes white dwarfs stable, dead stellar remnants rather than continuing to collapse.

The Chandrasekhar limit (about 1.4 solar masses) is the maximum mass a white dwarf can have while still being supported by non-relativistic electron degeneracy pressure. Above this mass, the electrons are forced to move at relativistic speeds, degeneracy pressure grows more slowly with density than gravity requires, and the star can no longer be supported, collapsing further into a neutron star or exploding as a Type Ia supernova.

It uses the non-relativistic formula, a good approximation for typical white dwarf cores where electron speeds are still well below relativistic. Near the Chandrasekhar limit, electrons become relativistic and the pressure-density relationship changes (P scales as n^(4/3) instead of n^(5/3)), which this calculator does not model.

Degeneracy pressure scales as n^(5/3), and a white dwarf core has an electron density around 10⁷ to 10⁸ times higher than a typical metal, translating into a pressure roughly 10¹² to 10¹³ times larger. Metal degeneracy pressure, while large by everyday standards (tens of thousands of atmospheres), is dwarfed by the pressures inside a collapsed star.

In the idealized zero-temperature limit used here, no. At finite but modest temperatures, degeneracy pressure is only very weakly modified, because the thermal energy kT is typically far smaller than the Fermi energy for both metals and white dwarfs, so the T=0 formula remains an excellent approximation.

It is a direct consequence of the Pauli exclusion principle: no two electrons (with the same spin) can occupy the same quantum state. To fit more electrons into a smaller volume, they must occupy progressively higher-momentum states, since the lower ones are already full, and that forced high-momentum motion is what produces the outward pressure.

Any fermion can exhibit degeneracy pressure. Neutron degeneracy pressure supports neutron stars in a similar way to how electron degeneracy pressure supports white dwarfs, once the star is dense enough that electrons and protons merge into neutrons via inverse beta decay.