A white-dwarf star is thought to constitute a degenerate...

A white-dwarf star is thought to constitute a degenerate electron gas system at a uniform temperature much below the Fermi temperature. This system is stable against gravitational collapse so long as the electrons are non-relativistic. (a) Calculate the electron density for which the Fermi momentum is one-tenth of the electron rest mass $\times c$. (b) Calculate the pressure of the degenerate electron gas under these conditions.

A white-dwarf star is thought to constitute a degenerate electron gas system at a uniform temperature much below the Fermi temperature. This system is stable against gravitational collapse so long as the electrons are non-relativistic.
(a) Calculate the electron density for which the Fermi momentum is one-tenth of the electron rest mass $\times c$.
(b) Calculate the pressure of the degenerate electron gas under these conditions.

Key Concepts

Electron Degeneracy Pressure

Electron degeneracy pressure arises from the Pauli exclusion principle, which prevents electrons from occupying the same quantum state. In dense systems like white dwarfs, this pressure provides the necessary support against gravitational collapse. The calculation of degeneracy pressure relies on integrating the contributions of electrons up to the Fermi momentum, especially under conditions where thermal effects are negligible compared to the Fermi energy.

Electron Density

Electron density is a key quantity that determines the spacing of quantum states in a degenerate electron gas. It is linked to the Fermi momentum through the density-of-states in momentum space. Knowing the electron density is essential for calculating both the Fermi energy and the resulting pressure in the system.

Fermi Momentum

The Fermi momentum is the maximum momentum of electrons in a degenerate system at zero temperature. It is directly related to the electron density and sets the scale for the energy distribution of the electrons. Defining conditions in terms of a fraction of the electron rest mass times the speed of light allows one to assess whether the system is in the non-relativistic or relativistic regime.

Degenerate Electron Gas

A degenerate electron gas is a state in which electrons obey Fermi–Dirac statistics and occupy energy states up to the Fermi level, even at zero temperature. This concept is central in understanding systems such as white dwarf stars, where quantum effects play a dominant role in providing pressure support as electrons fill momentum states up to the Fermi momentum.

Non-Relativistic Limit

The non-relativistic limit applies when the typical particle speeds are much smaller than the speed of light, meaning that the kinetic energies involved are far below the energy scale set by the particle rest mass energy. This allows the pressure and energy calculations to be performed using classical kinetic energy relations rather than their relativistic counterparts.

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