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When a low mass stars core mostly consists of helium (after years and years of fusion reactions) its thermal pressure decreases and so the core gets denser due to gravitational pressure.

Why is it that for extremely dense gas, where the separation between electrons is smaller than the de Broglie Wavelength, matter can be no longer described as an Ideal Gas? and what has this got to do with the electrons average momentum?

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Controversial - I think you can have an ideal degenerate gas and many texts I have read refer to the "ideal degenerate gas". The word ideal here refers to point-like, non-interacting particles.

Most real degenerate gases are not "ideal" to some extent. For example there are Coulomb interactions between degenerate electrons and themselves and the ions in the interior of a white dwarf - at the level of interaction energies of a couple of per cent of the Fermi energy. These non-ideal interactions are much more important in neutron stars where the assumption of an ideal degenerate gas is very poor. The n-n interactions actually dominate the pressure.

Perhaps what you mean is that the gas can no longer be treated as a perfect Maxwell-Boltzmann gas where $P = nk_B T$? The reason is indeed that the spatial part of the electron wavefunctions overlap and because they are fermions, the Pauli-Exclusion-Principle demands that they have net anti-symmetric wavefunctions such that no two electrons can occupy the same quantum state. This means that the way electrons fill up quantised energy states in the gas can no longer be described with Maxwell-Boltzmann statistics but instead by Fermi-Dirac statistics. As a result, the relationship between pressure, density and temperature is no longer that of an ideal Maxwell-Boltzmann gas.

The connection with the average electron momentum is that since all the low energy states are filled in a degenerate gas, even if the temperature is lowered, then the average momentum of an electron in a degenerate gas is much higher than would be expected in a Maxwell-Boltzmann gas of the same temperature.

In a (non-relativistic) degenerate gas, the average magnitude of the electron momentum is $\sim 0.85 \sqrt{m_e E_f}$, where $E_f$ is the (kinetic) Fermi energy, and $E_f \gg k_B T$. In a M-B gas, the average momentum magnitude is $\sim 1.6\sqrt{m_e k_B T}$, so much smaller.

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