# White dwarf without Pauli

In a national pre-university physics exam pupils are asked to explain that the white dwarf Sirius B exhibits (unspecified) quantum properties by comparing the debroglie wavelength of the electrons to the average distance d between neighboring electrons. The Heisenberg x,p-uncertainty relation and a simple one-dimensional particle-in-a-box model are part of the curriculum. The Pauli exclusion principle isn't part of the curriculum, but in the exam it is mentioned that "not all electrons can occupy the same energy level; more electrons present means more (and higher) energy levels are occupied".

According to the official answer, the debroglie wavelength of the electrons is of the same order as the distance between them; and the scale of the particle-in-a-box confinement can be regarded as the distance d. The first bit seems correct to me.

My question: can the distance d be regarded as "the scale of the confinement" of the electrons in the interior of Sirius B?

I would say no. Because the average energy of the electrons (3/5 of the Fermi energy) is much higher than the coulomb energy, the electrons can move almost freely and there is no confinement (other than the white dwarf as a whole). Am I correct to say it's impossible to explain the quantum properties of the electrons in Sirius B without using the Pauli exclusion principle (and the Heisenberg uncertainty relation)?

• Is "3/5 of the Fermi energy" exact for relativistic electrons? But one could argue that bulk properties do not depend on volume, only on density. Then it is fine to treat one electron in a Wigner-Seitz-size sphere. – Pieter May 25 '17 at 21:57

It's a bit hard to understand what you are asking. If electrons were bosons then they would still become affected by quantum statistics when the phase space they occupied $\sim \hbar^3$, but there would be no degeneracy pressure without the Pauli Exclusion Principle that operates for fermions.