# How does the Heisenberg Uncertainty Principle relate to electron degeneracy pressure?

##### 1 Answer

From the **Heisenberg Uncertainty Principle**, we know that

#DeltaxDeltap >= ℏ/2#

i.e. *the momentum and position cannot be simultaneously observed to the same precision*.

Now, imagine compressing two electrons into a confined space.

If a second electron is placed into a half-filled orbital, the electron already there necessarily *has to move to a new, higher energy level*. If one further contracts the orbital size, the electrons are going to have **more defined positions**,

That implies **higher uncertainties in their momenta**, i.e. on average they are moving at very high momentum. This gives rise to something called "degeneracy pressure":

#P_(deg) = ((3pi^2)^"2/3"ℏ^2)/(5m_e) rho_N^"5/3"# where:

#ℏ# has units of#"J"cdot"s"# , or#"kg"cdot"m"^2"/s"# .#m_e# is the mass of the electron in#"kg"# .#rho_N -= N/V# is the number of free electrons per unit volume, with#"m"^3# for the volume.#P_(deg)# is the degeneracy pressure in#"Pa"# .#P_(deg)# here is onlyweaklydependent on temperature, unlike the ordinary pressure in the gas laws.

Degeneracy pressure pertains to *the amount of energy involved in condensing matter to make it degenerate*, i.e. all the same energy levels.

Usually, degeneracy pressure is negligible compared to the kind of pressure we all are familiar with, i.e. at ordinary matter densities,

#P_(deg)# #"<<"# #[P -= (Nk_BT)/V]# where

#k_B# is the Boltzmann constant in#"J/K"# and#T# is temperature in#"K"# .

When electron density is **extraordinarily packed**, i.e.

#P_(t ot) = P_(deg) + P ~~ P_(deg)# .

When that is the case, the electrons in the space are considered **degenerate**, since energy at constant entropy (particle distributions in each energy level) is a function of the volume, i.e. ** smaller** volumes leads to

**energy levels.**

*more-degenerate*(Although this is easiest at