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# Pauli Exclusion Principle

Monday, March 14, 2022

## Energy States

Initially, it seems as though atoms with $Z$ electrons will have all those electrons eventually cascade down to the $1s$ state (lowest energy state). This would result in smooth variations between neighboring elements. However, by studying the properties of neighboring atoms, this theory does not match observations.

To fix this problem, Wolfgang Pauli proposed the following Pauli exclusion principle in 1925:

No two electrons in a single atom can have the same set of quantum numbers ($n$, $l$, $m_l$, $m_s$).

## Examples

Consider helium ($Z=2$): it has one electron in the $1s$ state, where $n=1$, $l=0$, $m_l=0$, $m_s=+1/2~\text{or}-1/2$. The second electron have have the same quantum numbers for $n$, $l$, and $m_l$ but not $m_s$. If the first electron has $m_s=-1/2$, the second one will have $m_s=+1/2$ and vice versa.

Now consider lithium ($Z=3$). Two of its three electrons can occupy the $1s$ state just as in helium. However, its third electron cannot go into the $n=1$ level, so it must use the $2s$ energy level.

Continuing to beryllium ($Z=4$), two electrons can occupy the $1s$ state and two can occupy the $2s$ state. However, for boron ($Z=5$), the fifth electron must use one of the $2p$ states, giving very different properties than both lithium and beryllium.