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Pauli Exclusion Principle

Monday, March 14, 2022

Energy States

Initially, it seems as though atoms with ZZ electrons will have all those electrons eventually cascade down to the 1s1s 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 (nn, ll, mlm_l, msm_s).


Consider helium (Z=2Z=2): it has one electron in the 1s1s state, where n=1n=1, l=0l=0, ml=0m_l=0, ms=+1/2 or1/2m_s=+1/2~\text{or}-1/2. The second electron have have the same quantum numbers for nn, ll, and mlm_l but not msm_s. If the first electron has ms=1/2m_s=-1/2, the second one will have ms=+1/2m_s=+1/2 and vice versa.

Now consider lithium (Z=3Z=3). Two of its three electrons can occupy the 1s1s state just as in helium. However, its third electron cannot go into the n=1n=1 level, so it must use the 2s2s energy level.

Continuing to beryllium (Z=4Z=4), two electrons can occupy the 1s1s state and two can occupy the 2s2s state. However, for boron (Z=5Z=5), the fifth electron must use one of the 2p2p states, giving very different properties than both lithium and beryllium.