Two fermions cannot coexist

A rule that structures matter

The Pauli exclusion principle (1925) states a prohibition of modest appearance, with immense consequences:

Two identical fermions (two electrons, for example) can never occupy the same quantum state at the same time.

In other words, in an atom, two electrons cannot have exactly the same quantum characteristics. At least one must distinguish them.

The state of an electron in the atom

An electron in an atom is described by a set of "quantum coordinates" (the quantum numbers): its energy level, the shape of its orbital, its orientation, and its spin (up or down). The Pauli principle says: not two electrons with the identical complete set.

   Une "case quantique" (meme energie, meme orbitale) :

        ___________
       | ↑   ↓     |   <- OK : deux electrons, mais spins OPPOSES (up + down)
        -----------      ils different par le spin -> Pauli respecte

        ___________
       | ↑   ↑     |   <- INTERDIT : meme etat ET meme spin
        -----------      Pauli viole

Each "box" (an orbital) can therefore contain at most two electrons, provided they have opposite spins. A third is impossible: it would necessarily duplicate one of the two.

Why electrons "stack up"

Without the Pauli principle, all the electrons of an atom would tumble down toward the lowest-energy state (the ground state) — this is what any system does: minimize its energy. All atoms would look alike, and chemistry would not exist.

Pauli forbids it. Once a box is full (two electrons), the next ones are forced to climb into higher-energy states, for lack of room at the bottom:

   Remplissage (sans Pauli)      Remplissage (avec Pauli)
   -----------------------       ------------------------
   E3 ______                     E3 ______
   E2 ______                     E2 __↑↓__   <- forces de monter
   E1 __(tous ici)__             E1 __↑↓__   <- rempli, donc on monte
        ↑↓↑↓↑↓↑↓                       (2 par niveau max)
   (absurde)                     (structure en couches !)

It is this forced stacking that creates the shell structure of atoms — and therefore all of chemistry.

In summary

The Pauli exclusion principle forbids two identical fermions from occupying the same quantum state. In an atom, each orbital therefore holds only two electrons, of opposite spins (up + down). Unable to pile into the ground state, the electrons are forced to stack up into levels of increasing energy — which gives atoms their shell structure, the foundation of all chemistry.