The four quantum numbers

The state of an electron: four labels

Each electron in an atom is entirely described by a set of four quantum numbers. They are like a complete address: they set the energy level, the shape of the orbital, its orientation, and the spin. The Pauli principle will require that no two electrons ever have all four identical.

n — the principal quantum number

n sets the energy level and the "size" of the orbital (the average distance from the nucleus at which the electron is found). It is the equivalent of Bohr's shell number.

   n = 1, 2, 3, 4, ...       (entier positif)

   n = 1 : couche la plus proche du noyau (la plus liee)
   n = 2 : couche suivante, plus etendue
   ...

l — the secondary (azimuthal) quantum number

l sets the shape of the orbital. For a given n, l can range from 0 to n-1. Each value corresponds to a traditional letter:

   l = 0  ->  orbitale "s"  (spherique)
   l = 1  ->  orbitale "p"  (bilobee, en haltere)
   l = 2  ->  orbitale "d"  (quadrilobee)
   l = 3  ->  orbitale "f"  (plus complexe)

m — the magnetic quantum number

m sets the orientation of the orbital in space. For a given l, m takes all the integer values from -l to +l, i.e. 2l + 1 orientations.

   l = 0 (s)  ->  m = 0                 -> 1 orientation
   l = 1 (p)  ->  m = -1, 0, +1         -> 3 orientations (px, py, pz)
   l = 2 (d)  ->  m = -2,-1, 0,+1,+2    -> 5 orientations

s — the spin quantum number

Finally, s (or m_s) gives the spin of the electron, which takes only two values:

   s = +1/2  (up)   ou   s = -1/2  (down)

The summary table

   Nombre  Symbole  Ce qu'il fixe         Valeurs permises
   ------  -------  --------------------  ----------------------------
   principal  n     energie, taille       1, 2, 3, ...
   secondaire l     forme                 0 a n-1
   magnetique m     orientation           -l a +l
   spin       s     spin de l'electron    +1/2 ou -1/2

How they nest together

The rules nest in cascade: n limits l, which limits m. Let us count the states for n = 2:

   n=2
   +-- l=0 (s) : m=0                 + spin(2)  -> 2 etats
   +-- l=1 (p) : m=-1,0,+1 (3 orient.) x spin(2) -> 6 etats
                                                  ---------
                                        total n=2 : 8 etats

These 8 states explain why the 2nd shell contains at most 8 electrons (one per state, Pauli principle) — and why the 2nd row of the periodic table has 8 elements. Everything fits together.

In summary

Four quantum numbers completely describe an electron: n (energy/size, integer ≥ 1), l (shape: s, p, d, f; from 0 to n-1), m (orientation; from -l to +l), and s (spin; ±1/2). Their rules nest together (n limits l, l limits m), which sets the number of possible states per shell — and determines, via Pauli, the capacity of the shells and the structure of the periodic table.