Orbitals and quantum numbers

The modern picture of the atom comes from Schrödinger's equation. Its solutions are wave functions, ψ, called atomic orbitals. ψ itself has no direct meaning, but ψ² gives the probability density: how likely the electron is to be found at each point.

Each electron in an atom is described by four quantum numbers, like an address:

• Principal quantum number, n = 1, 2, 3 … It gives the shell (K, L, M, N …), and fixes the size and energy of the orbital. A shell has n² orbitals.
• Azimuthal quantum number, l = 0, 1, 2 … up to n – 1. It gives the subshell and the shape of the orbital: l = 0, 1, 2, 3 are called s, p, d, f.
• Magnetic quantum number, mₗ = –l … 0 … +l. It gives the orientation of the orbital in space. So there is one s orbital (mₗ = 0), three p, five d and seven f orbitals.
• Spin quantum number, mₛ = +½ or –½: the two possible directions of the electron's spin.

Each orbital can hold at most two electrons, and they must have opposite spins.

An example: for n = 3, l can be 0, 1 or 2, giving the 3s, 3p and 3d subshells. For the 3p subshell (l = 1), mₗ can be –1, 0 or +1: three orbitals, holding up to 6 electrons. In all, the third shell has 3² = 9 orbitals.

To check whether a set of quantum numbers is possible, test each rule in turn: n at least 1; l from 0 to n – 1; mₗ from –l to +l; mₛ either +½ or –½.