the four quantum numbers

The four quantum numbers are a set of labels that together pin down the exact state of an electron in an atom, like a four-part address. The principal number n sets the main energy level and overall size. The azimuthal number, written ℓ, sets the shape of the orbital and how much angular momentum the electron carries. The magnetic number m fixes how that orbital is oriented in space. The fourth, the spin number, records which way the electron's intrinsic spin points.

Each number can only take certain whole-number or half-number values, and they are nested: ℓ runs from zero up to one less than n, m runs across a range set by ℓ, and spin takes one of two values. This stepwise quantization is why electrons cannot occupy just any state, and counting up all the allowed combinations tells you exactly how many electrons fit in each shell — two, eight, eighteen, and so on.

These four labels are the master key to atomic structure. Together with the rule that no two electrons in an atom may share all four values, they explain the entire architecture of the periodic table: the lengths of its rows, the families in its columns, and the chemical personalities of the elements. Almost everything about how atoms combine traces back to which quantum-number addresses are filled and which remain open.