The quantum numbers label one electron's states. Atoms with many electrons need one more rule, found by Wolfgang Pauli in 1925, and it is the reason chemistry exists: without it, every electron would drop into the 1s state and all atoms would act alike.
Key idea
The exclusion principle: no two electrons in an atom can have the same four quantum numbers $(n, l, m_l, m_s)$. So each orbital $(n, l, m_l)$ holds at most two electrons, with opposite spins. Electrons fill the lowest available states first (1s, 2s, 2p, 3s, 3p, 4s, 3d, …), and the outermost electrons set an element's chemistry. Electron spin is an intrinsic angular momentum with $|S| = \frac{\sqrt{3}}{2}\hbar$ and $S_z = \pm\frac{1}{2}\hbar$.
Why it works
Spin was discovered in experiments. In 1922 Otto Stern and Walther Gerlach sent silver atoms through a non-uniform magnetic field, and the beam split cleanly into two spots, not a smear: a magnetic moment that takes only two orientations. Spin is not a tiny ball turning: an electron behaves as a point particle in every experiment so far, and a ball small enough to match would need its surface to move faster than light to carry that much angular momentum. Spin is simply a built-in property, like charge. The exclusion principle then builds the periodic table: each new element adds one electron in the lowest open state, so shells fill in turn, and elements whose outer subshells are filled alike (like lithium, sodium and potassium, each with one s electron outside a closed shell) behave alike.
Worked examples
Worked example: Sodium's electrons
Write the ground-state electron configuration of sodium ($Z = 11$).
- 1s holds 2, 2s holds 2, 2p holds 6: that is 10 electrons.
- The 11th goes into the next lowest state, 3s.
Answer: $1s^2\,2s^2\,2p^6\,3s^1$: one outer electron, like lithium and potassium
Worked example: A count with the rules
In an atom with a completely filled $n = 3$ shell, how many electrons have $m_l = 0$?
- $n = 3$ has $l = 0, 1, 2$, and each $l$ has exactly one $m_l = 0$ orbital.
- Three orbitals, two electrons each.
Answer: 6 electrons
Worked example: How big is spin?
Find the magnitude of an electron's spin angular momentum and its $z$-component, in J·s.
- $|S| = \sqrt{s(s + 1)}\,\hbar$ with $s = \frac{1}{2}$: $\sqrt{3/4}\,\hbar = 0.866\hbar = 9.13 \times 10^{-35}$ J·s.
- $S_z = \pm\frac{1}{2}\hbar = \pm 5.27 \times 10^{-35}$ J·s.
Answer: $|S| = 9.13 \times 10^{-35}$ J·s; $S_z = \pm 5.27 \times 10^{-35}$ J·s
Common mistake
Thinking spin means the electron is a tiny ball literally turning. The name is historical. Spin acts like angular momentum (it adds to orbital angular momentum and gives the electron a magnetic moment), but it takes exactly two values and has nothing to do with the electron's size. A check: a classical spinning ball could spin at any rate; the electron's spin is always $\frac{\sqrt{3}}{2}\hbar$.