Orbitals, Quantum Numbers and Electron Configuration

From Orbits to Orbitals

De Broglie proposed that matter is also wave-like, with $\lambda = h/mv$. Heisenberg then showed that position and momentum cannot both be sharp: $\Delta x \, \Delta p \geq h/4\pi$. Together these destroy the idea of an electron on a definite path. Schrödinger's equation replaces the orbit with a wavefunction $\psi$, whose square $|\psi|^2$ gives the probability of finding the electron at a point. An orbital is that probability cloud, not a track.

Solving the equation for hydrogen produces exactly three quantum numbers, plus spin:

So a shell $n$ contains $n$ subshells, a subshell $l$ contains $2l+1$ orbitals, and each orbital holds 2 electrons, giving $2n^2$ electrons per shell.

Filling obeys three rules. The Aufbau principle fills the lowest-energy orbital first. The Pauli exclusion principle forbids two electrons in one atom from sharing all four quantum numbers, capping any orbital at two electrons of opposite spin. Hund's rule says degenerate orbitals each take one electron, all with parallel spin, before any of them pairs, because paired electrons repel.

In a hydrogen atom, energy depends on $n$ alone. In every other atom, electron-electron repulsion splits the subshells, so $2s$ lies below $2p$ and, famously, $4s$ fills before $3d$.

Common pitfall: treating an orbital as a path. Nothing orbits. An orbital is a region where the electron is likely to be found, and a $p$ orbital's two lobes are one orbital, not two, the electron is not travelling between them.