Standing Waves · Orbital Shapes · Quantum Numbers · Electron Configuration
Only discrete standing-wave modes persist on a plucked string. Fixed endpoints require an integer number of half-wavelengths:
Energy is therefore quantised — only specific values are permitted. The integer n is a mode index.
An electron bound to a nucleus obeys a 3-D wave equation. Atomic orbitals are the 3-D standing-wave solutions confined by the Coulomb potential:
The quantum numbers n, ℓ, mℓ are the 3-D mode indices. Each unique triple defines one orbital "mode shape."
Blue — positive phase (ψ > 0)
Orange — negative phase (ψ < 0)
Axes: x (red) · y (green) · z (blue)
Cr & Cu anomalies included · toggle for noble gas notation
Aufbau Lowest-energy subshells first: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p
Pauli Max 2 electrons per orbital, opposite spins (↑↓)
Hund Half-fill all orbitals in a subshell with ↑ before any pairing