At the atomic scale, classical mechanics breaks down. The electron is no longer a ball following a trajectory but a quantum object described by a wavefunction.
de Broglie's postulate
In 1924, Louis de Broglie proposed that every particle of momentum p = mv has an associated wavelength:
λ = h / p
where h is Planck's constant (≈ 6.626 × 10⁻³⁴ J·s). For an electron, this wavelength is of the order of the atom's size — so it cannot be treated as a classical point.
The Schrödinger equation
Erwin Schrödinger wrote in 1926 the time-independent quantum equation:
H ψ = E ψ
where H is the Hamiltonian operator (kinetic + potential energy), ψ is the wavefunction, and E its eigen-energy. The wavefunction itself has no direct physical meaning, but |ψ|² gives the probability density.
For the hydrogen atom (proton + electron), the equation is exactly solvable. Solutions are the atomic orbitals, parameterized by three quantum numbers:
- n (principal): 1, 2, 3, ... — fixes energy and average size.
- ℓ (azimuthal): 0, 1, ..., n−1 — fixes shape (s, p, d, f).
- mₗ (magnetic): −ℓ, ..., +ℓ — fixes orientation.
Hydrogen's energy depends only on n:
E_n = −13.6 eV / n²
Spin
Spin is an extra quantum degree of freedom. The electron carries spin mₛ = ±1/2. With it, an electron's full state is given by four quantum numbers (n, ℓ, mₗ, mₛ).
The Pauli exclusion principle forbids two electrons in the same atom from sharing all four numbers. This forces shell-by-shell filling and explains the structure of the periodic table.
Many-electron atoms
For Z ≥ 2, Schrödinger's equation is no longer exactly solvable because of electron-electron repulsion. We use approximations:
- Orbital approximation: each electron sits in an average field produced by the others.
- Hartree-Fock: orbitals are iteratively optimized by minimizing the energy.
- DFT (density functional theory): replaces the wavefunction with the electron density. The modern computational chemistry workhorse.
Energy is no longer a function of n alone: ℓ becomes relevant (4s and 3d orbitals are nearly degenerate, for instance).
Chemical consequences
p, d, f orbitals have specific geometries (lobes along x, y, z axes). This geometry dictates:
- bond angles in molecules (VSEPR rationalized by sp/sp²/sp³ hybridization).
- directionality of covalent bonds (σ vs π overlap).
- magnetic properties (nonzero total spin → paramagnetism).
- optical transitions (color of d-block complexes).
Quantum mechanics is not an aesthetic refinement — it is the only theoretical framework that accounts for real chemistry. Everything else is approximation.