Quantum mechanical model and quantum numbers

Quantum mechanics (Heisenberg and Schrödinger, 1926) treats particles with wave-particle duality. Schrödinger's equation, Ĥψ = Eψ, gives the allowed energies E and wave functions ψ; it can be solved exactly only for one-electron systems.

An atomic orbital is a one-electron wave function ψ. ψ itself has no physical meaning; |ψ|² is the probability density, always positive, and |ψ|² times a small volume gives the chance of finding the electron there.

An orbit (Bohr's fixed path) and an orbital are not the same thing; an orbit cannot be observed, while an orbital describes where the electron is likely to be found.

Features of the model: electron energies are quantised as a direct result of the electron's wave nature; only probabilities of location can be given; an orbital holds at most two electrons.

Principal quantum number n = 1, 2, 3 … sets the shell (K, L, M, N …), the size and, largely, the energy. Shell n has n² orbitals; for H and hydrogen-like ions energy depends on n alone.

Azimuthal quantum number l = 0 to n − 1 sets the subshell and shape: l = 0, 1, 2, 3 are s, p, d, f. Shell n has n subshells.

Magnetic quantum number mₗ = −l … 0 … +l, so 2l + 1 values: one s, three p, five d, seven f orbitals; it sets the orientation. n = 3 has 1 + 3 + 5 = 9 orbitals.

Spin quantum number mₛ = +½ or −½ (proposed in 1925 by Uhlenbeck and Goudsmit), needed to explain doublets and triplets in spectra; two electrons in one orbital must have opposite spins.