Quantum numbers and atomic orbitals

The early Bohr model pictured an electron orbiting the nucleus like a planet around a star, on a definite circular path at a definite radius. Solving the Schrödinger equation for an electron in a hydrogen-like atom replaces that picture entirely. The solutions are wavefunctions called orbitals, and the electron does not follow any definite trajectory at all — the square of the orbital's magnitude gives a probability density, a three-dimensional cloud describing where the electron is more or less likely to be found on measurement, with no path connecting one location to the next.

Each orbital is labelled by a set of quantum numbers, which arise directly from the boundary conditions and symmetries of the Schrödinger equation rather than being assigned by hand. The principal quantum number n sets the overall energy and size of the orbital, taking values 1, 2, 3, ... The orbital angular momentum quantum number l (values 0 to n−1, conventionally labelled s, p, d, f, ...) sets the orbital's angular momentum and controls its overall shape — s orbitals are spherically symmetric, p orbitals have a two-lobed shape, and so on. The magnetic quantum number ml (values −l to +l) sets the orbital's orientation in space. None of these three depend on time; they are properties of the standing-wave solutions to the equation, in direct analogy to how a vibrating string supports only certain standing-wave patterns.

A fourth quantum number, the spin quantum number ms, does not come from the spatial Schrödinger equation at all — it describes an intrinsic form of angular momentum the electron carries regardless of its orbital motion, with only two allowed values, spin-up and spin-down. All four numbers together — n, l, ml, ms — are needed to fully specify a single electron's state in an atom, and, as the Pauli exclusion principle establishes, no two electrons in the same atom can share an identical set of all four.