semiclassical approximation

The semiclassical approximation treats a quantum system as nearly classical, valid when the actions involved are large compared with the quantum of action, Planck's constant divided by two pi. In this regime the wavelike nature of matter is still present but very fine-grained, like ripples so closely spaced that from a distance they look like a smooth classical flow. Quantum mechanics does not switch off; it merely hides its graininess.

The guiding intuition is that classical paths re-emerge as the routes along which a quantum wave's phase changes most slowly, so contributions add up rather than cancel. The wave's phase is the classical action measured in units of the action quantum, and when that action is huge, tiny changes in path produce wild swings in phase, leaving only the classical trajectories to survive the averaging. Classical mechanics is thus the skeleton around which the quantum flesh is thin.

This viewpoint unifies several familiar tools — the WKB method, the correspondence principle, and the way Bohr's old quantum rules approximate the truth — under one banner. It also makes plain when classical thinking is safe: heavy, fast, large objects with enormous action behave classically, while light particles in small traps, where the action is only a few times the quantum, demand the full quantum treatment and refuse to be tamed by classical pictures.