Semiclassical approximation

A semiclassical approximation describes a quantum system in a regime where classical actions are large compared with $\hslash$, while retaining selected quantum corrections. In one-dimensional WKB theory, a local wavefunction has phase approximately ${\hslash }^{-1}\int p(x)dx$ and amplitude proportional to $|p(x){|}^{-1/2}$ away from turning points.

The approximation exposes classical trajectories and phases without discarding interference. It fails near a turning point where $p=0$, requiring a different local solution and matching. Semiclassical therefore names a controlled approximation regime and method, rather than simply replacing every quantum quantity by a classical one.