Classical limit

The classical limit is a regime in which specified observables of a quantum system are approximated by classical mechanics or classical field equations. It is a statement about scales, states and measurements, rather than simply about an object being large.

For a narrow wave packet in a smoothly varying potential energy, Ehrenfest theorem gives $d⟨P⟩/dt=-⟨V^{\prime }(X)⟩$. Approximating the force average by $-V^{\prime }(⟨X⟩)$ yields a classical trajectory while the packet remains sufficiently narrow. Strong spreading or quantum tunnelling can invalidate it. For highly occupied bosonic modes, occupation-number discreteness can likewise become relatively small, yet quantum interference can remain observable when relative quantum phases are preserved. A useful classical limit must therefore specify which predictions the approximation reproduces.

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.