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A potential barrier is a region of higher potential energy separating accessible regions of a physical system. A classical particle cannot pass through a barrier higher than its total energy along its trajectory, but a quantum particle can have nonzero transmission probability through a finite barrier by quantum tunnelling.

In a one-dimensional region with constant $V_0>E$, the Schrödinger equation gives exponential solutions with tunnelling decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The wavefunction can therefore penetrate a finite barrier and match an outgoing wave on the other side. The rectangular potential barrier is an exactly solvable model; smooth barriers lead naturally to a semiclassical approximation. Even for $E>V_0$, quantum reflection can occur where the potential changes.

The reduced Planck constant is $\hslash =h/(2\pi )$, with the same action units as the Planck constant. Since angular frequency is $\omega =2\pi \nu$, a photon's energy can be written $E=\hslash \omega$ instead of $h\nu$.

The reduced Planck constant occurs naturally in the Schrödinger equation, the canonical commutation relation $[X,P]=i\hslash I$, and the Heisenberg uncertainty principle $\Delta X\Delta P\ge \hslash /2$. Comparing a characteristic action with $\hslash$ helps assess a semiclassical approximation, although a large ratio alone does not guarantee that all quantum interference becomes unobservable.

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.