Semiclassical

Semiclassical describes methods or regimes in physics and mathematics that treat quantum systems using classical concepts augmented by quantum corrections. The term typically refers to situations in which the action is large compared with Planck's constant or where quantum numbers are high, so that classical trajectories or phases provide a leading description with quantization or interference effects introduced as refinements.

In quantum mechanics and chemistry, semiclassical methods include the WKB (Wentzel–Kramers–Brillouin) approximation, EBK quantization, and related path integral techniques where the dominant contributions come from classical paths. Other approaches use stationary-phase approximations and, in some settings, the Gutzwiller trace formula to relate quantum spectra to classical periodic orbits. Semiclassical dynamics describe wave packet motion using classical equations of motion with quantum corrections, and they are applied in solid state physics and quantum optics to model transport, tunneling, and interference phenomena.

In mathematics, semiclassical analysis studies asymptotic problems with a small parameter h, often identified with Planck's constant. It uses microlocal analysis and pseudodifferential operators to connect quantum evolution to classical Hamiltonian flows. Key tools include Weyl quantization, Egorov's theorem, and the study of semiclassical measures (such as Wigner measures) describing high-frequency limits.

Other contexts include semiclassical gravity, where spacetime is treated classically while matter fields are quantum, and semiclassical optics, which uses ray-based approximations supplemented by phase information to describe wave propagation in complex media.

The semiclassical approach provides a bridge between quantum and classical descriptions and is widely used in quantum chaos, atomic and molecular physics, nanostructures, and the analysis of partial differential equations.