Classical and semi-classical limits in phase space

Abstract

A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schr\"odinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by $\hbar$. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of $2\pi$, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.