Non-oscillating solutions to uncoupled Ermakov systems and the semi-classical limit

Abstract

The amplitude-phase formulation of the Schr\"{o}dinger equation is investigated within the context of uncoupled Ermakov systems, whereby the amplitude function is given by the auxiliary nonlinear equation. The classical limit of the amplitude and phase functions is analyzed by setting up a semiclassical Ermakov system. In this limit, it is shown that classical quantities, such as the classical probability amplitude and the reduced action, are obtained only when the semiclassical amplitude and the accumulated phase are non-oscillating functions respectively of the space and energy variables. Conversely, among the infinitely many arbitrary exact quantum amplitude and phase functions corresponding to a given wavefunction, only the non-oscillating ones yield classical quantities in the limit $\hbar \to 0$.