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

The amplitude–phase formulation of the Schrödinger equation is investigated within the context of uncoupled Ermakov systems, whereby the amplitude function is given by the auxiliary non-linear equation. The classical limit of the amplitude and phase functions is analysed by setting up a semi-classical 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 semi-classical 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 h̄ → 0. PACS numbers: 02.30.Jr, 02.30.−f, 03.65.−w