Ermakov and non-Ermakov systems in quantum dissipative models

Abstract

Via the hydrodynamical formulation of quantum mechanics, a unified protocol to treat the quantum time-dependent harmonic oscillator with friction is presented, described by two different models: an explicitly time-dependent, linear Schrödinger equation (Caldirola–Kanai model) and a logarithmic nonlinear Schrödinger equation (Kostin model). For the former model, an Ermakov system that makes it possible to obtain an invariant of Ermakov–Lewis-type is derived. For the latter model, a non-Ermakov system is derived instead and it is shown that neither an exact nor an approximate invariant of Ermakov–Lewis-type exists.