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Quantum hydrodynamics is a formulation of quantum mechanics based on theprobability density and flux (current) density of a quantum system. It can beused to define trajectories which allow for a particle-based interpretation ofquantum mechanics, commonly known as Bohmian mechanics. However, quantumhydrodynamics rests on the usual time-dependent formulation of quantummechanics where time appears as a parameter. This parameter describes thecorrelation of the state of the quantum system with an external system -- aclock -- which behaves according to classical mechanics. With the ExactFactorization of a quantum system into a marginal and a conditional system,quantum mechanics and hence quantum hydrodynamics can be generalized forquantum clocks. In this article, the theory is developed and it is shown thattrajectories for the quantum system can still be defined, and that thesetrajectories depend conditionally on the trajectory of the clock. Suchtrajectories are not only interesting from a fundamental point of view, butthey can also find practical applications whenever a dynamics relative to anexternal time parameter is composed of "fast" and "slow" degrees of freedom andthe interest is in the fast ones, while quantum effects of the slow ones (likea branching of the wavepacket) cannot be neglected. As an illustration, time-and clock-dependent trajectories are calculated for a model system of anon-adiabatic dynamics, where an electron is the quantum system, a nucleus isthe quantum clock, and an external time parameter is provided, e.g. via aninteraction with a laser field that is not treated explicitly.

Time in quantum mechanics: A fresh look on quantum hydrodynamics and quantum trajectories