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

TL;DR

This paper develops a generalized quantum hydrodynamics framework incorporating quantum clocks, enabling the definition of conditional trajectories that depend on clock dynamics, with potential applications in complex quantum systems involving multiple time scales.

Contribution

It introduces a novel approach to quantum hydrodynamics using the Exact Factorization, allowing trajectories to depend on quantum clock states, extending the Bohmian interpretation.

Findings

Trajectories depend conditionally on the quantum clock's trajectory.

The framework applies to non-adiabatic dynamics with fast and slow degrees of freedom.

Illustrative model demonstrates time- and clock-dependent trajectories.

Abstract

Quantum hydrodynamics is a formulation of quantum mechanics based on the probability density and flux (current) density of a quantum system. It can be used to define trajectories which allow for a particle-based interpretation of quantum mechanics, commonly known as Bohmian mechanics. However, quantum hydrodynamics rests on the usual time-dependent formulation of quantum mechanics where time appears as a parameter. This parameter describes the correlation of the state of the quantum system with an external system -- a clock -- which behaves according to classical mechanics. With the Exact Factorization of a quantum system into a marginal and a conditional system, quantum mechanics and hence quantum hydrodynamics can be generalized for quantum clocks. In this article, the theory is developed and it is shown that trajectories for the quantum system can still be defined, and that these trajectories depend conditionally on the trajectory of the clock. Such trajectories are not only interesting from a fundamental point of view, but they can also find practical applications whenever a dynamics relative to an external time parameter is composed of "fast" and "slow" degrees of freedom and the interest is in the fast ones, while quantum effects of the slow ones (like a branching of the wavepacket) cannot be neglected. As an illustration, time- and clock-dependent trajectories are calculated for a model system of a non-adiabatic dynamics, where an electron is the quantum system, a nucleus is the quantum clock, and an external time parameter is provided, e.g. via an interaction with a laser field that is not treated explicitly.