  Quantum trajectory calculations for electrons are a useful tool in the field
of molecular dynamics, e.g. to understand processes in ultrafast spectroscopy.
They have, however, two limitation: On the one hand, such calculations are
typically based on the Born-Oppenheimer approximation (BOA) and the electron
dynamics for stationary nuclei is considered, thus neglecting quantum effects
of the nuclei. On the other hand, even if the quantum nuclear motion would be
taken into account, a BOA dynamics on a single potential energy surface would
not provide any electron trajectories because the electronic part is treated as
a stationary problem. By using the exact factorization method, we overcome
these limitations and generalize the theory of electronic quantum trajectories
to a fully quantum-mechanical treatment of the nuclei. After reviewing the
time-dependent theory of quantum hydrodynamics and quantum trajectories, we
show that the nuclei can be viewed as a quantum clock for the electronic motion
and we develop a fully quantum-mechanical clock-dependent version of quantum
hydrodynamics. This theory is used to obtain electronic trajectories for
quantum nuclei, as is exemplified for a model system of a proton-coupled
electron transfer dynamics. Our work generalizes the concept of quantum
trajectories and lays the foundations for the development of trajectory-based
simulation methods of electron dynamics beyond the BOA.
