  We consider a Finslerian type geometrization of the non-relativistic quantum
mechanics in its hydrodynamical (Madelung) formulation, by also taking into
account the effects of the presence of the electromagnetic fields on the
particle motion. In the Madelung representation the Schr\"{o}dinger equation
can be reformulated as the classical continuity and Euler equations of
classical fluid mechanics in the presence of a quantum potential, representing
the quantum hydrodynamical evolution equations. The equation of particle motion
can then be obtained from a Lagrangian similar to its classical counterpart.
After the reparametrization of the Lagrangian it turns out that the Finsler
metric describing the geometric properties of quantum hydrodynamics is a
Kropina metric. We present and discuss in detail the metric and the geodesic
equations describing the geometric properties of the quantum motion in the
presence of electromagnetic fields. As an application of the obtained formalism
we consider the Zermelo navigation problem in a quantum hydrodynamical system,
whose solution is given by a Kropina metric. The case of the Finsler
geometrization of the quantum hydrodynamical motion of spinless particles in
the absence of electromagnetic interactions is considered in detail, and the
Zermelo navigation problem for this case is also discussed.
