  We propose a new systematic fibre bundle formulation of nonrelativistic
quantum mechanics. The new form of the theory is equivalent to the usual one
and is in harmony with the modern trends in theoretical physics and potentially
admits new generalizations in different directions. In it the Hilbert space of
a quantum system (from conventional quantum mechanics) is replaced with an
appropriate Hilbert bundle of states and a pure state of the system is
described by a lifting of paths or section along paths in this bundle. The
evolution of a pure state is determined through the bundle (analogue of the)
Schr\"odinger equation. Now the dynamical variables and the density operator
are described via liftings of paths or morphisms along paths in suitable
bundles. The mentioned quantities are connected by a number of relations
derived in this work.
