  We propose a new systematic fibre bundle formulation of nonrelativistic
quantum mechanics. The new form of the theory is equivalent to the usual one
but it is in harmony with the modern trends in theoretical physics and
potentially admits new generalizations in different directions. In it a pure
state of some quantum system is described by a state section (along paths) of a
(Hilbert) fibre bundle. Its evolution is determined through the bundle
(analogue of the) Schr\"odinger equation. Now the dynamical variables and the
density operator are described via bundle morphisms (along paths). The
mentioned quantities are connected by a number of relations derived in this
work.
  In the second part of this investigation we derive several forms of the
bundle (analogue of the) Schr\"odinger equation governing the time evolution of
state sections. We prove that up to a constant the matrix-bundle Hamiltonian,
entering in the bundle analogue of the matrix form of the conventional
Schr\"odinger equation, coincides with the matrix of coefficients of the
evolution transport. This allows to interpret the Hamiltonian as a gauge field.
Here we also apply the bundle approach to the description of observables. It is
shown that to any observable there corresponds a unique Hermitian bundle
morphism (along paths) and vice versa.
