purpose one can use, for instance, the quantum-mechanical formalism described in [@Prugovecki-QMinHS] or in [@Neumann-MFQM] (see also ).

In Sect. \[III\] we recall the notion of a linear transport along paths in vector fibre bundles and make certain remarks concerning the special case of a Hilbert bundle.

Sect. \[new-I\] begins the building of the new bundle approach to quantum mechanics. Here the concept of a *Hilbert fibre bundle of the states* corresponding to a quantum system is introduced. The analogue of the state vector now is the *state section (along paths)*. We present here also some technical (mathematical) details, such as ones concerning (Hermitian) bundle metric, Hermitian and unitary maps etc.

In Sect. \[IV\] is proved that in the new description the evolution operator of a quantum system is (equivalently) replaced by a suitable linear transport along paths, called *evolution transport* or a *bundle evolution operator*.

The paper closes with Sect. \[conclusion-I\].

Evolution of pure quantum states (review) {#II}
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In quantum mechanics  a pure state of a quantum system is described by a state vector $\psi(t)$ (in Dirac’s [@Dirac-PQM] notation $|t\rangle$) generally depending on the time $t\in\mathbb R$ and belonging to a Hilbert space ${\mathcal{F}}$ (specific to any concrete 