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The first of the two related papers analising and explaining the origin,manifestations and parodoxical features of the quantum potential (QP) from thenon-relativistic and relativistic point of view. QP arises in the quantumHamiltonian, under various procedures of quantization of the systems whoseHamilton functions are the positive-definite quadratic forms in momenta withcoefficients depending on the coordinates in (n-dimensional) configurationalspace (natural systems). Owing to the Riemannian structure thus introduced inthe space, the result of quantization is considered as quantum mecanics (QM) ofa particle. Contradiction of QP to the Principles of General Covariance andEquivalence is discussed. It is found that actually the historically firstHilbert space based quantization by E. Schr\"odinger (1926), after revision inthe modern framework of QM, also leads to QP in the form that B. DeWitt hadbeen found 26 years later. Efforts to avoid QP or reduce its drawbacks arediscussed. The general conclusion is that some form of QP and a violation ofthe principles of general relativity induced by it are apparently inevitable inthe non-relativistic QM. It is shown also that Feynman quantization singles outtwo versions of QP, which both determine two bi-scalar propagators which fixtwo different algorithms of path integral calculation. In the accompanyingpaper under the same general title and the subtitle "The Relativistic Point ofView", relation of the non-relativistic QP to the quantum theory of the scalarfield non-minimally coupled to the curved space-time metric is considered.

Unfinished History and Paradoxes of Quantum Potential. I. Non-Relativistic Origin, History and Paradoxes