We present a new non-Archimedean realization of the Fock representation of
    the q-oscillator algebras where the creation and annihilation operators act on
    complex-valued functions, which are defined on a non-Archimedean local field of
    arbitrary characteristic, for instance, the field of p-adic numbers. This new
    realization implies that a large number of quantum models constructed using
    q-oscillator algebras are non-Archimedean models, in particular, p-adic quantum
    models. In this framework, we select a q-deformation of the Heisenberg
    uncertainty relation, and construct the corresponding q-deformed Schr\”odinger
    equations. In this way we construct a p-adic quantum mechanics which is a
    p-deformed quantum mechanics. We also solve the time-independent Schr\”odinger
    equations for the free particle, and a particle in a non-Archimedean box. In
    the last case we show the existence of a discrete sequence of energy levels. We
    determine the eigenvalues of Schr\”odinger operator for a general radial
    potential. By choosing the potential in a suitable form we recover the energy
    levels of the q-hydrogen atom.

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