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.
