The formulation of the generally eovariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect to $c^{-2}$) operators of the spatial position of a spirdess particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of the $V_{1,3}$ Newton–Wigner operator is established. It is shown that the projection of the momentum onto the $4$-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrödinger equation.