  This paper is the first of two papers devoted to formulation of quantum
mechanics of a particle in a normal geodesic frame of reference in the general
Riemannian space-time. Here canonical quantization of geodesic motion in the
1+3-formalism is considered, the result of which will be compared in the
subsequent paper II with that of the field-theoretical approach, see also
gr-qc/9807030. The Schr\"odinger representation of quantum-mechanical
kinematics and dynamics is presented in the general-covariant form and a
physical interpretation of the state vectors and the position operators is
discussed.
