  The aim of these three papers (I, II, and III) is to develop a q-deformed
version of non-relativistic Schroedinger theory. Paper I introduces the
fundamental mathematical and physical concepts. The braided line and the
three-dimensional q-deformed Euclidean space play the role of position space.
For both cases the algebraic framework is extended by a time element. A short
review of the elements of q-deformed analysis on the spaces under consideration
is given. The time evolution operator is introduced in a consistent way and its
basic properties are discussed. These reasonings are continued by proposing
q-deformed analogs of the Schroedinger and the Heisenberg picture.
