  Starting on the basis of $q$-symmetric oscillator algebra and on the
associate $q$-calculus properties, we study a deformed quantum mechanics
defined in the framework of the basic square-integrable wave functions space.
In this context, we introduce a deformed Schroedinger equation, which satisfies
the main quantum mechanics assumptions and admits, in the free case, plane wave
functions that can be expressed in terms of the q-deformed exponential,
originally introduced in the framework of the basic-hypergeometric functions.
