  Starting on the basis of the non-commutative q-differential calculus, we
introduce a generalized q-deformed Schr\"odinger equation. It can be viewed as
the quantum stochastic counterpart of a generalized classical kinetic equation,
which reproduces at the equilibrium the well-known q-deformed exponential
stationary distribution. In this framework, q-deformed adjoint of an operator
and q-hermitian operator properties occur in a natural way in order to satisfy
the basic quantum mechanics assumptions.
