Free motion of $q$-deformed quantum particle

The paper considers $q$ deformation of the Heisenberg algebra and a representation for it by finite-difference operators. It is shown that for the automorphism of the algebra corresponding to free motion there does not exist a polynomial Harniltonian generating it. The propertiesofplanewaves, i.e., the eigenfunctions of the $q$-deformed momentum operator, are also considered. It is noted that for $q=\exp(\pi i\vartheta)$, where $\vartheta$ is irrational, the eigenfunctions can be umneasurable.