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 Hamiltonian generating it. The properties of plane waves, i.e., the eigenfunctions of the q-deformed momentum operator, are also considered. It is noted that for q = exp(πi var-theta), where var-theta is irrational, the eigenfunctions can be unmeasurable. 5 refs