  The algebra of quantum differential operators on graded algebras was
introduced by V. Lunts and A. Rosenberg. D. Jordan, T. McCune and the second
author have identified this algebra of quantum differential operators on the
polynomial algebra with coefficients in an algebraically closed field of
characteristic zero. It contains the first Weyl algebra and the quantum Weyl
algebra as its subalgebras. In this paper we classify irreducible weight
modules over the algebra of quantum differential operators on the polynomial
algebra. Some classes of indecomposable modules are constructed in the case of
positive characteristic and q root of unity.
