  We consider algebras of quantum differential operators, for appropriate
bicharacters on a polynomial algebra in one indeterminate and for the
coordinate algebra of quantum $n$-space for $n\geq 3$. In the former case a set
of generators for the quantum differential operators was identified in work by
the first author and T. C. McCune but it was not known whether the algebra is
Noetherian. We answer this question affirmatively, setting it in a more general
context involving the behaviour Noetherian condition under localization at the
powers of a single element. In the latter case we determine the algebra of
quantum differential operators as a skew group algebra of the group
$\mathbb{Z}^n$ over a quantized Weyl algebra. It follows from this description
that this algebra is a simple right and left Noetherian domain.
