  Following the definition of quantum differential operators given by Lunts and
Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of
quantum differential operators on the affine line is the ring generated by x
and \del, the familiar differential operators on the line, along with two
additional operators which we call \del^\beta^1 and \del^\beta^-1. We describe
this ring both as a subring of the ring of graded endomorphisms and as a ring
given by generators and relations. From this starting point, we are able to
describe the ring of quantum differential operators on affine n-space and to
construct the ring of global quantum differential operators on the projective
line.
