Does anyone have a suggestion for the best computer program to perform calculations in the 2nd Weyl algebra?
I use GAP (http://www.gap-system.org/) with the GBNP package (http://www.win.tue.nl/~amc/pub/grobner/chap0.html). You define Weyl algebra as a quotient of free associative algebra by the ideal $<x_1 x_2 - x_2 x_1 - 1>$. Easy to generalize to arbitrary dimension and can handle Clifford algebras (of arbitrary signatures) in the same setting.
Singular claims to be wise in the ways of Weyl algebra.