I have managed to show that for the one-dimensional classical system (with two dimensional phase space) we have
$$\left\{ q^{a}p^{b},q^{c}p^{d}\right\} =\left(ad-bc\right)q^{a+c-1}p^{b+d-1}$$
Where $a,b,c,d$ are non-genative integer exponents (not labelling different coordinates). Equivalently
$$\left\{ q^{a},p^{b}\right\} =abq^{a-1}p^{b-1}$$
Can we get similar results for the quantum CCR with generators $q,p$ satisfying $\left[q,p\right]=i\hbar$? That is, general (and hopefully fairly simple) formulae for the commutator of arbitrary elements of the CCR?
It will certainly be more complicated, and we now have to worry about ordering ambiguities of course.
