Normal-ordered equivalent of the Weyl ordering of ${\hat{q}}^j{\hat{p}}^k$

Abstract

The problem of quantizing a bivariate dynamical system can be reduced to evaluating the ordering of ${\hat{q}}^j{\hat{p}}^k$. Here, we consider the Weyl ordering of ${\hat{q}}^j{\hat{p}}^k$ that is then expressed in term of the annihilation $\hat{a}$ and creation ${\hat{a}}^{†}$ operator. The explicit formula for the normal-ordered equivalent (all ${\hat{a}}^{†}$'s preceeding all $\hat{a}$'s) of the resulting expression is then given, and some relations are discussed.

8 pages, with Supplementary Material; no plan for journal submission