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

TL;DR

This paper derives an explicit formula for converting Weyl-ordered expressions of ^j ^k into their normal-ordered form using annihilation and creation operators, aiding quantum system quantization.

Contribution

It provides a new explicit formula for the normal-ordered equivalent of Weyl-ordered ^j ^k expressed in terms of annihilation and creation operators.

Findings

Derived explicit formula for normal ordering of Weyl-ordered ^j ^k

Discussed relations between different orderings in quantum operators

Facilitates quantization of bivariate dynamical systems

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 $\hat{a}^\dagger$ operator. The explicit formula for the normal-ordered equivalent (all $\hat{a}^\dagger$'s preceeding all $\hat{a}$'s) of the resulting expression is then given, and some relations are discussed.