Abstract: Using the Weyl ordering of operators expansion formula (Hong-Yi Fan, J. Phys. A 25 (1992) 3443) this paper finds a kind of two-fold integration transformation about the Wigner operator $\varDelta \left( q',p'\right) $ ($\mathrm{q}$-number transform) in phase space quantum mechanics, $$\iint_{-\infty}^{\infty}\frac{{\rm d}p'{\rm d}q'}{\pi }\varDelta \left( q',p'\right) {\rm e}^{-2{\rm i}\left( p-p'\right) \left( q-q'\right) }=\delta \left( p-P\right) \delta \left( q-Q\right),$$ and its inverse% $$ \iint_{-\infty}^{\infty}{\rm d}q{\rm d}p\delta \left( p-P\right) \delta \left( q-Q\right) {\rm e}^{2{\rm i}\left( p-p'\right) \left( q-q'\right) }=\varDelta \left( q',p'\right),$$ where $Q,$ $P$ are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among $Q$--$P$ ordering, $P$--$Q$ ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.

Key words: Wigner operator, Weyl ordering, two-fold integration transformation