New approach to Q — P (P — Q) ordering of quantum mechanical operators and its applications

In this paper, we introduce a new way to obtain the Q — P (P — Q) ordering of quantum mechanical operators, i.e., from the classical correspondence of Q — P (P — Q) ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single (two-) mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator's Q — P (P — Q) ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q — P ordering of operators.