Studying bi-partite entangled state representations via the integration over ket-bra operators in \mathfrak{Q}-ordering or \mathfrak{P}-ordering

For two particles’ relative position and total momentum we have introduced the entangled state representation |η〉, and its conjugate state |ξ〉. In this work, for the first time, we study them via the integration over ket-bra operators in \(\mathfrak{Q}\) -ordering or \(\mathfrak{P}\) -ordering, where \(\mathfrak{Q}\) -ordering means all Qs are to the left of all Ps and \(\mathfrak{P}\) -ordering means all Ps are to the left of all Qs. In this way we newly derive \(\mathfrak{P}\) -ordered (or \(\mathfrak{Q}\) -ordered) expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket-bra operators within normally ordered, but also within \(\mathfrak{P}\) -ordered (or \(\mathfrak{Q}\) -ordered) are feasible and useful in developing quantum mechanical representation and transformation theory.