We construct four linear composite operators for a two-particle system and give common eigenvectors of those operators. The technique of integration within an ordered product （IWOP） of operators is employed to prove that those common eigenvectors are complete and orthonormal. Therefore, a new two-mode intermediate momentum-coordinate representation which involves quantum entanglement for a two-particle system is proposed and applied to some twobody dynamic problems. Moreover, the pure-state density matrix ｜ξ1,ξ2｜ C,D C,D（ξ1, ξ2｜ is a Radon transform of Wigner operator.
Based on the generalized Weyl quantization scheme, which relies on the generalized Wigner operator Ok （p, q） with a real k parameter and can unify the P-Q, Q-P, and Weyl ordering of operators in k = 1, - 1,0, respectively, we find the mutual transformations between 6 （p - P） （q - Q）, （q - Q） 3 （p - P）, and （p, q）, which are, respectively, the integration kernels of the P-Q, Q-P, and generalized Weyl quantization schemes. The mutual transformations provide us with a new approach to deriving the Wigner function of quantum states. The - and - ordered forms of （p, q） are also derived, which helps us to put the operators into their - and - ordering, respectively.
A new kind of four-mode continuous variable coherent-entangled state is proposed in the Fock space by using the technique of integration within an ordered product, which exhibits both the properties of a coherent state and an entangled state, and spans a complete and orthonormal representation. The conjugate state of the four-mode continuous variable coherent-entangled state is derived by using the Fourier transformation. Moreover, a simple experimental protocol of generating a four-mode continuous variable coherent-entangled state is proposed by using beam splitters. As applications of this four-mode continuous variable coherent-entangled state, a four-mode entangled state and a four-mode squeezing-Fresnel operator are constructed.
A new entangled state ｜η 0） is proposed by the technique of integral within an ordered product. A generalized Hadamaxd transformation is derived by virtue of η; θ）, which plays a role of Hadamard transformation for （a1 sinθ - a2 cosθ） and （a1 cosθ ＋ a2 sin θ）.
The coherent-entangled state |α, x; λ> with real parameters λ is proposed in the two-mode Fock space, which exhibits the properties of both the coherent and entangled states. The completeness relation of |α, x; λ> is proved by virtue of the technique of integral within an ordered product of operators. The corresponding squeezing operator is derived, with its own squeezing properties. Furthermore, generalized P-representation in the coherent-entangled state is constructed. Finally, it is revealed that superposition of the coherent-entangled states may produce the EPR entangled state.
