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Based on the idea of transition from classical optics to quantum optics we deduce the natural expressions of optical beam splitter (BS) and 2-cascaded BS operators in coherent state representation and also obtain their normally ordered forms. Moreover, we theoretically prepare some desired quantum entangled states and investigate several carried experiments. Finally, our work shall be focused on the following two aspects: the quantum-optical catalysis of the BS and quantum state truncation of the 2-cascaded BS. [ABSTRACT FROM AUTHOR]

By virtue of integration technique within ordered product of operators and Dirac's representation theory we find a new general formula for normally ordering coordinate-momentum operator functions, that is $f(g\hat {{Q}}+h\hat {P})= :\exp [\textstyle {g^{2}+h^{2} \over 4}\textstyle {{\partial ^{2}} \over {\partial (g\hat {{Q}}+h\hat {P})^{2}}}]f(g\hat {{Q}}+h\hat {P})$ :, where $\hat {Q}$ and $\hat {P}$ are the coordinate operator and momentum operator respectively, the symbol :: denotes normal ordering. Using this formula we can derive a series of new relations about Hermite polynomial and Laguerre polynomial, as well as some new differential relations. [ABSTRACT FROM AUTHOR]

The new intermediate entangled state | η; θ〉 is proposed by virtue of IWOP technique, which is the common eigenvector of $[(\hat{x}_{1} - \hat{x}_{2})\cos\theta -(\hat{p}_{1} - \hat{p}_{2})\sin\theta ]$ and $[(\hat{x}_{1} +\hat{x}_{2})\sin\theta + (\hat{p}_{1} + \hat{p}_{2})\cos\theta ]$. The squeezing transformation operator, Hadamard transformation operator, Fresnel transformation operator and Radon transform operator are constructed by | η; θ〉. [ABSTRACT FROM AUTHOR]

The new coherent-entangled state | z, x; θ〉 is proposed in the two-mode Fock space, which exhibits both the properties of coherent and entangled states. The completeness relation of | z, x; θ〉 is proved by virtue of the technique of integral within an ordered product of operators. A generalized Hadamard-Fresnel complementary transformation derived by virtue of the coherent-entangled state | z, x; θ〉, which is unitary. The new unitary operator plays the role of both Hadamard transformation for $(\hat{a}_{1}\sin\theta -\hat{a}_{2}\cos\theta)$ and Fresnel transformation for $(\hat{a}_{1}\cos\theta +\hat{a}_{2}\sin\theta)$, respectively. [ABSTRACT FROM AUTHOR]

ket-bra form quantum operator is constructed and its transformation properties are researched. Furthermore, we quantize two L-C circuits with coupling and investigate what happens to the two circuits when this system is quantized. [ABSTRACT FROM AUTHOR]

Some unitary operators are derived using quantum state by depending on the technique of integration within an ordered product of operators, for example parity operator, displacement operator, squeezed operator, etc. The characteristics of these operators are analyzed. Their unitary transformations play an essential role in some transformations. As applications, the dynamic problems of the double momentum coupling harmonic oscillators are solved exactly. [ABSTRACT FROM AUTHOR]

We construct eight operators for a four-particle system, namely one center-of-mass coordinate operator, three relative coordinate operators, one total momentum operator and three mass-weighted relative momentum operators, and give common eigenvectors of four compatible observables $\{\sum_{i=1}^{4}\hat{p}_{i},\hat{x}_{1}-\hat{x}_{2},\hat{x}_{2}-\hat{x}_{3},\hat{x}_{3}-\hat{x}_{4}\}$ , which are composed of four particles’ coordinate $\hat{x}_{i}$ and momentum $\hat{p}_{i}$ . By compatible we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product (IWOP) of operators, we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation. [ABSTRACT FROM AUTHOR]