Developing The Phase Space Theory In Quantum Mechanics By Virtue Of The Entangled State Representation And The Technique Of Integration Within An Ordered Product (IWOP) Of Operators

Based on the technique of integration within an ordered product (IWOP) of operators and the quantum entangled state representation we develop the quantum phase space theory. The phase-space distribution function allows one to describe the quantum aspects of a system with as much classical language allowed and have been used as representation tools for quantum-mechanical operators. They provide the ideal link to explore and understand the transition to classical mechanics and to display in phase-space quantum effects. They have also recently proposed as a useful tool for studies related to quantum information and computation. The Wigner function theory can be applied to dealing with all kinds of problems in quantum optics and quantum chemistry, especially be used to study the density matrix of the light field, the quantum interference and the laser theory. The Husimi functions are known to be good tools to study the quantum-classical correspondence and quantum chaos. All these work develop the quantum phase space theory further. So our study is significant. The main content includes:We explore how to solve the P represenrtation from the Wigner function and present a formula to obtain the P represenrtation from the known Wigner function. We also show how to use the formula through several examples. In doing so, the Weyl-Wigner corresponding theory can be enriched and developed. Based on the entangled Wigner operator we bring forward the new concept of the entangled Husimi operator, find that the entangled Husimi operator is a pure two-mode squeezed coherent state density matrix, which provides us with a neat and concise operator version to calculate the Husimi distribution function of the two-mode quantum state.Based on the explicit Weyl-ordered form of Wigner erator and the technique of integration within Weyl ordered product of operators we derive the Weyl-ordered operator product formula. The formula is then generalized to the entangled form with the help of entangled state representations. In so doing the Weyl-Wigner correspondence theory gets enriched and developed. We find a new application of the Weyl correspondence in studying the Husimi operator and present a simple way to find the Husimi operator, i.e. regarded exp as the classical Weyl correspondence function connecting the Husimi operator.Through the Radon transformation of the normally ordered Wigner operator we introduce two mutually conjugate intermediate coordinate- momentum representations. Based on them we construct the appropriate quantum phase space theory which includes the new Wigner operator adapting to this space and construct the appropriate generalized Fredholm operator equation and then find its solution. We then deriving the Hermite polynomials operator identities by applying the Fredholm equation. We also reveal the connection between the generalized Wigner operator and the 2-dimension normal distribution in statistics, which is useful to study the quantum tomogram.As the application of the entanglement Husimi operator theory we calculate the Wigner function and the Husimi function of the one- and two-mode combination squeezed state ( OTCSS ) , study their characters through drawing the three-dimensional graphics. we also study the Husimi distribution of the excited squeezed vacuum state ( ESVS ).The theoretical calculation of Husimi function can help experimentalists to judge the quality of the experiment. We solve the problem of the density matrix for two interacting particles with kinetic coupling by using the bipartite entangled state representation. Such kind of interaction often appear as describing internal potential in molecular physics theory, the mutual magnetic inductance between two quantized coupling circuits, etc.For the first time we introduce the operator for studying Husimi distribution function in phase space (γ,ε) for electron's states in uniform magnetic field by virtue of the IWOP technique and the entanglement state representation. Using the Wigner operator in the entangled state <λ| representation we find that is just a pure squeezed coherent state density operator, which brings much convenience for studying the Husimi distribution of various electron's states . We also demonstrate that the marginal distributions of the Husimi function are Gaussian-broadened version of the Wigner marginal distributions.