The Physical Meaning Of Normal Distribution In Quantum Phase Spase

There also exist a common used phase space representation in quantum mechanics besides Schr(o|¨)dinger's wave mechanics representation,Heisenberg's matrix mechanics representation(these two representation are summarized by Dirac as symbolic method) and Feyman's routing integration,It is to correspond quantum operators to classical coordinate-momentum function in p-q phase space(such as Weyl correspondence rule) to deduce Wigner function of quantum state and constract the time evolutionary equation simiar to Schr(o|¨)dinger's wave equation of Wigner function. The phase-space distribution function allows one to describe the quantum aspects of a system with classical language as much as possible and have been used as representation tools for quantum-mechanical operators.They have also recently proposed as a useful tool for studies related to quantum optics,quantum information and quantum computation.The phase-space distribution function is the most important constitution of quantum phase-space theory.It is not only the base of quantum phase-space theory,but also the main tool in practical application.The description of the state of quantum mechanics state is supposed as probability.And that the random phenomenon in mathmatics also has identities of probability statistics.Then it urge us to search such a theory which can connect probability statistics in mathmatics with quantum mechanics,which is this paper's origin.Recently,the reseach on Wigner function is a hotspot.Wigner function used in international is usually followed by Wigner's definition in 1932 and developed in many quantum optics books.Based on review of usual Wigner operator,we will give generalized Wigner operator.With the help of IWOP,we find the connection beteewn the generalized Wigner operator and the bivariate normal distribution in statistics.Its marginal distribution is discussed and physical meaning is gived.Applying Wigner operator theory to Fourier slice theorem,we naturally derive a projection operator of pure state based on the technique of integration within an ordered product(IWOP) of operators and the Weyl ordering theory.By further studies, we find that this pure state is a new complete representation and helpful to study quantum tomography.All these work not only enrich the theory of phase-space distribution function,but also a new approach for obtaining new quantum mechanical representation.The technique of integration within an ordered product(IWOP for short) of operators can be used to study the completeness relation of pure states,such as the coordinate eigenstate,the momentum eigenstate and the coherent state.We find that all these completeness relation can be recast into the normally ordered Gaussian forms. Based on this idea and employing the technique of integration within an ordered product of operators and Weyl ordering invariance under similar transformations,we discuss the density matrices of single-mode mixed states and two-mode mixed states with entanglement involved.We find that these matrices can be recast into the normally orded bivariate normal distribution.The marginal distribution and variance are analysed.In this way,the density operators theory in quantum statistics can be connected to mathematical statistics.All these enrich and develop the distribution function theory in quantum phase space.Based on the idea that annihilation operator's eigenstate have P representation, We use the creation operator's eigenkets and completeness relation in contour integration to study special function.We derive new contour integration expression of associated Laguerre polynomial and its generalized generating function formula,and a series of recursive relations have been derived in our new approach.These work enrich the content of generalized phase space and representation theory of quantum mechanics.