量子相空间中正态分布的物理意义
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摘要
量子力学除了有薛定谔的波动力学表述、海森堡的矩阵力学表述(这两种被狄拉克总结为符号法)和费曼的路径积分表述外,还有一种常用的是相空间表述,即把量子力学算符以一定的规则(例如Weyl对应规则)对应到q—p相空间的经典的坐标—动量函数,导出量子态的Wigner函数,建立相似于薛定谔波动方程的Wigner函数的时间演化方程。量子相空间分布函数允许人们用尽可能多的经典语言来描述系统的量子特性,并作为量子力学算符的表象工具来使用,最近被作为用来研究量子光学、量子信息和量子计算机的有用工具。量子相空间分布函数是量子相空间理论中最重要的组成部分,它既是量子相空间理论的基础,也是实际应用中的最主要的工具之一。
量子力学中态的描述是基于几率的假定,而数学上的随机现象也具有概率统计的特性,于是这促使我们寻找这样一种理论基础,该理论能够把概率统计这一数学工具与量子力学中联系起来,这就是本文的出发点。
最近关于Wigner函数的研究一直是个热点,在国际上研究相空间理论中的Wigner函数一般是沿用Wigner于1932年提出的Wigner函数定义式.发展新的更具有普遍意义的量子相空间分布函数是目前量子相空间理论的重要研究课题之一。我们将给出广义Wigner算符,借助IWOP技术,可以发现其正规乘积形式正好与统计中的二维正态形式相对应,并且讨论其边缘分布,最后给出其合理的物理意义。
利用有序算符内的积分技术和Weyl编序理论,把Wigner算符理论应用到傅立叶切片定理中,我们得到了某个纯态的投影算符。经过进一步研究发现,此态是完备的,构成量子力学新表象,且该态有助于量子力学中的层析成像(Tomography)理论的研究。这不仅丰富和发展了量子相空间分布函数理论,而且开辟了寻找量子力学表象的新途径。
有序算符内的积分技术可以用来研究许多纯态的完备关系,例如坐标本征态,动量本征态和相干态,研究发现其完备关系都可化成在正规乘积内的高斯形式,基于以上思想,借助有序算符内的积分技术(IWOP)和相似变换下Weyl变换的不变性,我们讨论一类单模混态和具有纠缠性质的两模混态的密度矩阵。经过研究,我们发现它们都可化成正规乘积内的二维正态分布形式,并且分析其边缘分布情况和方差。通过这种方法,我们可以把量子统计中的密度算符理论与数学统计联系起来,这大大丰富和发展了量子相空间分布函数理论。
基于湮灭算符的本征态存在p表示的思想,我们利用产生算符的本征态及其围道积分形式的完备关系来研究若干特殊函数的性质,最后得到了因变量为|z|~2的连带Laguerre多项式的围道积分形式及其新的母函数和递推公式,这些工作大大丰富了广义相空间理论和量子力学表示理论。
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.
量子力学中态的描述是基于几率的假定,而数学上的随机现象也具有概率统计的特性,于是这促使我们寻找这样一种理论基础,该理论能够把概率统计这一数学工具与量子力学中联系起来,这就是本文的出发点。
最近关于Wigner函数的研究一直是个热点,在国际上研究相空间理论中的Wigner函数一般是沿用Wigner于1932年提出的Wigner函数定义式.发展新的更具有普遍意义的量子相空间分布函数是目前量子相空间理论的重要研究课题之一。我们将给出广义Wigner算符,借助IWOP技术,可以发现其正规乘积形式正好与统计中的二维正态形式相对应,并且讨论其边缘分布,最后给出其合理的物理意义。
利用有序算符内的积分技术和Weyl编序理论,把Wigner算符理论应用到傅立叶切片定理中,我们得到了某个纯态的投影算符。经过进一步研究发现,此态是完备的,构成量子力学新表象,且该态有助于量子力学中的层析成像(Tomography)理论的研究。这不仅丰富和发展了量子相空间分布函数理论,而且开辟了寻找量子力学表象的新途径。
有序算符内的积分技术可以用来研究许多纯态的完备关系,例如坐标本征态,动量本征态和相干态,研究发现其完备关系都可化成在正规乘积内的高斯形式,基于以上思想,借助有序算符内的积分技术(IWOP)和相似变换下Weyl变换的不变性,我们讨论一类单模混态和具有纠缠性质的两模混态的密度矩阵。经过研究,我们发现它们都可化成正规乘积内的二维正态分布形式,并且分析其边缘分布情况和方差。通过这种方法,我们可以把量子统计中的密度算符理论与数学统计联系起来,这大大丰富和发展了量子相空间分布函数理论。
基于湮灭算符的本征态存在p表示的思想,我们利用产生算符的本征态及其围道积分形式的完备关系来研究若干特殊函数的性质,最后得到了因变量为|z|~2的连带Laguerre多项式的围道积分形式及其新的母函数和递推公式,这些工作大大丰富了广义相空间理论和量子力学表示理论。
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.
引文
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[3]H,W.Lee,Phys.Rep.,1995,259,147.
[4]Y.S.Kim and M.E.Noz,Phase Space Picture of Quantum Mechanics,Lecture Notes in Physics Series,Vol.40,World Scientific,Singapore,1991.
[5]Y.S.Kim and W.W.Zachary,edited,the Physics of Phase Space,Lecture Notes in Physics,Vol.278,Springer-Verlag,New York.1987.
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[7]J.E.Moyal,Proc.Cambridge Phios.Soc.,1949,45,99.
[8]R,J.Glauber,Phs.Rev.,1963,130,2529;131,2767.
[9]R,J.Glauber,in Quantum Optics and Electronics,edited by D.C.Dewitt,A.Balandin,and C.Cohen-Tannoudji,Gordon and Breach,New York,1965.
[11]k.Takahashi,Prog.Theor.Phys.Suppl.,1989,98.109.
[12]E.Prugovecki,Ann.Phys.,1978,110,102.
[13]范洪义,量子力学表象与变换论-狄拉克符号法进展,上海:上海科学技术出版社,1997
[14]H.Weyl,Z.Phys.46(1927) 1;E.Wigner,Phys.Rev.40(1932) 749
[15]Dirac PAM.The Principles of Quantum Mechanics,Oxford:Clarendon Press,1930.
[16]J.R.Klauder and B.S.Skargerstam,Coherent States,World Scientific,1985
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[18]Marian O,Scully M and Suhail Z,Quantum Optics,北京-世界图书出版公司,2000
[19]P A M Dirac,The Principles of Quantum Mechanics,Oxford:Clarendon Press,1930;
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