量子力学表象新应用及构造的新方法
|
摘要
量子力学的语言是Dirac符号法,也称为q数理论,而q数理论核心内容之一就是表象理论。量子力学表象不但能作为“坐标架”表述量子力学的若干基本规律,而且在研究特定的动力学问题时,选择一个合适的表象,对于问题的解决往往是关键的,所以表象具有运动学和动力学的双重意义。我国学者创造的有序算符内的算符积分技术(The technique of integration within an ordered product (IWOP) of operators)(简称:IWOP技术)成功地实现了对Dirac的ket-bra型算符的积分,不但为Newton-Leibniz积分的发展开拓了一个新的方向,并且也为实现经典变换到量子幺正变换寻找显示形式的q数(幺正算符)提供了自然过渡的捷径。IWOP技术不但能够更好地解读量子力学和发展量子力学数理基础,而且开拓了Dirac符号法及表象理论应用潜力。我们发现,有了IWOP技术,一些已知的表象就有了新的应用,新的表象就可被方便的找出。
本文内容分三个主题:1.将量子力学和量子光学中一些常见的表象应用到一些具体的物理问题中,并借助IWOP技术使这些原本解决起来较为困难的问题能够方便的得以解决。2.鉴于量子力学表象的重要性,我们提出了一个能够推导出连续表象的新方法;3.我们还构造并提出了一些q变形量子力学理论方面的新表象,并讨论了它们的一些性质及应用。
本文主要内容安排如下:
第一章,我们简要回顾了正规乘积内的算符积分技术和Weyl量子化方案及Weyl编序内的算符积分技术。
第二章,借助玻色相干态表象和角动量的Schwinger玻色算符实现,我们推导出了SU(2)转动的量子哈密顿量形式,及角速度与SU(2)变换的关系表达式。虽然,作为客观测量的自旋没有经典对应,但是,我们仍可以假设将其作为一个刚体来处理,并计算了SU(2)转动的准经典配分函数。
第三章,我们主要借助EPR纠缠态表象和纠缠Wigner算符来研究含动能耦合项的两体量子系统的Winger函数的时间演化方程,这恰好表明了,选择一个恰当的表象,确实能够为解决动力学问题提供极大的方便。
第四章,借助热纠缠态表象,我们求解腔阻尼Raman耦合模型的密度主方程,获得了密度矩阵元的正规乘积形式,并推导出相应的Wigner函数。
第五章,鉴于Wigner算符在量子相空间中的重要性,我们通过将EPR纠缠态表象推广至多模情况,在多模质心坐标和质量权重相对动量的共同本征态表象及其共轭表象中构造出多模Wigner算符。
第六章,借助Wigner算符的完备性关系和Weyl对应,我们构造一个能够获得纯态密度算符的新方程,即:寻找到了一个获得连续量子力学表象的新方法;作为这方程的应用,量子力学和量子光学中一些重要的表象都能够利用此方程被推导出。
第七章,我们提出一个激发双模广义压缩真空态,发现它可以被看作是一个广义压缩双变量厄米多项式激发在真空态上,证明它的归一化系数恰好是一个雅可比多项式。并研究了它的统计性质,如:光子数分布和相应的Wigner函数。
第八章,利用δ函数的围道积分表示和有序算符内的积分技术,我们指出q变形产生算符具有本征右矢,揭示了一组由非厄米共轭的左右矢构成的新的正交完备性关系;作为应用我们讨论了密度算符的广义P表示。
第九章,利用q变形理论和q变形玻色子真空投影算符的正规乘积形式,我们引入q变形纠缠态,同样得到了一组由非厄米共轭的左右矢构成的正交完备性关系;进一步研究了由q变形纠缠态表象描述的压缩算符的压缩性质。
第十章,同样使用q变形理论和q变形玻色子真空投影算符的正规乘积形式,我们介绍q变形坐标本征态,并利用此表象的正交完备性关系实现并研究了一些连续变量量子门算符。
Language of quantum mechanics is Dirac symbol method, also known as q-number theory, in which the one of the most important contents is quantum mechanics representation which can not only describe some fundamental law on quantum mechanics as coordinate frame, but also bring great convenience for choosing an appropriate representation to study specific problem of dynamics. Thus representation has the double meaning of kinematics and dynamics. The technique of integration within an ordered product (IWOP) of operators has successfully realized the integral over ket-bra operators, which means that it exploits a new direction for Newton-Leibniz integral, and provides a new approach to finding directly evident form of q-number for performing the natural transition from classical transformation to quantum unitary transformation. It can not only interpret quantum mechanics and develop foundation of the mathematical physics of quantum mechanics, but also exploit the potential applications of Dirac symbol method and representation theory. We find that IWOP technique can help us to explore new application for some known representations and find out some new representations.
The main content in this paper includes the three aspects: 1. we shall mainly apply some usual representations to some concrete physical problems, in which we show that the IWOP technique is a very powerful tool for solving these quantum problems difficult to deal with. 2. based on the significance of quantum mechanics representation, we propose a new method or equation to obtain continuous quantum mechanics representation. 3. we construct and introduce several new representations on q deformed case and discuss their some properties and applications.
The main content of this paper is arranged as follows:
In chapter 1, We briefly review the Technique of Integration Within normal ordered product and the Weyl quantization scheme, as well as the Technique of Integration Within Weyl ordering.
In chapter 2, by virtue of the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum, we find the formula for the quantum Hamiltonian for SU(2) rotation, we further specify the angular velocity. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.
In chapter 3, for the bipartite Hamiltonian system with kinetic coupling, we derive time evolution equation of Wigner functions by virtue of the EPR entangled state representation and entangled Wigner operator, which just indicates that choosing a good representation indeed provides great convenience for us to deal with the dynamics problem.
In chapter 4, by virtue of the thermo entangled state representation, we exhibit a novel approach to deriving density operator for a Raman-coupled model with damping of the cavity mode. The normal ordering forms of density matrix elements can be obtained, and the corresponding Wigner functions are also derived.
In chapter 5, based on the significance in quantum phase space, by extending EPR entangled state representation to multipartite case. We construct n-mode Wigner operator in the common eigenstate of the multipartite centre-of-mass coordinate and two mass-weighted relative momenta, as well as its canonical conjugate state.
In chapter 6, using the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators, that is to say, we find a new method to derive continuous quantum mechanics representation. Several important examples in quantum mechanics and quantum optics are considered as the applications of this equation.
In chapter 7, we construct a new state called excited two-mode generalized squeezed vacuum states and find that it is just regarded as a generalized squeezed two-variable Hermite polynomial excitation on the vacuum state, as well as its normalization constant is just proved as a Jacobi polynomial. Their statistical properties are investigated such as squeezing properties and the form of the corresponding Wigner function.
In chapter 8, by using the contour integral representation ofδ-function and the technique of integration within an ordered product of operators, we point out that the q-deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized P-representation of density operator is demonstrated.
In chapter 9, the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator. Similarly, the new completeness and orthogonality relations composed of the bra and ket, which are not mutually Hermitian conjugates are obtained. Furthermore, the property of squeezing operator represented by the q-deformed entangled states is exhibited.
In chapter 10, similarly using the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator, we introduce the q deformed coordinate representation. Further, we utilize the eigenket and eigenbra for this representation to realize and study some important quantum gate operators for continuum variables.
本文内容分三个主题:1.将量子力学和量子光学中一些常见的表象应用到一些具体的物理问题中,并借助IWOP技术使这些原本解决起来较为困难的问题能够方便的得以解决。2.鉴于量子力学表象的重要性,我们提出了一个能够推导出连续表象的新方法;3.我们还构造并提出了一些q变形量子力学理论方面的新表象,并讨论了它们的一些性质及应用。
本文主要内容安排如下:
第一章,我们简要回顾了正规乘积内的算符积分技术和Weyl量子化方案及Weyl编序内的算符积分技术。
第二章,借助玻色相干态表象和角动量的Schwinger玻色算符实现,我们推导出了SU(2)转动的量子哈密顿量形式,及角速度与SU(2)变换的关系表达式。虽然,作为客观测量的自旋没有经典对应,但是,我们仍可以假设将其作为一个刚体来处理,并计算了SU(2)转动的准经典配分函数。
第三章,我们主要借助EPR纠缠态表象和纠缠Wigner算符来研究含动能耦合项的两体量子系统的Winger函数的时间演化方程,这恰好表明了,选择一个恰当的表象,确实能够为解决动力学问题提供极大的方便。
第四章,借助热纠缠态表象,我们求解腔阻尼Raman耦合模型的密度主方程,获得了密度矩阵元的正规乘积形式,并推导出相应的Wigner函数。
第五章,鉴于Wigner算符在量子相空间中的重要性,我们通过将EPR纠缠态表象推广至多模情况,在多模质心坐标和质量权重相对动量的共同本征态表象及其共轭表象中构造出多模Wigner算符。
第六章,借助Wigner算符的完备性关系和Weyl对应,我们构造一个能够获得纯态密度算符的新方程,即:寻找到了一个获得连续量子力学表象的新方法;作为这方程的应用,量子力学和量子光学中一些重要的表象都能够利用此方程被推导出。
第七章,我们提出一个激发双模广义压缩真空态,发现它可以被看作是一个广义压缩双变量厄米多项式激发在真空态上,证明它的归一化系数恰好是一个雅可比多项式。并研究了它的统计性质,如:光子数分布和相应的Wigner函数。
第八章,利用δ函数的围道积分表示和有序算符内的积分技术,我们指出q变形产生算符具有本征右矢,揭示了一组由非厄米共轭的左右矢构成的新的正交完备性关系;作为应用我们讨论了密度算符的广义P表示。
第九章,利用q变形理论和q变形玻色子真空投影算符的正规乘积形式,我们引入q变形纠缠态,同样得到了一组由非厄米共轭的左右矢构成的正交完备性关系;进一步研究了由q变形纠缠态表象描述的压缩算符的压缩性质。
第十章,同样使用q变形理论和q变形玻色子真空投影算符的正规乘积形式,我们介绍q变形坐标本征态,并利用此表象的正交完备性关系实现并研究了一些连续变量量子门算符。
Language of quantum mechanics is Dirac symbol method, also known as q-number theory, in which the one of the most important contents is quantum mechanics representation which can not only describe some fundamental law on quantum mechanics as coordinate frame, but also bring great convenience for choosing an appropriate representation to study specific problem of dynamics. Thus representation has the double meaning of kinematics and dynamics. The technique of integration within an ordered product (IWOP) of operators has successfully realized the integral over ket-bra operators, which means that it exploits a new direction for Newton-Leibniz integral, and provides a new approach to finding directly evident form of q-number for performing the natural transition from classical transformation to quantum unitary transformation. It can not only interpret quantum mechanics and develop foundation of the mathematical physics of quantum mechanics, but also exploit the potential applications of Dirac symbol method and representation theory. We find that IWOP technique can help us to explore new application for some known representations and find out some new representations.
The main content in this paper includes the three aspects: 1. we shall mainly apply some usual representations to some concrete physical problems, in which we show that the IWOP technique is a very powerful tool for solving these quantum problems difficult to deal with. 2. based on the significance of quantum mechanics representation, we propose a new method or equation to obtain continuous quantum mechanics representation. 3. we construct and introduce several new representations on q deformed case and discuss their some properties and applications.
The main content of this paper is arranged as follows:
In chapter 1, We briefly review the Technique of Integration Within normal ordered product and the Weyl quantization scheme, as well as the Technique of Integration Within Weyl ordering.
In chapter 2, by virtue of the bosonic coherent state representation and the Schwinger bosonic operator realization of angular momentum, we find the formula for the quantum Hamiltonian for SU(2) rotation, we further specify the angular velocity. Though the spin as a quantum observable has no classical correspondence, we may still mimic it as a rigid body rotation characterized by 3 Euler angles, and calculate its Pseudo-classical rotational partition function of spin one-half.
In chapter 3, for the bipartite Hamiltonian system with kinetic coupling, we derive time evolution equation of Wigner functions by virtue of the EPR entangled state representation and entangled Wigner operator, which just indicates that choosing a good representation indeed provides great convenience for us to deal with the dynamics problem.
In chapter 4, by virtue of the thermo entangled state representation, we exhibit a novel approach to deriving density operator for a Raman-coupled model with damping of the cavity mode. The normal ordering forms of density matrix elements can be obtained, and the corresponding Wigner functions are also derived.
In chapter 5, based on the significance in quantum phase space, by extending EPR entangled state representation to multipartite case. We construct n-mode Wigner operator in the common eigenstate of the multipartite centre-of-mass coordinate and two mass-weighted relative momenta, as well as its canonical conjugate state.
In chapter 6, using the completeness of Wigner operator and Weyl correspondence we construct a general equation for deriving pure state density operators, that is to say, we find a new method to derive continuous quantum mechanics representation. Several important examples in quantum mechanics and quantum optics are considered as the applications of this equation.
In chapter 7, we construct a new state called excited two-mode generalized squeezed vacuum states and find that it is just regarded as a generalized squeezed two-variable Hermite polynomial excitation on the vacuum state, as well as its normalization constant is just proved as a Jacobi polynomial. Their statistical properties are investigated such as squeezing properties and the form of the corresponding Wigner function.
In chapter 8, by using the contour integral representation ofδ-function and the technique of integration within an ordered product of operators, we point out that the q-deformed creation operator possesses the eigenkets. A set of new completeness and orthogonality relations composed of the kets and bras which are not mutually Hermitian conjugates are derived. Application of the completeness relation in constructing the generalized P-representation of density operator is demonstrated.
In chapter 9, the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator. Similarly, the new completeness and orthogonality relations composed of the bra and ket, which are not mutually Hermitian conjugates are obtained. Furthermore, the property of squeezing operator represented by the q-deformed entangled states is exhibited.
In chapter 10, similarly using the q-deformed entangled states are introduced by using q deformed theory and new normal ordering of the vacuum projection operator for q-deformed boson oscillator, we introduce the q deformed coordinate representation. Further, we utilize the eigenket and eigenbra for this representation to realize and study some important quantum gate operators for continuum variables.
引文
[1]. M. Planck, Verh. D. Phys. Ges. 2 (1900) 202.
[2]. A. Einstein, Ann. Der Physik, 17 (1905) 132.
[3]. N. Bohr, Phil. Mag. 26 (1913) pp. 1~25, 471~502, 857~875.
[4]. L. de Broglie, Compus Rendus, 177 (1923) 507; Nature, 112 (1923) 540.
[5]. W. Heisenberg, Zeit. Physik, 33 (1925) 879.
[6]. E. Schr?dinger, Ann. der Physik, 79 (1926) 36, 489; 80 (1926) 437; 81 (1926) 109.
[7]. M. Born, Zeit Physik, 38 (1926) 803.
[8]. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford: Clarendon Press, 1930.
[9]. H. Y. Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35, (1987)1831.
[10]. H. Y. Fan, J. R. Klauder, J. Phys. A 21, (1988) L725.
[11]. H. Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147.
[12]. W. H. Louisell, Quantum Statistical Properties of Radiation, New York: Wiley, 1973.
[13]. H. Weyl, Z. Phys. 46 (1927) 1.
[14]. E. Wigner, Phys. Rev. 40 (1932) 749.
[1]. R. J. Glauber, Phys. Rev. 130, (1963) 2529.
[2]. R. J. Glauber, Phys. Rev. 131, (1963) 2766.
[3]. J. R. Klauder, B. S. Skargerstam, Coherent States, World Scientific, Singapore, 1985.
[4].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.
[5]. See e.g. H. Goldstein, Classical Mechanics, 2nd ed. Addison-Wesley, New York 1980.
[6]. J. Schwinger, Quantum Theory of Angular Momentum, Academic, New York, 1995.
[7].钱学森,物理力学讲义,上海,上海交通大学出版社, 2007.
[1]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[2]. W. P. Schleich, Quantum Optics in Phase Space, Berlin, New York: Wiley-VCH, c2001.
[3]. D. F. Walls, G. Milburn, Quantum Optics, Berlin: Springer, 1994.
[4]. H. Y. Fan, H. R. Zaidi, Phys. Lett. A, 124 (1987) 303.
[5]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[6]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[1]. L. Schoendorff, H. Risken, Phys. Rev. A, 41 (1990) 5147.
[2]. J. B. Xua, et al., Eur. Phys. J. D, 10(2000) 295.
[3]. B. Deb, et al., Phys. Rev. A, 48 (1993) 1400.
[4]. Y. Takahashi, H. Umezawa, Collect. Phenom., 2 (1975) 55.
[5]. L. Y. Hu, H. Y. Fan, Int. J. Theor. Phys., 48 (2009) 3396.
[6]. H. Y. Fan, L. Y. Hu, Mod. Phys. Lett. B, 21 (2007) 183.
[7]. H. Y. Fan, A. Wünsche, J. Opt. B, Quantum Semiclass. Opt., 7 (2005) R88.
[8]. S. Chaturvedi, V. Srinvasan, Phys. Rev. A, 43 (1991) 4054.
[9]. S. Chaturvedi, V. Srinvasan, J. Mod. Opt. 38(1991) 777.
[10]. K. E. Hellwig, K. Kraus, Commun. Math. Phys.. 11 (1969) 214.
[1]. H. Y. Fan, et al., Ann. Phys., 321 (2006) 480.
[2]. H. Y. Fan, J. R. Klauder Phys. Rev. A, 49(1994) 704.
[3]. H. Y. Fan, B. Z. Chen, Phys. Rev A, 53 (1996) 2948.
[4]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[5]. H. Y. Fan, X. Ye, Phys. Rev A, 51 (1995) 3343.
[6]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[7]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[1].范洪义,量子学纠缠态表象及应用,上海:上海交通大学, 2001.
[2]. S. Helgason, The Randon Transform, Boston, Massachusetts: Birkhauser, 1980.
[3]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[4]. H. Y. Fan, Y. Fan, Phys. Rev. A, 54(1996) 958.
[5]. H. Y. Fan, B. Z. Chen, Phys. Rev. A, 53 (1996) 2948.
[6]. G. J. Milburn, S. L. Braunstein, Phys. Rev. A, 60 (1999) 937.
[7]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[8]. H. Y. Fan, X. Ye, Phys. Rev. A, 51 (1995) 3343.
[9]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[10]. H. Y. Fan, H. R. Zaidi, Phys. Lett. A, 124 (1987) 303.
[1] S. Helgason, The Randon Transform, Boston, Massachusetts: Birkhauser,1980
[2]. G. Schrade, et al., Phys. Rev. A, 48 (1993) 2398.
[3]. T. Gantsog, R. Tana′s, Phys. Lett. A, 152 (1991) 251.
[4]. A. V. Chizhov, B. K. Murzakhmetov, Phys. Lett. A, 176 (1993) 33.
[5]. M. Selvadoray, M. S. Kumar, Opt. Commun., 136 (1997) 125.
[6]. T. Hiroshima, Phys. Rev. A, 63 (2001) 022305.
[7]. M. Ban, J. Opt. B Quantum Semiclass. Opt., 1(1999) L9.
[8]. L. Y. Hu, H. Y. Fan, Commun. Theor. Phys., 50 (2008) 965.
[9]. M. Ban, J. Opt. B Quantum Semiclass. Opt., 7 (2005) L4.
[10]. G. S. Agarwal, K.Tara, Phys. Rev. A, 43(1991) 492.
[11]. L. Y. Hu, H. Y. Fan, Phys. Scr., 79 (2009) 035004.
[12]. Z. X. Zhang, H. Y. Fan, Phys. Lett. A, 165 (1992) 14.
[13]. X. G. Meng, et al., Phys. Lett. A, 361(2007) 183.
[14]. Meng X G, et al., Chin. Phys. B, 17 (2008) 1791.
[15]. H. C. Yuan, et al., Int. J. Theor. Phys., 48 (2009) 3596.
[16]. Q. Dai, H. Jing, Int. J. Theor. Phys., 47 (2008) 2716.
[17]. H. M.Li, et al., Int. J. Theor. Phys., 48 (2009) 2849.
[18]. H. Y. Fan, H. L. Lu, Phys. Lett. A, 334 (2005) 132.
[19]. H. Y. Fan, J. H. Chen, Commun. Theor. Phys., 40(2003) 589.
[20] R. Loudon, P. L. Knight, J. Mod. Opt., 34 (1987) 709.
[21]. V. Buzek, J. Mod. Opt., 37 (1990) 303
[22] W.Magnus, et al., Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. Berlin: Springer, 1966.
[23]. H. Y. Fan, J. Phys. A Math. Gen., 25(1992) 3443.
[24] A. Erdelyi, Higher Transcendental Functions, New York: McGraw-Hill. 1995.
[1]. A. S. Davydov, Quantum Mechanics 2nd ed, Oxford: Pergamon Press, 1976.
[2]. H. Y. Fan, et al., Commun. Theor. Phys., 3 (1984) 175.
[3]. L. C. Biedenharn, J. Phys. A, 22 (1989) L873.
[4]. A. J. Macfarlane, J. Phys. A, 22 (1989) 4581.
[5]. C. P. Sun, H. C. Fu, J. Phys. A, 22 (1989) L983.
[6]. S. Chaturvedi, et al., Phys. Rev. A, 43 (1991) 4555.
[7]. S. Chaturvedi, V. Srinivasan, Phys. Rev. A, 44(1991) 8020.
[8]. S. S. Avancini, et al., J. Phys. A, 29(1996) 5559.
[9]. C. Quesne, J. Phys. A, 35 (2002) 9213.
[10]. M. Schork, J. Phys. A, 36 (2003) 4651.
[11]. T.Mansour, et al., J. Phys. A: Math. Theor., 40(2007) 8393.
[12]. M. Schork, Phys. Lett. A, 355 (2006) 293.
[13]. T. Mansour, et al., Phys. Lett. A, 364 (2007) 214.
[14]. R. W. Gray, C. A. Nelson, J. Phys. A, 23 (1990) L945.
[15]. H. Y. Fan, M. Xiao, Phys. Lett. A, 219 (1996) 175.
[1]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[2]. H. Y. Fan, B. Z. Chen, Phys. Rev A, 53 (1996) 2948.
[3]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[4]. H. Y. Fan, X. Ye, Phys. Rev A, 51 (1995) 3343.
[5]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[6]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[1]. C. H. Bennett, Phys. Today, 48 (1995) 24.
[2]. P. W. Shor, Phys. Rev. A, 52 (1995) R2493.
[3]. A. Ekert, R.Jozsa, Rev. Mod. Phys., 68 (1996) 733.
[4]. Y. He, N. Q. Jiang, Opt. Commun., 283 (2010) 1558.
[5]. See e.g. J. Preskill, Quantum Information and Computation, California: Institute of Technology, Pasadena, 1998.
[6]. S.Parker, et al., Phys. Rev. A, 61 (2000) 032305.
[7]. S. L.Braunstein, Phys. Rev. Lett., 80 (1998) 4084.
[2]. A. Einstein, Ann. Der Physik, 17 (1905) 132.
[3]. N. Bohr, Phil. Mag. 26 (1913) pp. 1~25, 471~502, 857~875.
[4]. L. de Broglie, Compus Rendus, 177 (1923) 507; Nature, 112 (1923) 540.
[5]. W. Heisenberg, Zeit. Physik, 33 (1925) 879.
[6]. E. Schr?dinger, Ann. der Physik, 79 (1926) 36, 489; 80 (1926) 437; 81 (1926) 109.
[7]. M. Born, Zeit Physik, 38 (1926) 803.
[8]. P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford: Clarendon Press, 1930.
[9]. H. Y. Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35, (1987)1831.
[10]. H. Y. Fan, J. R. Klauder, J. Phys. A 21, (1988) L725.
[11]. H. Y. Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147.
[12]. W. H. Louisell, Quantum Statistical Properties of Radiation, New York: Wiley, 1973.
[13]. H. Weyl, Z. Phys. 46 (1927) 1.
[14]. E. Wigner, Phys. Rev. 40 (1932) 749.
[1]. R. J. Glauber, Phys. Rev. 130, (1963) 2529.
[2]. R. J. Glauber, Phys. Rev. 131, (1963) 2766.
[3]. J. R. Klauder, B. S. Skargerstam, Coherent States, World Scientific, Singapore, 1985.
[4].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.
[5]. See e.g. H. Goldstein, Classical Mechanics, 2nd ed. Addison-Wesley, New York 1980.
[6]. J. Schwinger, Quantum Theory of Angular Momentum, Academic, New York, 1995.
[7].钱学森,物理力学讲义,上海,上海交通大学出版社, 2007.
[1]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[2]. W. P. Schleich, Quantum Optics in Phase Space, Berlin, New York: Wiley-VCH, c2001.
[3]. D. F. Walls, G. Milburn, Quantum Optics, Berlin: Springer, 1994.
[4]. H. Y. Fan, H. R. Zaidi, Phys. Lett. A, 124 (1987) 303.
[5]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[6]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[1]. L. Schoendorff, H. Risken, Phys. Rev. A, 41 (1990) 5147.
[2]. J. B. Xua, et al., Eur. Phys. J. D, 10(2000) 295.
[3]. B. Deb, et al., Phys. Rev. A, 48 (1993) 1400.
[4]. Y. Takahashi, H. Umezawa, Collect. Phenom., 2 (1975) 55.
[5]. L. Y. Hu, H. Y. Fan, Int. J. Theor. Phys., 48 (2009) 3396.
[6]. H. Y. Fan, L. Y. Hu, Mod. Phys. Lett. B, 21 (2007) 183.
[7]. H. Y. Fan, A. Wünsche, J. Opt. B, Quantum Semiclass. Opt., 7 (2005) R88.
[8]. S. Chaturvedi, V. Srinvasan, Phys. Rev. A, 43 (1991) 4054.
[9]. S. Chaturvedi, V. Srinvasan, J. Mod. Opt. 38(1991) 777.
[10]. K. E. Hellwig, K. Kraus, Commun. Math. Phys.. 11 (1969) 214.
[1]. H. Y. Fan, et al., Ann. Phys., 321 (2006) 480.
[2]. H. Y. Fan, J. R. Klauder Phys. Rev. A, 49(1994) 704.
[3]. H. Y. Fan, B. Z. Chen, Phys. Rev A, 53 (1996) 2948.
[4]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[5]. H. Y. Fan, X. Ye, Phys. Rev A, 51 (1995) 3343.
[6]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[7]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[1].范洪义,量子学纠缠态表象及应用,上海:上海交通大学, 2001.
[2]. S. Helgason, The Randon Transform, Boston, Massachusetts: Birkhauser, 1980.
[3]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[4]. H. Y. Fan, Y. Fan, Phys. Rev. A, 54(1996) 958.
[5]. H. Y. Fan, B. Z. Chen, Phys. Rev. A, 53 (1996) 2948.
[6]. G. J. Milburn, S. L. Braunstein, Phys. Rev. A, 60 (1999) 937.
[7]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[8]. H. Y. Fan, X. Ye, Phys. Rev. A, 51 (1995) 3343.
[9]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[10]. H. Y. Fan, H. R. Zaidi, Phys. Lett. A, 124 (1987) 303.
[1] S. Helgason, The Randon Transform, Boston, Massachusetts: Birkhauser,1980
[2]. G. Schrade, et al., Phys. Rev. A, 48 (1993) 2398.
[3]. T. Gantsog, R. Tana′s, Phys. Lett. A, 152 (1991) 251.
[4]. A. V. Chizhov, B. K. Murzakhmetov, Phys. Lett. A, 176 (1993) 33.
[5]. M. Selvadoray, M. S. Kumar, Opt. Commun., 136 (1997) 125.
[6]. T. Hiroshima, Phys. Rev. A, 63 (2001) 022305.
[7]. M. Ban, J. Opt. B Quantum Semiclass. Opt., 1(1999) L9.
[8]. L. Y. Hu, H. Y. Fan, Commun. Theor. Phys., 50 (2008) 965.
[9]. M. Ban, J. Opt. B Quantum Semiclass. Opt., 7 (2005) L4.
[10]. G. S. Agarwal, K.Tara, Phys. Rev. A, 43(1991) 492.
[11]. L. Y. Hu, H. Y. Fan, Phys. Scr., 79 (2009) 035004.
[12]. Z. X. Zhang, H. Y. Fan, Phys. Lett. A, 165 (1992) 14.
[13]. X. G. Meng, et al., Phys. Lett. A, 361(2007) 183.
[14]. Meng X G, et al., Chin. Phys. B, 17 (2008) 1791.
[15]. H. C. Yuan, et al., Int. J. Theor. Phys., 48 (2009) 3596.
[16]. Q. Dai, H. Jing, Int. J. Theor. Phys., 47 (2008) 2716.
[17]. H. M.Li, et al., Int. J. Theor. Phys., 48 (2009) 2849.
[18]. H. Y. Fan, H. L. Lu, Phys. Lett. A, 334 (2005) 132.
[19]. H. Y. Fan, J. H. Chen, Commun. Theor. Phys., 40(2003) 589.
[20] R. Loudon, P. L. Knight, J. Mod. Opt., 34 (1987) 709.
[21]. V. Buzek, J. Mod. Opt., 37 (1990) 303
[22] W.Magnus, et al., Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. Berlin: Springer, 1966.
[23]. H. Y. Fan, J. Phys. A Math. Gen., 25(1992) 3443.
[24] A. Erdelyi, Higher Transcendental Functions, New York: McGraw-Hill. 1995.
[1]. A. S. Davydov, Quantum Mechanics 2nd ed, Oxford: Pergamon Press, 1976.
[2]. H. Y. Fan, et al., Commun. Theor. Phys., 3 (1984) 175.
[3]. L. C. Biedenharn, J. Phys. A, 22 (1989) L873.
[4]. A. J. Macfarlane, J. Phys. A, 22 (1989) 4581.
[5]. C. P. Sun, H. C. Fu, J. Phys. A, 22 (1989) L983.
[6]. S. Chaturvedi, et al., Phys. Rev. A, 43 (1991) 4555.
[7]. S. Chaturvedi, V. Srinivasan, Phys. Rev. A, 44(1991) 8020.
[8]. S. S. Avancini, et al., J. Phys. A, 29(1996) 5559.
[9]. C. Quesne, J. Phys. A, 35 (2002) 9213.
[10]. M. Schork, J. Phys. A, 36 (2003) 4651.
[11]. T.Mansour, et al., J. Phys. A: Math. Theor., 40(2007) 8393.
[12]. M. Schork, Phys. Lett. A, 355 (2006) 293.
[13]. T. Mansour, et al., Phys. Lett. A, 364 (2007) 214.
[14]. R. W. Gray, C. A. Nelson, J. Phys. A, 23 (1990) L945.
[15]. H. Y. Fan, M. Xiao, Phys. Lett. A, 219 (1996) 175.
[1]. H. Y. Fan, J. R. Klauder, Phys. Rev. A, 49 (1994) 704.
[2]. H. Y. Fan, B. Z. Chen, Phys. Rev A, 53 (1996) 2948.
[3]. H. Y. Fan, Phys. Lett. A, 286 (2001) 81.
[4]. H. Y. Fan, X. Ye, Phys. Rev A, 51 (1995) 3343.
[5]. H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[6]. H. Y. Fan, Phys. Lett. A, 294 (2002) 253.
[1]. C. H. Bennett, Phys. Today, 48 (1995) 24.
[2]. P. W. Shor, Phys. Rev. A, 52 (1995) R2493.
[3]. A. Ekert, R.Jozsa, Rev. Mod. Phys., 68 (1996) 733.
[4]. Y. He, N. Q. Jiang, Opt. Commun., 283 (2010) 1558.
[5]. See e.g. J. Preskill, Quantum Information and Computation, California: Institute of Technology, Pasadena, 1998.
[6]. S.Parker, et al., Phys. Rev. A, 61 (2000) 032305.
[7]. S. L.Braunstein, Phys. Rev. Lett., 80 (1998) 4084.