非对易空间物理系统若干问题的研究
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摘要
非对易空间中的量子力学理论与通常量子力学有许多的不同之处,因此处理非对易空间的量子力学问题也需要一些新的方法和技巧.本文主要从两方面来探讨非对易空间的量子力学问题：第一部分是讨论如何将变形量子化方法推广到非对易空间.我们分别考察了只有空间部分非对易以及空间和动量都非对易的相空间中的变形量子化的表达形式,对于一般的哈密顿系统和非哈密顿系统,我们都得到了相应的*-本征方程和Wigner函数的表达式,并且讨论了不同方法的等价性和区别.然后我们将变形量子化方法应用到非对易空间的具体物理模型上,首先我们考虑了非对易平面上的耦合谐振子,通过计算得到了相应的能谱和Wigner函数,并且得到了关于一类特殊哈密顿量的变形量子化的公式.然后我们还考察了非对易空间的阻尼系统和Bateman系统这些典型的非哈密顿系统,也求解得到了系统的能谱和广义的Wigner函数.第二部分我们主要介绍了非对易空间中的几种量子表象的构建以及表象的性质和应用.首先我们基于广义非对易相空间的坐标的对易关系构造了一类变形产生湮灭算符,这类算符之间满足变形玻色代数关系,基于这些算符和变形玻色代数,我们构造了非对易的相干态和压缩态,通过计算发现这些态是归一的但不是正交的,而且是超完备的,因此可以构成相干态表象和压缩态表象.我们进一步还得到了这两组态之间的变换关系.然后我们分析了单模和双模正交算符在这些表象上的海森堡不确定关系,通过分析我们发现海森堡不确定关系可以给出空间非对易参数和普朗克常数之间的一个限制关系.另外我们还构造了非对易相空间的纠缠态表象,并且计算了非对易空间坐标算符在纠缠态表象上的矩阵元,我们计算了纠缠态的Wigner算符的表达式.最后我们通过二维非对易平面上的耦合谐振子演示了我们所构建的纠缠态表象的应用.
The theory of quantum mechanics in noncommutative space has some differences with that of normal quantum mechanics, so there are some new methods and technolo-gies to deal with the quantum mechanics problems in noncommutative space. In this thesis we discuss the quantum mechanics problems in noncommutative space in two ways. First we discuss how to extend the deformation quantization methods of the normal space to that of the noncommutative space. Respectively, we investigate the case of only the spatial coordinates is noncommutative and also that of both spatial and momentum coordinates are noncommutative. We derive the expressions of the corre-sponding *-genvalue equations and Wigner functions of both the Hamiltonian systems and non-Hamiltonian systems, and discuss the equivalence and differences between dif-ferent methods. Then we apply the deformation quantization method to some physical models in noncommutative space. First we investigate the coupled harmonic oscillators on noncommutative plane, and calculate the corresponding energy spectra and Wigner function, and derive a formula for a class of Hamiltonian with special form in defor-mation quantization. We also investigate some typical non-Hamiltonian systems such as the damped systems and Bateman system in noncommutative space, and obtain the energy spectra and generalized Wigner function of these systems. In the second part of the thesis we discuss the construction of some quantum representations in noncom-mutative space and the properties and applications of these representations. Firstly, based on the commutation relations of the coordinates of generalized noncommutative phase space we construct a type of deformed creation and annihilation operators, and these operators satisfy the deformed boson algebraic relations. Based on these operators and the deformed boson algebra, we construct the noncommutative coherent states and squeezed states, and after some calculations we find that these states are normalized but not orthogonal to each other, and besides, they are over-complete, so we can construct the coherent representation and squeezed representation. Furthermore we derive the transformation relations between these representations. We investigate the Heisenberg uncertainty relations of the one-and two-mode quadrature operators, and we find that the Heisenberg uncertainty relation can give a restricted relation between the parame-ters of the noncommutativity of the space and the Planck constant. We also construct the entangled state representations for the noncommutative phase space, and calculate the matrix elements of the coordinate operators of noncommutative space on entangled state representations. We also obtain the expression of the Wigner operator on entan-gled state. Finally we show some applications of the entangled state representation by the coupled harmonic oscillators on 2D noncommutative plane.
The theory of quantum mechanics in noncommutative space has some differences with that of normal quantum mechanics, so there are some new methods and technolo-gies to deal with the quantum mechanics problems in noncommutative space. In this thesis we discuss the quantum mechanics problems in noncommutative space in two ways. First we discuss how to extend the deformation quantization methods of the normal space to that of the noncommutative space. Respectively, we investigate the case of only the spatial coordinates is noncommutative and also that of both spatial and momentum coordinates are noncommutative. We derive the expressions of the corre-sponding *-genvalue equations and Wigner functions of both the Hamiltonian systems and non-Hamiltonian systems, and discuss the equivalence and differences between dif-ferent methods. Then we apply the deformation quantization method to some physical models in noncommutative space. First we investigate the coupled harmonic oscillators on noncommutative plane, and calculate the corresponding energy spectra and Wigner function, and derive a formula for a class of Hamiltonian with special form in defor-mation quantization. We also investigate some typical non-Hamiltonian systems such as the damped systems and Bateman system in noncommutative space, and obtain the energy spectra and generalized Wigner function of these systems. In the second part of the thesis we discuss the construction of some quantum representations in noncom-mutative space and the properties and applications of these representations. Firstly, based on the commutation relations of the coordinates of generalized noncommutative phase space we construct a type of deformed creation and annihilation operators, and these operators satisfy the deformed boson algebraic relations. Based on these operators and the deformed boson algebra, we construct the noncommutative coherent states and squeezed states, and after some calculations we find that these states are normalized but not orthogonal to each other, and besides, they are over-complete, so we can construct the coherent representation and squeezed representation. Furthermore we derive the transformation relations between these representations. We investigate the Heisenberg uncertainty relations of the one-and two-mode quadrature operators, and we find that the Heisenberg uncertainty relation can give a restricted relation between the parame-ters of the noncommutativity of the space and the Planck constant. We also construct the entangled state representations for the noncommutative phase space, and calculate the matrix elements of the coordinate operators of noncommutative space on entangled state representations. We also obtain the expression of the Wigner operator on entan-gled state. Finally we show some applications of the entangled state representation by the coupled harmonic oscillators on 2D noncommutative plane.
引文
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