《量子力学》自学辅导之七—表象理论（续）

《量子力学》自学辅导之七──表象理论（续）曾心愉，宋宇辰，斐文杰（复旦大学物理系，复旦大学理论物理骨干教师班，上海２００４３３）４量子力学公式的矩阵表述（分立谱为例）实质上是把ｘ表象中的达式变换到Ｑ表象中，故以Ｑ表象基矢ｕ（ｘ）为基础，以态矢及算符（...     

《量子力学》自学辅导之八：狄拉克符号和占有数表象（续）

《量子力学》自学辅导之八──狄拉克符号和占有数表象（续）曾心愉，宋宇辰，裴文杰（复旦大学物理系，复旦大学理论物理骨干教师班上海２００４３３）２关于一维线性谐振子的讨论２．１坐标表象一维线性谐振子Ｈ算符及其本征函数在坐标表象中为式中的本征值为由厄米多项...     

研究宇称算符的一种新方法

在量子力学中,宇称算符P^的3种表象积分分别表示为∫d2zπ|z〉〈-z|、∫dq|q〉〈-q|和∫dp|p〉〈-p|.借助Dirac符号法和量子力学表象的完备性,找到了一种非常新颖的方法,给出了宇称算符的积分显式.     

X表象中态和算符两种不同表示的一致性

本文从量子力学教学角度认为，有必要讲清楚在Ｘ表象中态和算符两种不同表示是完全一致的，以免学生的误解。     

也谈量子力学的基础

本文通过在不同边界条件下动量算符的厄密性和自伴性的区分,给出了对量子力学的一维无限深方势阱中基态粒子的动量分布问题的严格分析,论证了量子力学基本假设的严谨性和自洽性.     

研究热相干态的Wigner函数

利用相干态表象下的Wigner算符和有序算符内的积分(IWOP)技术,首先得到了热相干态(量子纯态)的Wigner函数;同时借助相干态表象和算符的正规乘积形式给出了相应混合态的Wigner函数.结果表明,热相干态与相应混合态的Wigner甬数是相一致的,支持了热场动力学(TFD)理论.且采用相干态表象下的Wigner算符、IWOP技术和算符的正规乘积形式来研究量子态的Wigner函数非常简捷方便.研究结果加深了人们对量子统计中相空间技术和热场动力学(TFD)理论的认识,且对于其它量子纯态与相应混合态相空间分布函数一致性的研究具有很好的理论指导意义.     

从二维正态分布密度函数的角度研究量子力学基本表象

利用二维正态分布密度函数和有序算符内的积分技术,简捷地得到了坐标本征态、动量本征态、坐标-动量中介表象和相干态在Fock表象中的表达式.     

利用IWOP技术构建量子力学新表象

基于对坐标表象、动量表象及相干态表象完备性关系式的正规乘积内纯高斯积分形式的分析,阐述了利用有序算符内的积分技术构建量子力学新表象的思路和方法,并具体以单模坐标-动量中介表象、双模纠缠态表象和双模相干纠缠态表象的构建为例进行了论述.     

曲面上的动量和动能算符

对约束在曲面上粒子运动的描述可以在内部坐标即曲面局部坐标下进行,也可以在外部坐标即在笛卡尔坐标下进行.在量子力学中,动量和动能算符的表示在这两种描述中各有不同,前者的动量算符仅包含内禀几何量,后者的动量算符包含了曲面的平均曲率.考虑到算符次序问题,动能算符对动量算符的依赖关系也不同,前者的依赖关系仅发现存在一种,后者的依赖关系已经发现有两种. 关键词： 量子力学 微分几何     

对突破工科近代物理教学一个瓶颈问题的探讨

在工科近代物理教学中,量子力学部分所占比例非常少,而量子力学基础教学中的一个瓶颈在于算符和算符本征值与力学量概率分布的关系.介绍了为突破这一瓶颈,在教材建设和教学实践中所做的改进.     

Representation of the coherent state for a beam splitter operator and its applications

A beam splitter operator is a very important linear device in the field of quantum optics and quantum information. It can not only be used to prepare complete representations of quantum mechanics, entangled state representation, but it can also be used to simulate the dissipative environment of quantum systems. In this paper, by combining the transform relation of the beam splitter operator and the technique of integration within the product of the operator, we present the coherent state representation of the operator and the corresponding normal ordering form. Based on this, we consider the applications of the coherent state representation of the beam splitter operator, such as deriving some operator identities and entangled state representation preparation with continuous-discrete variables. Furthermore, we extend our investigation to two single and two-mode cascaded beam splitter operators, giving the corresponding coherent state representation and its normal ordering form. In addition, the application of a beam splitter to prepare entangled states in quantum teleportation is further investigated, and the fidelity is discussed. The above results provide good theoretical value in the fields of quantum optics and quantum information.     

用纠缠态表象推导广义Ehrenfest定理

利用两粒子相对坐标和总动量的共同本征态|η〉构成的连续变量纠缠态表象,推导了转动系统的Ehrenfest定理,从而说明了在满足某些近似的情况下,量子力学不但能够与经典物理完美自洽,而且利用量子力学的规律可以导出相应的经典物理规律.     

基于坐标表象的脊波变换研究

在小波变换量子力学机制的启发下, 通过采用Fock空间里双模坐标本征态改写经典Ridgelet变换, 定义了量子光学态的Ridgelet变换. 然后利用IWOP技术给出不对称积分算符的显式, 并推导出了两个有用的双模算符正规乘积公式. 在此基础上, 通过选取双模“墨西哥帽”母小波函数, 分析了相干态、特殊压缩相干态、中介纠缠态表象的Ridgelet变换. 关键词： IWOP技术 Ridgelet变换 相干态     

Teleportations of Mixed States and Multipartite Quantum States

In this paper, we propose a protocol to deterministically teleport an unknown mixed state of qubit by utilizing a maximally bipartite entangled state of qubits as quantum channel. Ira non-maximally entangled bipartite pure state is employed as quantum channel, the unknown mixed quantum state of qubit can be teleported with 1 -√ 1- C^2 probability, where C is the concurrence of the quantum channel. The protocol can also be generalized to teleport a mixed state of qudit or a multipartite mixed state. More important purpose is that, on the basis of the protocol, the teleportation of an arbitrary multipartite （pure or mixed） quantum state can be decomposed into the teleportation of each subsystem by employing separate entangled states as quantum channels. In the case of deterministic teleportation, Bob only needs to perform unitary transformations on his single particles in order to recover the initial teleported multipartite quantum state.     

Quantum Mechanics in Curved Configuration Space

Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasiclassical, and path integration formalisms are considered for quantization of geodesic motion on the Riemannian configuration spaces. A unique rule of ordering of operators in the canonical formalism and a unique definition of the path integral are established and, thus, a part of ambiguities in the quantum counterpart of geodesic motion is removed. A geometric interpretation is proposed for noninvariance of the quantum mechanics on coordinate transformations. An approach alternative to the quantization of geodesic motion is surveyed, which starts with the quantum theory of a neutral scalar field. Consequences of this alternative approach and the three formalisms of quantization are compared. In particular, the field theoretical approach generates a deformation of the canonical commutation relations between operators of coordinates and momenta of a particle. A cosmological consequence of the deformation is presented in short.     

Entropy of Quantum States

Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.     

一个描述金融投资项目演化的量子力学状态方程

本文提出了一个描述金融投资项目演化的量子力学状态方程,该方程的参数比较好地模拟了金融市场的基本要素,包括投资(输入)、资产损耗(缩水)、资产增益、和收益(输出)等,方程中的量子力学算符也能够反映该项目的动力学过程与特征,所以能代表一类金融投资项目(其状态用Dirac符号表示)在市场中的演化模式.通过引入纠缠态表象得出了该方程的量子解(算符之和形式),该量子解给出了初态与终态的联系,也就给出了投资项目的动态过程.作为例子,求出单一投资项目在金融市场中的演化,不但得出了符合市场演化趋势的解,而且提出了二项-负二项纠缠态.在推导过程中,充分利用了Dirac符号和有序算符内的积分技术(IWOP).     

Quantum deleting and cloning in a pseudo-unitary system

In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.     

Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure

In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.