利用特殊函数和类比法有序化排列正负指数幂算符

研究算符函数的有序化排列是一项重要的数理任务.本文利用特殊函数和正规乘积排序与反正规乘积排序间的互换法则法导出了幂算符(αα^+)^±n和(α^+α)^±n的正规与反正规乘积排序.进一步,利用类比法得到了算符(XP)^±n和(PX)^±n的坐标-动量排序与动量-坐标排序式.最后,对新得到的这些算符结果的应用进行一些讨论.     

曲线坐标下的动量算符和动能算符

本文从量子力学算符必须满足厄密性这一基本原理出发，导出了曲线坐标下的动量算符的普适形式　，得到了球坐标及柱坐标等曲线坐标中的动量算符的正确形式，又根据量子力学中的动能算符应是微分算符这一特点，导出了曲线坐标下动能算符的普适形式．     

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     

关于量子力学中正则对易关系与对应原理自洽性的讨论

正则对易关系与对应原理是从经典力学建立量子力学的数学体系所必须遵守的两个基本假定.正则对易关系决定了坐标与动量算符的具体形式.例如,从直角坐标与动量算符的对易关系即可得出在坐标表象中至于一般力学量的算符形式,则由对应原理给出.例如,设某经典力学量为F(x,px)(总可表为直角坐标与动量的函数),如将x与px分别换成x与px(当然要按一定的规则厄米化),就得到量子力学中相应的力学量算符F(x,px).特别是,由于经典力学中的广义坐标(一般为曲线坐标)及其共轭动量均可表为直角坐标与动量的函数,那么量子力学中,相应的广义坐标及其共轭动量…     

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.     

广义坐标中动量算符的自伴性

本文旨在藉助希耳伯特空间算符理论方面的知识，澄清在量子力学书籍上通常出现的一些概念．在量子力学书籍里常常对有界和无界算符之间的基本区别不予理会．一个无界算符是对称（厄密）的条件并不足以使它成为自伴的，这一点经常被忽视．遗憾的是，量子力学里差不多所有的算符都是无界的．时常看到这样的叙述：对于任意线性算符Ａ，我们可以写出厄密算符ＨＡ＝（Ａ＋Ａ＋） ／２，其中的厄密性被设想为具有自伴性的意义．沿着这条思路，在广义坐标里采取那种表述的动量算符的自伴性是有问题的．本文特别研究了运用球极坐标的重新描述，并且引出了与这些坐标相共轭的动量失去了自伴性的结论．     

势模型中排序问题的初步研究

由动量空间计算得到的夸克和反夸克之间的散射振幅推导坐标空间等效势时,总要出现算符排序问题,这是一个量子力学中长期存在而又未能得到很好解决的重要问题.以重夸克偶素中常用的Cornell势为例,重新进行了研究,给出了一些初步的结果.指出即使考虑了厄米性的要求,各种排序方案仍然会导致有很大差别的数值结果,会严重影响谱的拟合精度.对这些结果的意义及其相关问题作了一些讨论.     

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.     

广义坐标正则动量的量子化

本文由直角坐标系坐标算符和正则动量算符的明显形式及它们的对易关系出发，导出了广义坐标下正则动量算符的一般公式，并由此求出几种常用正交曲线坐标下广义正则动量算符的具体形式．     

谈谈量子力学里的动量算符

讨论了在量子力学里建立动量算符的一些原则性问题。     

New operator-ordering identities and associative integration formulas of two-variable Hermite polynomials for constructing non-Gaussian states

For directly normalizing the photon non-Gaussian states(e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product(IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.     

Simple approach to deriving operator identities and mathematical formulas regarding to two-variable Hermite polynomials

By virtue of operator ordering technique and the generating function of polynomials, we provide a simple and neat approach to studying operator identities and mathematical formulas regarding to two-variable Hermite polynomials, which differs from the existing mathematical ways. We not only derive some new integration formulas and summation relations about two-variable Hermite polynomial, but also draw a conclusion that two-variable Hermite polynomial excitation of two-mode squeezed vacuum state is a squeezed two-mode number state. This may open a new route of developing mathematics by virtue of the quantum mechanical representations and operator ordering technique.     

New Bosonic Operator Ordering Identities Gained by the Entangled State Representation and Two-Variable Hermite Polynomials

Based on the technique of integration within an ordered product of operators, we derive new bosonicoperators‘ ordering identities by using entangled state representation and the properties of two-variable Hermite poly-nomials H and vice versa. In doing so, some concise normally (antinormally) ordering operator identities, such asa man =:Hm,n(a ,a):, ana m = (-i)m n:Hm,n(ia ,ia): are obtained.     

New operator identities with regard to the two-variable Hermite polynomial by virtue of entangled state representation

By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial （TVHP）. By them and the technique of integration within an ordered product （IWOP） of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.     

A new kind of special function and its application

By virtue of the operator Hermite polynomial method and the technique of integration within the ordered product of operators we derive a new kind of special function, which is closely related to one- and two-variable Hermite polynomials.Its application in deriving the normalization for some quantum optical states is presented.     

New Integration Transformation Connecting Coherent State and Entangled State

With the help of technique of integration within an ordered product (IWOP) of operators we find new integration transformation connecting the coherent state and the biparticle entangled state. We also point out that under this kind of integration transformation the direct product of two single-variable Hermite polynomials behaves quite different from the two-variable Hermite polynomials, in this way we show that the latter is intrinsic to the phase space of quantum entanglement. As a byproduct, some operator identities for theoretical quantum optics can also be neatly expressed in terms of the two-variable Hermite polynomials.     

Weyl Ordering Expansion of Power Product of Coordinate and Momentum Operators

By virtue of the technique of integration within Weyl ordered of operatorswe derive the formula of Weyl ordering expansion of power product ofcoordinate and momentum operators (2^1/2Q)^m (2^1/2iP )^r=::H_m,r(2^1/2Q,2^1/2iP ) ::, the introduction of two-variable Hermite polynomial H_m,r brings much convenience to the study of Weyl correspondence.     

Solution to Operator Fredholm Equation in Bipartite Entangled State Representation and Its Applications

Based on the bipartite entangled state representation and using the technique of integration within an ordered product （IWOP） of operators we construct the corresponding operator Fredholm equations and then derive their solutions. As its application we deduce some new bosonic operator identities and new relations about the two-variable Hermite polynomials.     

Normal ordering and antinormal ordering of the operator （fQ＋gP）n and some of their applications

In this paper by virtue of the technique of integration within an ordered product (IWOP) of operators and the intermediate coordinate-momentum representation in quantum optics, we derive the normal ordering and antinormal ordering products of the operator (f Q + gP )n when n is an arbitrary integer. These products are very useful in calculating their matrix elements and expectation values and obtaining some useful mathematical formulae. Finally, the applications of some new identities are given.