涉及双变数Hermite多项式的二项式定理及其在量子光学中的应用

我们用量子力学算符Hermite多项式方法发现了涉及双变数Hermite多项式的二项式的两个定理,它们在计算若干量子光场的物理性质时有实质性的应用.该方法不但具有简捷的优点,而且能导出很多新的算符恒等式,成为发展数学物理理论的一个重要分支.若干常用的特殊函数的宗量也由普通数变为算符.     

用压缩态理论导出偶-奇数阶厄密多项式的无穷和

利用有序算符内积分技术,用压缩态理论导出偶数阶厄密多项式H2n(x)和奇数阶H2n+1(x)的无穷和。并提出用量子力学算符Hermite多项式方法计算奇-偶相干态的波函数。我们用的新途径具有物理意义鲜明的特点。     

由新二项式定理导出在Schwinger玻色实现下的原子相干态的波函数

用相干态表象和有序算符内的积分技术,我们导出了一个关系到含有双变量厄米多项式的二项式定理,用它可以导出在Schwinger玻色实现下的原子相干态在纠缠态表象中的波函数,并且得到一个新的算符恒等式。     

构造阶梯算符的定理及其应用

给出一个基本定理，提供了构造二阶线性常微分算符相应的阶梯算符的普遍适用的方法.利用这个定理得到量子力学中常见的本征问题的解.     

奇偶二项式光场态的小波变换

利用有序算符内积分方法,小波变换可以表示为被转换态矢|f〉在压缩平移算符U(μ,s)作用下向母小波态矢〈ψ|转换的矩阵元〈ψ|U(μ,s)|f〉. 在此基础上,计算了奇、偶二项式态的小波变换,得到小波变换谱. 结果表明,小波变换谱可以起到识别这些量子力学态的作用,具有直观易辨的优点. 关键词： 小波变换 奇偶二项式态 有序算符内积分     

量子力学坐标-动量算符幂次排序互换的简捷公式与新推导方法

量子力学坐标-动量算符幂次排序的相互转换是一个基本的量子力学课题, 本文提出了一个十分简捷有效的方法处理此问题, 即利用双变量厄米特多项式的母函数性质及有序算符记号内的算符特点, 给出一系列关于坐标-动量算符幂次排序的恒等式, 它们具有广泛的应用.     

量子光学理论中双模厄米多项式的新应用

强调双模厄米多项式在量子光学理论中的地位,认为它是研究连续变量纠缠态和压缩态的必要函数,具有明确的物理意义。利用双模厄米多项式,结合有序算符内的积分技术,给出了若干新的算符恒等式和互逆的积分变换公式,证明了压缩双模粒子数态恰好是双变量厄米多项式激发压缩真空态。     

基于算符正规乘积的若干厄米多项式关系式证明

主要利用量子力学的方法证明了厄米多项式的递推关系.另外,根据厄米多项式的表达式,得出了两个与厄米多项式有关的积分公式.     

一种新的Glauber公式的证明方法

Glauber公式在量子力学中有着重要的应用,教科书中证明它的方法可以分为两种。一种是构造一个含有参变量的算符指数的函数,然后对参变量微分并利用Baker-Hausdorff公式,最后得到微分方程并积分求解得证。此方法存在一点瑕疵,因为在积分的过程中需要将算符放在分母,然而算符所对应的矩阵是没有除法的。另一种是先证明考虑算符对易性质的两个算符相加的二项式定理与不考虑算符对易性质的两个算符相加的二项式定理之间的关系,然后直接将Glauber公式中两个算符和的指数做展开并利用上述关系直接证明。此方法的证明过程略显复杂。本文通过构造、利用Baker-Hausdorff公式和算符的指数展开公式,给出了一种新的Glauber公式的证明方法。     

拉盖尔多项式算符激发相干态的非经典性质

将拉盖尔多项式算符作用在相干态上,构造了拉盖尔多项式算符激发相干态.利用有序算符积分技术,导出了它的归一化系数以及〈a^la^+m〉的计算表达式.采用数值计算方法,讨论了相干态相位角和平均光子数对它的非经典性质的影响.研究结果表明:一阶拉盖尔多项式算符激发相干态呈现出压缩效应、反聚束效应、亚泊松分布和Wigner函数负性等量子特性,并且相干态的相位角对它的量子特性有重要影响;另一方面,随相干态平均光子数增大,它的反聚束效应和亚泊松分布性质逐渐减弱,压缩效应和Wigner函数的负性却先增强,而后又逐渐减弱.     

Quantum mechanical operator realization of the Stirling numbers theory studied by virtue of the operator Hermite polynomials method

Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.     

Identities involving Laguerre polynomials derived from umbral calculus

In this paper, we investigate some identities for Laguerre polynomials involving Bernoulli and Euler polynomials derived from umbral calculus.     

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.     

The Calogero-Sutherland Model and Generalized Classical Polynomials

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr?dinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. Received: 16 August 1996 / Accepted: 21 January 1997     

Deriving new operator identities by alternately using normally,antinormally,and Weyl ordered integration technique

Dirac's ket-bra formalism is the language of quantum mechanics.We have reviewed how to apply Newton-Leibniz integration rules to Dirac's ket-bra projectors in previous work.In this work,by alternately using the technique of integration within normal,antinormal,and Weyl ordering of operators we not only derive some new operator ordering identities,but also deduce some new integration formulas regarding Laguerre and Hermite polynomials.This may open a new route of directly deriving some complicated mathematical integration formulas by virtue of the quantum mechanical operator ordering technique,without really performing the integrations in the ordinary way.     

Multi-variable special polynomials using an operator ordering method

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.     

Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉_q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e.,

$ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $

and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉_q is stronger for odd q than for even q.

     

Tomographic probability and entropic inequalities for special functions

We discussed some aspects of the tomographic-probability representation of quantum mechanics. Using known generic inequalities for Shannon and relative entropies, we obtain some new inequalities for special functions such as Laguerre, Legendre, and two-variable Hermite polynomials.     

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.     

基于纠缠态表象和自旋相干态的新复变函数空间

本文利用有序算符内的积分(IWOP)技术,构造了一个基于单变量厄米多项式H2j(ξ*+τξ/2√τ)的新复变函数空间,该空间与纠缠态表象及施温格玻色环境下的自旋相干态有关。推导出了包含二元厄米多项式的二项式定理,有助于构造新的复变函数空间。同时还提出了一种新的基于H2j(ξ*+τξ/2√τ)的积分变换及其逆变换,这对于推导某些算符恒等式是很有用的。