量子光学中的算符Hermite多项式理论及其应用

发布时间：2018-07-24 19:28

【摘要】：近代物理理论的发展毋庸置疑地需要先进的数学,理论物理学家应该根据物理需要自己发明相应的数学,展示自己独到的思维方式。Hermite多项式作为一种特殊函数,在量子力学和量子光学中有广泛的应用。例如,量子谐振子的本征态的波函数就是由Hermite多项式来描述。特殊函数之谓是它们有其特殊的递推规律、并容易被记忆。Hermite多项式H_m(x)的集合组成了完备正交的函数空间,在数学物理领域也有一定的地位。本论文提出把Hermite多项式的x宗量(x是实数)以量子力学的算符(坐标算符X)来代替,称H_m(X)为算符Hermite多项式,由于量子力学中的算符一般是不对易的,所以算符特殊函数本身的编序问题是一个全新的数理问题。我们结合有序算符内的积分技术,较系统地研究它的各种性质,给出了H_m(X)在算符排序方面的新特点,在此基础上可以导出若干新的算符恒等式,它们在构建量子光学态矢量方面有重要的应用。当进一步将有关H_m(X)的排序规则过渡到经典情形,我们又得到有关Hermite多项式H_m(x)的广义二项式定理和若干新母函数公式。算符Hermite多项式理论包括:1.借助算符特殊函数的新公式可找出沟通各种特殊函数之间的新关系。2.利用算符特殊函数各种编序形式及量子表象完备性可以导出许多有用的新积分公式(而并不需要真正地做积分)。3.可用算符特殊函数恒等式丰富量子论的表象论与变换论。4.更深刻地建立经典函数的量子对应,有助于量子相空间理论的发展。5.发现若干特殊函数的新的幂级数展开及倒易关系。6.方便而直接地计算各种物理量,如矩函数、累积函数等。本文还将研究在量子纠缠中需要的双变数Hermite多项式H_m,n(x,y),将算符Hermite多项式理论推广到算符双变数Hermite多项式,研究其性质和应用。本论文从物理概念与物理要求来架构算符Hermite多项式理论并深入讨论,注意在数学公式中把握物理,体现数学和物理的相辅相成。要较顺利地探讨算符Hermite多项式理论,在第一章先对我国学者范洪义发明的有序算符内的积分技术做一介绍。在第二、三章引入算符Hermite多项式(单变数)的理论,其核心是算符Hermite多项式的正规乘积和反正规乘积展开。第四章介绍如何用算符Hermite多项式理论导出Laguerre多项式,而以往都认为它们是分别定义的。第五章陈述算符Hermite多项式的正规乘积Hn(X)= 2n::的物理应用。第六章介绍用算符Hermite多项式方法推导含Hn(q的新二项式定理。第七章用算符Hermite多项式方法推导偶-和奇-厄密多项式的母函数公式。从第八章到第十一章介绍引入双变数Hermitee多项式的途径和建立双变数算符Hermite多项式理论,并导出新的母函数公式等。在第十二章,鉴于Fock空间的完备性以往只能用纯态展开,探讨完备性能否用混合态展开?结果发现Fock空间可以用混合态(二项式态或负二项式态)来划分。总而言之,算符Hermite多项式理论和Fock空间的新划分对于进一步了解了 Fock空间的结构,丰富Fock空间的内涵起到了积极的作用。
[Abstract]:The development of modern physics theory needn't be doubted to need advanced mathematics. Theoretical physicists should invent corresponding mathematics according to their own physical needs, display their unique way of thinking.Hermite polynomials as a special function, and have extensive applications in quantum mechanics and quantum optics. For example, the eigenstates of quantum harmonic oscillator. The wave function is described by the Hermite polynomials. The predicate of the special functions is that they have their special recursive law, and they are easy to be memorized by the set of.Hermite polynomials H_m (x), which constitute a complete orthogonal function space, and have a certain position in the field of mathematics and physics. This paper proposes the X sect of the Hermite polynomial (x is real) as a quantum force. The operator (coordinate operator X) is a substitute for the H_m (X) as the operator Hermite polynomial, because the operator in the quantum mechanics is generally not easy, so the ordering problem of the operator's special function itself is a new mathematical problem. We combine the integral techniques in the ordered operator to systematically study its various properties and give H_m (X) in the system. On the basis of the new characteristics of the operator sorting, some new operator identities can be derived. They have important applications in the construction of quantum optical state vectors. When the H_m (X) ordering rules are transitioning to the classical case, we get the generalized binomial theorem and some new parent functions of the Hermite multinomial H_m (x). Formula. The operator Hermite polynomial theory includes: 1. the new relation between various special functions can be found by the new formula of the special function of the operator..2. can derive a number of useful new integral formulas (but without real integral), the.3. can be used as a special operator, using the order forms of the special functions of the operator and the completeness of the quantum representation. Function identities rich quantum theory and transformation theory.4. more deeply establish the quantum correspondence of classical functions, help the development of quantum phase space theory,.5. find new power series expansion of some special functions and invert.6. convenient and direct calculation of various physical quantities, such as moment function, cumulative function and so on. This paper will also study The dual variable Hermite polynomials H_m, n (x, y) needed in quantum entanglement, the operator Hermite polynomial theory is generalized to the operator's double variable Hermite polynomial, and its properties and applications are studied. In this paper, the theory of the operator Hermite polynomial is constructed and discussed from physical concepts and physical requirements. The theory of the operator Hermite polynomials should be discussed smoothly. In the first chapter, the integral technique in the ordered operator invented by Chinese scholar Fan Hongyi is introduced. In the second, third chapter, the theory of the operator Hermite polynomial (single variable) is introduced, the core of which is the regular multiplication and the unconventional multiplication of the operator Hermite polynomials. The fourth chapter introduces how to derive Laguerre polynomials using the operator Hermite polynomial theory. In the past, they are considered respectively. The fifth chapter describes the physical application of the normal product Hn (X) = 2n: of the operator Hermite polynomials. The sixth chapter introduces the derivation of the Hn (Q's new binomial theorem) by the operator Hermite multinomial method. The seventh chapter The formula of the parent function of even and odd Hermite polynomial is derived by the operator Hermite polynomial method. From eighth chapters to the eleventh chapter, the way to introduce the dual variable Hermitee polynomial is introduced and the dual variable operator Hermite polynomial theory is established, and the new mother function formula is derived. In the twelfth chapter, the completeness of the Fock space can only be used in the past. How can the completeness spread in the mixed state? The results show that the Fock space can be divided into a mixed state (binomial state or negative binomial state). In a word, the operator Hermite polynomial theory and the new division of the Fock space have played an active role in further understanding the structure of the Fock space and enriching the connotation of the Fock space.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O431.2

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