正规算子构成的量子逻辑

发布时间：2018-05-21 21:11

本文选题：量子逻辑 + 正规算子　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：量子力学是二十世纪物理学最重要的成果之一,是近代物理的主旋律,并且导致了物理学在观念和思想上的彻底变革,使物理学得到了全面的改观。量子逻辑正是伴随着量子理论的数学公理化而发展起来的一个数学分支,已有八十多年历史和丰富内容。众所周知,在微观世界中,粒子运动遵循的是薛定谔方程.薛定谔方程是线性的,因此其解可构成线性空间。另外,物理背景要求线性空间能够做投影和内积。所以,希尔伯特空间上的数学理论在此物理背景下有实际意义。比如,实验所测量的值一定是实数,那么我们用来表示可观测量的算子的谱应该是实数,而自共轭算子正有此性质。另一方面,量子力学公理化提出的基本假设指出,封闭的量子系统随时间演化的过程可以用一个酉算子来刻画。对于任意给定的酉算子,必有某个封闭的单量子比特系统在某段时间的演化可用此酉算子描述。在此背景下,酉算子作为在量子观测和量子计算中的工具,可用于计算两个乃至多个量子系统之间的某些物理关系。本文研究了同时包含自共轭算子和酉算子的逻辑结构,即由正规算子构成的量子逻辑。第一章介绍了本课题的来源与背景,并列举了近年来国内外学者对此课题相关领域的研究现状。第二章介绍了与本课题相关的一些基础知识,主要是相关代数结构的定义与简单性质。第三章我们将算子垂直的关系引入到在希尔伯特空间上的正规算子中。利用垂直关系在正规算子集合上定义二元关系和部分二元运算,得到由正规算子全体构成的量子逻辑。第四章定义了正规算子之间的两种偏序,研究了此二者在该量子逻辑中的性质,并研究了含偏序子集的结构。最后研究了两种此偏序之间的关系。
[Abstract]:Quantum mechanics is one of the most important achievements of physics in the 20th century, which is the main melody of modern physics, and has led to the thorough transformation of the concept and thought of physics, and has made a comprehensive change in physics. Quantum logic is a branch of mathematics developed with the mathematical axiom of quantum theory, which has a history of more than 80 years and rich content. It is well known that in the micro-world, particle motion follows the Schrodinger equation. The Schrodinger equation is linear, so its solution can form a linear space. In addition, the physical background requires the linear space to do projection and inner product. Therefore, the mathematical theory in Hilbert space has practical significance in this physical background. For example, the measured value of the experiment must be a real number, then the spectrum of the operator we use to denote observable measurements should be a real number, and the self-adjoint operator has this property. On the other hand, the axiomatic hypothesis of quantum mechanics indicates that the evolution of closed quantum systems over time can be characterized by a unitary operator. For any given unitary operator, the evolution of a closed single quantum bit system at a certain time can be described by this unitary operator. In this context, unitary operator, as a tool in quantum observation and quantum computation, can be used to calculate some physical relations between two or more quantum systems. In this paper, we study the logic structure of both self-conjugate operator and unitary operator, that is, quantum logic composed of normal operators. The first chapter introduces the origin and background of this topic, and lists the current research situation of domestic and foreign scholars on this subject in recent years. The second chapter introduces some basic knowledge related to this subject, mainly the definition and simple properties of the related algebraic structure. In chapter 3, we introduce the perpendicular relations of operators into normal operators on Hilbert spaces. By using the vertical relation to define the binary relation and partial binary operation on the set of normal operators, the quantum logic consisting of all normal operators is obtained. In chapter 4, we define two kinds of partial ordering between normal operators, study their properties in the quantum logic, and study the structure of subsets with partial ordering. Finally, the relationship between the two kinds of partial order is studied.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O177;O413

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