本文選題：量子邏輯 + 正規(guī)算子�。� 參考：《哈爾濱工業(yè)大學(xué)》2017年碩士論文

【摘要】：量子力學(xué)是二十世紀(jì)物理學(xué)最重要的成果之一,是近代物理的主旋律,并且導(dǎo)致了物理學(xué)在觀念和思想上的徹底變革,使物理學(xué)得到了全面的改觀。量子邏輯正是伴隨著量子理論的數(shù)學(xué)公理化而發(fā)展起來的一個數(shù)學(xué)分支,已有八十多年歷史和豐富內(nèi)容。眾所周知,在微觀世界中,粒子運動遵循的是薛定諤方程.薛定諤方程是線性的,因此其解可構(gòu)成線性空間。另外,物理背景要求線性空間能夠做投影和內(nèi)積。所以,希爾伯特空間上的數(shù)學(xué)理論在此物理背景下有實際意義。比如,實驗所測量的值一定是實數(shù),那么我們用來表示可觀測量的算子的譜應(yīng)該是實數(shù),而自共軛算子正有此性質(zhì)。另一方面,量子力學(xué)公理化提出的基本假設(shè)指出,封閉的量子系統(tǒng)隨時間演化的過程可以用一個酉算子來刻畫。對于任意給定的酉算子,必有某個封閉的單量子比特系統(tǒng)在某段時間的演化可用此酉算子描述。在此背景下,酉算子作為在量子觀測和量子計算中的工具,可用于計算兩個乃至多個量子系統(tǒng)之間的某些物理關(guān)系。本文研究了同時包含自共軛算子和酉算子的邏輯結(jié)構(gòu),即由正規(guī)算子構(gòu)成的量子邏輯。第一章介紹了本課題的來源與背景,并列舉了近年來國內(nèi)外學(xué)者對此課題相關(guān)領(lǐng)域的研究現(xiàn)狀。第二章介紹了與本課題相關(guān)的一些基礎(chǔ)知識,主要是相關(guān)代數(shù)結(jié)構(gòu)的定義與簡單性質(zhì)。第三章我們將算子垂直的關(guān)系引入到在希爾伯特空間上的正規(guī)算子中。利用垂直關(guān)系在正規(guī)算子集合上定義二元關(guān)系和部分二元運算,得到由正規(guī)算子全體構(gòu)成的量子邏輯。第四章定義了正規(guī)算子之間的兩種偏序,研究了此二者在該量子邏輯中的性質(zhì),并研究了含偏序子集的結(jié)構(gòu)。最后研究了兩種此偏序之間的關(guān)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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O177;O413

【參考文獻】

相關(guān)期刊論文 前1條

1 ;On the infimum problem of Hilbert space effects[J];Science in China(Series A:Mathematics);2006年04期

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本文編號：1920823