从一道典型试题的多种解法谈算符运算的基本技巧

算符运算在量子力学中十分重要,并且在各种考试中占有一定比重.要能正确而迅速地推证算符关系式,除了要了解各有关算符的意义、性质以及基本的对易关系等以外,还应掌握一些基本技巧.例如作用法、直接法、参数微分法、积分变换法、待定算子法、表象法等.限于篇幅,我们仅就其中几种最常用的方法通过一道算符运算方面的典型试题以一题多解的形式归纳如下. 题: 其中u为单位矢量,σ为Pauli算符.解法1[直接法:它仅仅是利用有关算符的一些已知性质和对易关系等,直接通过算符运算对有关算符关系式加以推证.] 因为(J·u)(0·N -aillja^ uk (重复指…     

关于傅里叶光学的描述问题(续)

四、抽象向量法上述算符法虽然将某些运算作为算符来处理,并建立了一些算符之间的代数关系,但没有利用向量空间的概念。在下面的方法中,基于与量子力学的类比,明确地提出了一个抽象向量空间,从而将量子力学的方法移到傅里叶光学中来。 1.单色波传播方程的傍轴近似     

从空间转动的几何性质推导角动量对易式

经典力学中把L=r×P叫做角动量.量子力学将r和P看作算符后得到算符(1)(V是微分算符),称L为角动量算符.由定义式(1)出发,经过微分运算可得到角动量算符不同分量间的对易关系(2a) (2b) (2c)这种关于角动量的定义和对易关系的推导方法,不具有普遍意义,它只适用于轨道角动量.而角动量这个量跟系统在转动下的变换性质有本质联系.角动量的对易关系,与体系在转动下的特性密切相关.笔者认为,在量子力学的教学中,如果在利用经典概念建立了量子力学的轨道角动量算符后,能再进一步从体系的转动变换性质推导角动量算符,并给出角动量的一般定义式,对提…     

一种角动量算符的简便推导方法

在量子力学教本中推导(L)2及(L)z的球坐标表达式一般都采用由直角坐标系到球坐标系的变量代换的方法,运算过程十分复杂,本文利用了球坐标系中坐标变量r、θ、ψ对正交基矢er、eθ、eψ微分性质,采用球坐标系直接进行推导,简化量子力学中关于角动量算符(L)2及(L)z的的推导.     

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.     

相干态法研究量子论的一类交换算符

利用相干态和正规乘积内的积分法，我们研究了量子力学中两个态的交换运算算符，得出了交换算符在相干态表象、粒子数表象和坐标表象中的表示．这一方法也可自然地推广到研究多状态之间的循环置换．     

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。     

自旋交换算符在量子力学中的应用

根据自旋交换算符的性质,讨论了自旋交换算符在量子力学中的某些应用,这一方法为两体自旋耦合体系的求解提供了方便的途径.     

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.     

论量子力学的公理基础

从三十年代冯·诺伊曼等人的工作开始,量子力学逐渐被看成是一些公理组成的数学体系.量子力学基本公理主要有以下几个: 1.一切微观状态为希尔伯特(Hilbert)空间中的矢量; 2.可观察量相应算符为希尔伯特空间中的线性厄密算符; 3.可观察量与相应算符存在如下关系(本征态情况):其中P──算符,p──相应可观察量,──希尔伯特空间矢量; 4.可观察量的平均值用下式表示(非本征态情况): 从这些公理出发,可以获得量子力学各种重要成果.但是,这些公理看来似乎没有直接的物理意义,因此,资产阶级学者们就利用这一点对量子力学进行歪曲,把量子力学说成…     

A unified quantum theory of mechanics and thermodynamics. Part I. Postulates

A unified axiomatic theory that embraces both mechanics and thermodynamics is presented in three parts. It is based on four postulates; three are taken from quantum mechanics, and the fourth is the new disclosure of the existence of quantum states that are stable (Part I). For nonequilibrium and equilibrium states, the theory provides general original results, such as the relation between irreducible density operators and the maximum work that can be extracted adiabatically (Part IIa). For stable equilibrium states, it shows for the first time that the canonical and grand canonical distributions are the only stable distributions (Part IIb). The theory discloses the incompleteness of the equation of motion of quantum mechanics not only for irreversible processes but, more significantly, for reversible processes (Part IIb). It establishes the operational meaning of an irreducible density operator and irreducible dispersions associated with any state, and reveals the relationship between such dispersions and the second law (Part III).     

Optimal measurements for general quantum systems

We give a general formulation of the theory of optimal quantum measurements, based on Gudder's [8] convex structure approaches to axiomatic quantum mechanics, which includes the case of Holevo's formulation [14] and operational quantum mechanics [3]. Simple and general conditions for existence of Bayes optimal measurements are obtained by a method without operator valued measure techniques. For this purpose, a representation of convex prestructures and a characterization of a class of loss functions are obtained. Finally, an application of the results to Wald's theory of statistical decision functions is shown.     

两个力学量算符具有共同本征态系的条件

量子力学是人们了解微观世界的一门重要的课程.在量子力学中,由于任何实物体都具有波粒二象性,必须用算符来表示力学量,因此算符理论显得尤为重要.在研究多个算符时,有一个非常重要的定理:两个力学量算符能够具有共同本征函数系的充分必要条件是这两个个力学量算符能够互相对易.本文分析了在使用该定理时可能存在的一些问题,并对这些问题进行了澄清.     

Symplectic Geometry of Quantum Noise

We discuss a quantum counterpart, in the sense of the Berezin–Toeplitz quantization, of certain constraints on Poisson brackets coming from “hard” symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.     

A Non-Associative Quantum Mechanics

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.     

Consistent histories and operational quantum theory

In this work a generalization of the consistent histories approach to quantum mechanics is presented. We first critically review the consistent histories approach to nonrelativistic quantum mechanics in a mathematically rigorous way and give some general comments about it. We investigate to what extent the consistent histories scheme is compatible with the results of the operational formulation of quantum mechanics. According to the operational approach, nonrelativistic quantum mechanics is most generally formulated in terms of effects, states, and operations. We formulate a generalized consistent histories theory using the concepts and the terminology which have proven useful in the operational formulation of quantum mechanics. The logical rule of the logical interpretation of quantum mechanics is generalized to the present context. The algebraic structure of the generalized theory is studied in detail.     

A probabilistic approach to quantum mechanics based on ‘tomograms’

It is usually believed that a picture of Quantum Mechanics in terms of true probabilities cannot be given due to the uncertainty relations. Here we discuss a tomographic approach to quantum states that leads to a probability representation of quantum states. This can be regarded as a classical‐like formulation of quantum mechanics which avoids the counterintuitive concepts of wave function and density operator. The relevant concepts of quantum mechanics are then reconsidered and the epistemological implications of such approach discussed.     

The C Operator in iϕ 3 Field Theory

The C operator defines a dynamically-determined positive-definite metric in PT-symmetric theories. We show how the operator formalism for the perturbative calculation of C can be extended from quantum mechanics to quantum field theory with a cubic self interaction.     

对突破工科近代物理教学一个瓶颈问题的探讨

在工科近代物理教学中,量子力学部分所占比例非常少,而量子力学基础教学中的一个瓶颈在于算符和算符本征值与力学量概率分布的关系.介绍了为突破这一瓶颈,在教材建设和教学实践中所做的改进.