量子力学中新的角动量阶梯算符

角动量阶梯算符在量子力学中有着极其广泛的应用,传统的教科书只给出角动量磁量子数的阶梯算符本文介绍一个新的总角动量阶梯算符,它可使总角动量量子数j上升(或下降). 在量子力学中,力学量用厄米算符表示,力学量之间的内在联系体现在对易关系中.因此,在一些问题中,不需解薛定谔方程,便可确定本征值及本征矢.其办法是构造出一个阶梯算符,例如对谐振子[1]、角动量[2]的处理.特别在处理角动量问题时,引入了阶梯算符L+(J+),由此推导出角动量的本征值、本征矢及有关矩阵元公式等. 那么,是否可以找出关于总角动量量子数的阶梯算符呢?目前的教科…     

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…     

标量算符本征值对g壳层LSQ耦合态的完全分类计算

按（Ｕ,Ｄ）Ｌ－ＬＳＱ格式构造ｌ壳层ＬＳＱ耦合态,这里Ｕ（Ｄ）是自旋向上（向下）电子的轨道角动量,Ｌ、Ｓ、Ｑ是总轨道角动量、总自旋和准旋。由４个产生－湮灭算符构造与轨道、自旋准旋算符均为易的标量算符并用基本征值对ＬＳＱ耦合态进上步分类,实现了对ｆ、ｇ壳层耦合态的完全分类,列出了ｇ壳层耦合态完全分类的主要结果。     

对量子力学中角动量的一点认识

指出了角动量算符Ｊ不存在本征值和本征矢量完全集,角动量算符Ｊ不描写一个可观察量以及角动量Ｊ不宜作为整体来讨论。     

角动量平方算符的矢量分析计算

量子物理是从经典物理中发展而来的,在其教学中有意识地挖掘现有教材以便与经典进行对比,指出两者的差异,并说明在什么条件下量子描述退化为经典描述,具有十分重要的教学价值,而角动量的平方算符的推导刚好提供了这样的契机.本文利用矢量算符分析的方法来推导出在球坐标系下角动量平方算符的表达式,同时与经典的角动量平方进行了比较,得到量子角动量平方算符比其经典对应量多出含普朗克常数的项,在经典极限下,前者退化为后者.作为拓展,最后用Bohm规则计算了角动量平方算符.     

电子自旋是角动量的讨论

从施特恩-盖拉赫实验测量的结果电子自旋只有两种取值出发,在不引入自旋算符的具体矩阵表示的情况下,利用矩阵求迹,导出自旋算符和轨道角动量算符满足完全相同的对易关系,从而说明自旋与轨道角动量同类,属于角动量.区别于历史上和国内外量子力学教材中直接将自旋归为角动量,本文尝试用"物以类聚"的逻辑对自旋是角动量进行讨论.     

用转动矩阵导出角动量算符的球坐标表示式

本文在推导出无限小角度转轨算符的形式后，用真转动（ｐｒｏｐｅｒ ｒｏｔａｔｉｏｎ）中转动矩阵的方法，得到角动量算符的球坐标表示式，使角动量算符的物理意义更为明确．     

柱矢量光束的角动量性质

介绍了非近轴光束的表示理论,利用该表示理论很好地解决了非近轴光束的角动量问题,发现非近轴光束的总角动量可以严格地分解成自旋和轨道两部分,但是两者都依赖于由偏振椭圆度表征的光束的偏振状态。主要研究了柱矢量光束的角动量问题。给出了动量空间和位形空间中的柱矢量光束表达式和角动量算符表达式。通过分析两个空间中的角动量算符及柱矢量光束表达式,发现在这两种空间中,具有螺旋型相位的柱矢量光束是角动量算符沿着传播方向的分量的本征态,其本征值与偏振椭圆度无关,这为计算这类特殊光束的角动量提供了一种新方法。     

交换超算符方法的李代数研究

本文讨论了交换超算符方法的理论基础,结果表明由交换超算符所定义的算符集合g是一个李代数,交换超算符的定义就是李代数中内导子的定义,由此得出一些交换超算符间的代数关系。证明了g中所有算符诱导的超算符集合也是一个李代数,指出了与g对应的是由复盖群派生的,有内积定义的李群,而角动量超算符是由矢量场的内禀角动量和单位算符的直积所生成。结论是交换超算符方法的理论基础是李代数。     

三粒子Calogero-Sutherland模型的相干态

利用赝角动量算符方法求解三粒子CSM(Calogero-Sutherland Model)的类径向方程,给出本征态和相干态的解析表达式. 关键词： 三粒子Calogero-Sutherland模型 赝角动量算符方法 相干态     

VECTOR LADDER OPERATORS FOR THE CENTRAL POTENTIALS

A new class of nonlinear Lie algebra has been found, which is generated naturally by the Hamiltonian operator, the square of the angular momentum operator and the ladder operator for the central potentials. According to the theory of nonlinear Lie algebra, without using the factorization method, we obtained the vector ladder operators for the three-dimensional isotropic harmonic oscillator and hydrogen atom. The radial components of these operators, which are independent of the quantum numbers, are just the radial ladder operators for the same potentials.     

Raising and Lowering Operators for Orbital Angular Momentum Quantum Numbers

Two vector operators aimed at shifting orbital angular momentum quantum number l successfully constructed based on the primary form proposed by Prof. X.L. Ka in 2001. The lowering operators can give the lowest angular momentum quantum numbers l for a given magnetic quantum number m in spherical harmonics |lm〉; and the state with minimum angular momentum quantum number in whole set of the spherical harmonics turns out to be |0,0〉. How to use the raising and lowering operators as acting on the state |0,0〉. to generate whole set of spherical harmonics is illustrated.     

The quantum groupSUq(2) andq-analog of angular momentum operators in quantum mechanics

We present three operators in quantum mechanics that obey the commutation relations of quantum groupSUq(2). These operators are nonlinear combinations of the conventional angular momentum operators and are called the quantumq-analog angular momentum operators. When the quantum deformation parameterr = Inq vanishes, these quantumq-analog angular momentum operators reduce to the usual angular momentum operators.     

Multiconfiguration Hartree-Fock method: Direct diagonalization for the construction of a multielectron basis

A method based on the direct diagonalization of basis states in the framework of the multiconfiguration Hartree-Fock procedure has been proposed for constructing the multielectron basis. With the use of the technique of ladder operators of the orbital angular momentum and spin, this method has been generalized to the case of arbitrary electron configurations. It has been shown that such an approach can be easily implemented on a computer and has low requirements for computational resources in the case of d and f-electrons. The calculations of the multielectron basis states have been exemplified for several electron configurations and various sets of quantum numbers.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

二维与三维氢原子的四类升、降算子

用因式分解法求出了二维和三维氢原子的角动量升、降算子，还利用合流超几何函数的递推关系求出了另外三类升、降算子．讨论了四类升、降算子的选择定则和守恒量子数．二维和三维氢原子有很大的相似性 关键词：     

Nucleon internal structure: a new set of quark, gluon momentum, angular momentum operators and parton distribution functions

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

Gauge invariance and quantization applied to atom and nucleon internal structure

The prevailing theoretical quark and gluon momentum, orbital angular momentum and spin operators, satisfy either gauge invariance or the corresponding canonical commutation relation, but one never has these operators which satisfy both except the quark spin. The conflicts between gauge invariance and the canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. To achieve such a proper decomposition the key point is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics, and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

An approach to the anomalous commutation relations of rotational angular momenta in molecules

In molecular quantum mechanics, angular momentum operators occur which obey anomalous commutation relations. A direct method of evaluating the matrix elements of such operators is presented. The method is particularly well suited to calculations which use spherical tensor techniques; it also avoids some confusing aspects of Van Vleck's reversed angular momentum method [2]. The relationship between results derived in space- and molecule-fixed axis systems is made explicit and a consistent phase convention is established. The transformation matrix relating coupled and decoupled basis sets for problems with molecular quantization is also derived.