Traces of products of angular momentum operators

Traces of products of angular momentum operators in a spherical or cartesian basis are common in the theory of atomic levels in fields, in the theory of nuclear orientation and of asymmetric top moments. Conventional angular momentum techniques lead to difficult and cumbersome calculations. In the present paper Schwinger's coupled boson representation is used in straightforward calculations of angular momentum and spherical tensor traces, of matrix elements and of asymmetric top moments. Only simple algebra, elementary multiplication and summation of integers are necessary. The method considerably simplifies calculations with angular momentum operators.     

Feynman Transformation Matrix Element for a type of Hamiltonian of Angular Momentum system

Using the technique of integration within ordered product (IWOP) of operators and the Schwinger boson representation for the angular momentum theory as well as the coherent state's method, we evaluate the Feynman transformation matrix element for the angular momentum system whose Hamiltonian is H = AJ_x + BJ_y + CJ_x (A, B, C real). The eigenstate of H is derived and the comparison between it and the SU(2) coherent state is made.     

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.     

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。     

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.     

Exact calculation,using angular momentum,of combined Zeeman and quadrupolar interactions in NMR

The NMR of nuclei with spins greater than ½ is often strongly influenced by the quadrupole interaction. This combination of Zeeman and quadrupole terms can usually be treated using perturbation theory, but an exact calculation is also needed. We explain an exact approach that eliminates the evaluation of commutators of complicated operators. Instead, the calculation is based on matrix elements of the Liouvillian, the commutator with the Hamiltonian. The spectrum can then be calculated directly from the eigenvectors and eigenvalues of the Liouvillian. With the aid of angular momentum methods, it can be shown that the quadupole interaction for spin I is fully determined by only (2I ?1) reduced matrix elements—for spin 3/2, this means only two quantities. The exact nature of the various basis operators is not needed, since the calculation only needs the angular momentum quantum numbers. The full Liouvillian matrix can be calculated from selection rules and the Wigner-Eckart theorem. Furthermore, we present an expression for these reduced matrix elements which is valid for any spin. This theory covers the whole range from quadrupole-perturbed NMR spectra to Zeeman-perturbed nuclear quadrupole resonance.     

An approximate projection technique for the calculation of electromagnetic properties of deformed nuclei

A three-dimensional angular momentum projection is carried out for cranking model wave functions. The projected matrix elements of electromagnetic operators are evaluated using a method originally developed by Kamlah for the case of projected energy, which is valid for large deformations and weakly triaxial nuclei. The calculated spectroscopic quadrupole moments deviate substantially from the predictions of a rigid rotor model with axial symmetry. For E2 transitions the deviations are small. Projected values of the magnetic moments are almost identical with those of a semiclassical calculation. Cranking model wave functions are decomposed into its components having good angular momentum.     

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.     

PHASE OPERRTORS FOR ATOMIC SYSTEM IN SCHWINGER BOSON REPRESENTATION AND THEIR PROPERTIES

In terms of the Schwinger boson representation of angular momentum We introduce suitable phase operators which exhibit similar properties to those of the phase operators in radiation field into atomic system. With the help of them angular momentum's third component phase uncertainty relation are discussed. Also, the eigenstates of the Hermitian phase operator are derived.     

A symmetry-based approach to the effective Hamiltonian for symmetric top molecules in degenerate vibronic states

The effective Hamiltonian for a symmetric top molecule in a degenerate vibronic state is obtained. Included in this Hamiltonian are the rotational, spin-rotational, spin-orbit coupling and electronic spin-spin interactions. The terms of the Hamiltonian are expressed as the product of molecular ‘constants,’ rotational angular momentum operators, and symmetry operators. A formalism is derived, and tables included, to determine whether or not a symmetry operator vanishes for a given vibronic state of a particular molecular symmetry. In this way, one can easily obtain all the non-vanishing Hamiltonian terms for a particular application.     

State reconstruction of molecular spatial rotation

We establish a reconstruction approach for the rotational quantum state of linear molecules possessing a magnetic manifold. Our approximate method contains an iteration with generalized matrix inverses, processing the tomographic integral of the time-dependent molecular-axis distribution in a polar angle. As shown in a simulated example for an alignment state, the density matrix is determined in a high fidelity. An analytic tomographic formula is also derived for the symmetric top rotation. The state coherent in the quantum space of both the angular momentum and its magnetic projection can be approximately retrieved from the observable time-resolved solid-angle distribution of the molecular axis.     

Remarks on the Configuration Space Approach to?Spin-Statistics

The angular momentum operators for a system of two spin-zero indistinguishable particles are constructed, using Isham’s Canonical Group Quantization method. This mathematically rigorous method provides a hint at the correct definition of (total) angular momentum operators, for arbitrary spin, in a system of indistinguishable particles. The connection with other configuration space approaches to spin-statistics is discussed, as well as the relevance of the obtained results in view of a possible alternative proof of the spin-statistics theorem.     

Partial-Wave Decomposition for Meson-Exchange Currents in Few-Nucleon Systems

 We develop an approach for calculating matrix elements of meson-exchange current operators between three-nucleon basis states in -coupling and a three-nucleon bound state. The contributions generated by π- and ρ-exchanges are included into the consideration. The matrix elements are expressed in terms of multiple integrals in momentum space. We apply a technique of partial-wave decompositions and carry out some angular integrations in closed form. Different representations appropriate for numerical calculations are derived for the matrix elements of interest. The momentum dependences of the matrix elements are studied and benchmark results are presented. The approach developed is of interest for investigations of deuteron-proton radiative capture and ³He photo- and electrodisintegration when the interactions in the initial or final states are taken into account by solving the Faddeev equations. Received April 2, 1999; accepted for publication December 30, 1999     

Equivalent angular momentum operators for second quantised operators

Formula useful for replacing second quantised operators by angular momentum operators are derived and commented on in the light of recent results on exchange interactions.     

通用的角动量阶梯算符

利用最新发展的非线性代数理论,给出了一般角动量阶梯算符所应满足的代数方程,并具体构造出了这些算符,所构造的北算符能对所有角动量本征态的解量子数和磁量子数起升降作用,具有很好的通用性。     

Can gluon spin contribute to that of nucleon?

The transformation of angular momentum to its Belinfante form requires the smooth behaviour of classical fields at infinity which for the case of quantum operators transforms to the smooth behaviour of matrix elements at small momentum transfers. For the case of quarks this provides the kinematical counterpart of U _A (1) problem while for gluons there is a contradiction between kinematics and dynamics governed by Kogut-Susskind pole. This may result in the violation of Equivalence Principle for nucleons or in the stringent constraints to the strange quark polarization in nucleons, while the most likely outcome would be the impossibility to separate gluon angular momentum to spin and orbital parts in the meaningful way.     

Angular momenta and eigenvectors of the weakly coupled T ⊗ t Jahn‐Teller system

The eigenvalues of the weakly coupled T ? t Jahn‐Teller problem are known for several decades, and the same holds also true for the eigenstates. These, however, are only given in the traditional position representation, which proves inconvenient if one attempts to extend the weak‐coupling treatment into the region of stronger coupling. Here the solution of the T ? t eigenvalue problem at weak coupling is derived in terms of creation and annihilation operators. This reformulation of the problem is nontrivial, since the algebraic form of the oscillator eigenvectors, being simultaneous angular‐momentum eigenstates, has been worked out only recently and is probably still widely unknown. The electronic and oscillator eigenstates are then coupled to form eigenvectors of the total angular momentum. Finally, in preparation for an extension of the weak‐coupling treatment, the action of the boson creation and annihilation operators on the oscillator eigenvectors is calculated, thus completing the algebraic approach to the weakly coupled T ? t system.     

Quantum mechanics in discrete space and angular momentum

Recently we have studied quantum mechanics of bounded operators with a discrete spectrum. In particular, we derived an expression for the commutator[Q, P] of two bounded operators whose spectrum is discrete, and we showed that in the limit of a continuous spectrum the commutator becomes the standard one of Heisenberg. In this paper we show that the angular momentum operator and the phase operator satisfy the new commutation relation. We also briefly discuss the problem of the canonical phase operator conjugate to the number operator.     

Elastic scattering of relativistic electrons by nuclei in the helicity representation

The partial-wave analysis of the scattering problem electron-nucleus is performed using helicities instead of the angular momentum of the electron. This proceeding leads to remarkable advantages in the treatment of the scattering process, especially in the numerical solution of the system of coupled radial differential equations: a) The matrix elements over the potential operators are independent of the total angular momentum. b) Instead of 6-j symbols etc., only very simple algebraic expressions are to be evaluated. c) Due to special properties of the system computing time is reduced for higher values of the nuclear spin by an amount proportional to about the order of the dimension of the system.