Relation between the projection operator formalism and the Faddeev theory

In an earlier paper it was shown by means of the Faddeev equations that the reduction of the three-body problem to formal two-body equations with energy-dependent complex potentials — a procedure which was already well known for separable potentials — can be generalized rigorously to arbitrary potentials. In the present paper we show that the projection operator approach of Feshbach appears as a special, and in a certain sense singular case of our general treatment. Optical potentials for rearrangement scattering are given explicitly and questions of how to calculate them are discussed, showing that the Faddeev approach is better suited in these respects.     

Covariant fuzzy observables and coarse-graining

A fuzzy observable is regarded as a smearing of a sharp observable, and the structure of covariant fuzzy observables is studied. It is shown that the covariant coarse-grainings of sharp observables are exactly the covariant fuzzy observables. A necessary and sufficient condition for a covariant fuzzy observable to be informationally equivalent to the corresponding sharp observable is given.     

Projection operator approach to the thermodynamic formalism of dynamical systems

An analytical perturbative treatment of characteristic exponents describing the fluctuations of temporal coarse-grained quantities in the context of nonlinear dynamical systems is proposed. It is based on the analysis of the resolvent of the corresponding transfer operator by a projection operator method similar to those used in statistical mechanics. Two different approximation schemes are presented and tested for the case of an exactly solvable but nontrivial model system.     

On the connection between quantum and classical descriptions

The review paper presents generalization of d??Alembert??s variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum force ?j, ??. Spatial attention is dedicated to the systems with (hidden) symmetries. It is shown how the symmetry reduces the number of quantum degrees of freedom down to the independent ones. The sin-Gordon model is considered as an example of such field theory with symmetry. It is shown why the particles S-matrix is trivial in that model.     

双模积分型投影算符及其应用

引进了幺正的双模积分型投影算符,利用有序算符内的积分(IWOP)技术分析了其变换特性;然后利用该积分型投影算符对角化了双模耦合量子谐振子体系的哈密顿量,从而求出了体系的本征能级与本征波函数;最后讨论了特例情形.     

也谈概率密度和概率流密度算符

针对<大学物理>2008年第8期上关于粒子概率密度算符和概率流密度算符的讨论一文进行了补充,对一般空间(如动量空间)中的概率和概率流进行了较系统的阐述,并纠正了该文中的有关错误论断.     

Systematic treatment of the anderson localization on the basis of the projection operator formalism

A derivation within the projection operator technique is given for the density and current response functions of a system of independent particles moving in a random potential. The essential point is the derivation of kinetic equations for the current relaxation kernel instead of for the density propagator as in a previous treatment on the basis of the projection operator formalism. In these equations the divergent contributions from the 2k _F -scattering mechanism can be systematically separated from those of diffusional scattering. Especially, both the self-consistent current relaxation theory, developed by Götze and the self-consistent treatment of Vollhardt and Wölfle are rederived from simple approximations of the kinetic equations. The outlined method represents a systematic approach to the Anderson localization. It may be applicable also to more realistic models for the Anderson localization as well as extended to the evaluation of general transport coefficients on a level far beyond the usual perturbation theories.     

Highly scalable discrete-particle simulations with novel coarse-graining: accessing the microscale

ABSTRACT

Simulating energetic materials with complex microstructure is a grand challenge, where until recently, an inherent gap in computational capabilities had existed in modelling grain-scale effects at the microscale. We have enabled a critical capability in modelling the multiscale nature of the energy release and propagation mechanisms in advanced energetic materials by implementing, in the widely used LAMMPS molecular dynamics (MD) package, several novel coarse-graining techniques that also treat chemical reactivity. Our innovative algorithmic developments rooted within the dissipative particle dynamics framework, along with performance optimisations and application of acceleration technologies, have enabled extensions in both the length and time scales far beyond those ever realised by atomistic reactive MD simulations. In this paper, we demonstrate these advances by modelling a shockwave propagating through a microstructured material and comparing performance with the state-of-the-art in atomistic reactive MD techniques. As a result of this work, unparalleled explorations in energetic materials research are now possible.     

Theory of resonance pressure broadening: Resolvent operator formalism and classical path approximation

The problem of resonance pressure broadening of spectral lines in monatomic gases is discussed using a resolvent operator formalism. A differential equation is developed to determine the resolvent, and it is shown how its solution for a limiting case yields the familiar classical path approximation for the translational motion of the atoms, and how quantum corrections may be systematically studied. Commonly used limiting cases within the classical path approximation (two-body static and impact approximations) are also exhibited as limiting cases, with methods for systematic evaluation of corrections. Closed form solutions are obtained for the two-body static and impact cases. The results are compared with available experimental data, and generally satisfactory agreement is obtained. Of some theoretical interest is the formalism, which embraces all the usual approximations and permits them to be studied together with corrections to them from a unified point of view. New results of more practical interest are the closed form solutions for the limiting cases, and the estimation of the lowest-order quantum corrections, which are appreciable under some experimental conditions.Supported in part by the National Science Foundation through Grant GP 7552.Alfred P. Sloan Fellow.     

经典菲涅耳变换的量子力学对应

介绍经典菲涅耳变换的量子力学对应,它是相干态在相空间中的代表点做辛运动所对应的量子菲涅耳算符.     

对一个算符公式的修正

指出近日出版的《高等量子力学》一书中给出的一个算符公式在其推导中存在漏洞，因而结果也不正确讨论了正确的推导方法和修正的结果。     

On the connection between the continuum eigenfunctions and the dispersion function of the vlasov operator

It is shown that a function may be extracted from the continuum eigenmode of the Vlasov operator which is related to the boundary values of the dispersion function on the real axis of the complex frequency plane. The precise content and possible consequences of this relationship are investigated in detail, especially the possibility of reconstructing the dispersion function and establishing stability criteria.     

On the excluded space in applications of Feshbach projection formalism

Various model applications in nuclear structure and reactions have been formulated starting with the Feshbach projection formalism. In recent studies a truncated excluded space has been enumerated to facilitate calculation and identify a convergence in expansions within that truncation. However, the effect of any remainder must be addressed before results from such can be considered physical.     

双模坐标-动量积分型投影算符及其在量子光学中的应用

引进了幺正的双模坐标-动量积分型投影算符,利用有序算符内的积分（IWOP）技术分析了其变换特性,并导出了其正规乘积展开式.然后利用该积分型投影算符对角化了双模耦合量子谐振子体系的哈密顿量,从而求出了体系的本征能级与本征波函数.最后讨论了特例情形. 关键词： 积分型投影算符 有序算符内的积分技术 坐标-动量耦合     

The operator formalism of quantum mechanics from the viewpoint of short disturbances in nonrelativistic classical motion

The effect of short disturbances on nonrelativistic motion is formulated in terms of operators. Analogies with quantum mechanics are developed and some disparities noted. For the one-dimensional particle we obtain analogues of the de Broglie wave commonly associated with particle motion, Heisenberg's commutation relation, Schrödinger's equation, and the statistical interpretation. Whether these results have any bearing on quantum mechanics itself is left an open question.     

Existence of the transfer matrix formalism for a class of classical continuous gases

For classical gases of particles interacting through nonnegative, many-body interactions of short range it is verified that the corresponding grand canonical Gibbs measures have the global Markov property for sufficiently low values of the chemical activity. This yields the existence of a (nonsymmetric in general) transfer matrix formalism for such systems.     

A comparison of projection operator formalisms for the study of self-diffusion

The utility of projection operator formalisms for describing the dynamics of many-body systems is studied, and the compatibility of these formalisms with certain approximation schemes is evaluated in the light of known behavior of such systems. For simplicity the investigation is limited to the study of Brownian motion. Specifically, a memory kernel formalism and a kinetic equation formalism are compared for the calculation of the time evolution of the momentum autocorrelation function. Both perturbation expansions and averaged propagator approximations are investigated. The results from these studies suggest that the long-time behavior of the momentum autocorrelation function is sensitive to the long-range nature of the interparticle potential.This work was supported in part by the National Science Foundation under Grant GK-19360X.     

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple (Mⁿ,g,) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second-order operator Ω acting on spinor fields. In case of a naturally reductive space and its canonical connection, our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated G₂-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of -parallel spinors.