四元数在量子力学中的应用

把双四元数推广到了二级双四元数，并设计了一种态函数的四元数表示法，从而用四元数表述了相对论量子力学，使四元数物理学形成了系统，用四元数表示的算符和状态，对于导出算符间的对易关系和状态的洛伦兹变换性质是方便的。     

Exactly solvable time-dependent models in quantum mechanics and their applications

Methods for obtaining exact and approximate solutions of the evolution of quantum-mechanical problems are discussed. The cyclic evolution of quantum systems described by time-periodic Hamiltonians is analyzed. A class of time-periodic Hamiltonians is constructed in the close analytical form. The corresponding cyclic solutions are calculated. Time-dependent Hamiltonians are generated whose expectation values calculated with cyclic solutions are time independent. It is shown that the expectation values of the spin projection calculated with the same cyclic solutions, as well as the probability density of finding a particle at a given space-time point, are also time independent. Therefore, the approach can be used to simulate quantum dynamic potential wells with the particle localization effect. Nonadiabatic geometric phases are expressed in terms of the cyclic solutions. Exactly solvable time-dependent problems are used to construct a universal set of gates for quantum computers. A method for obtaining entanglement operators is discussed.     

Relativistic quantum mechanics over stochastic phase space

Stochastic quantum mechanics is a quantum theory in which the basic limitations of real-world measuring instruments, due to their intrinsically quantum nature, are taken into account. Among other things this leads to a new operational definition of space-time, called quantum space-time. Fundamental to this approach is the formulation of quantum mechanics over phase space rather than just over position or momentum space. A concept of extended particle is a natural outgrowth of this development. Gauge and internal symmetry have a natural place within the theory, and preliminary computations combining some old ideas due to Born with more recent ideas on symmetry breaking suggest that the theory could lead to a mass formula compatible with known data on the low-lying baryons.Supported in part by NSERC Grant, No. A8403.     

广义经典力学与非完整力学的统一理论

建立广义经典力学与非完整力学的统一理论———广义非完整力学理论,构造其基本框架.提出广义非完整力学的Чeтаeв(Chetaev)定义,建立广义非完整力学系统的Routh方程和正则方程.给出广义非完整力学系统的非等时变分方程,证明由广义非完整力学系统的第一积分可以构造积分不变量.研究广义非完整力学系统作用量的非等时变分,给出系统的Poincar啨Cartan积分变量关系和积分不变量,进而给出等时变分下系统的Poincar啨积分变量关系和通用积分不变量.给出一些推论,表明广义经典力学和一阶至高阶非完整力学的相关结论均为广义非完整力学理论的特款. 关键词： 广义非完整力学 广义Чeтаeв定义 运动方程 积分不变量     

On the representation of quantum mechanics on phase space

It is shown that Hilbert-space quantum mechanics can be represented on phase space in the sense that the density operators can be identified with phase-space densities and the observables can be described by functions on phase space. In particular, we consider phase-space representations of quantum mechanics which are related to certain joint position-momentum observables.     

Geometrical setting of solid mechanics

The starting point in the geometrical setting of solid mechanics is to represent deformation process of a solid body as a trajectory in a convenient space with Riemannian geometry, and then to use the corresponding tools for its analysis. Based on virtual power of internal stresses, we show that such a configuration space is the (globally) symmetric space of symmetric positive-definite real matrices. From this unifying point of view, we shall analyse the logarithmic strain, the stress rate, as well as linearization and intrinsic integration of corresponding evolution equation.     

Geometrical characterization of reduced density matrices reveals quantum phase transitions in many-body systems

<正>Quantum phase transitions (QPTs) play a central role for understanding many-body physics [1]. Different from classical phase transitions which are driven by thermal fluctuations, QPTs are driven by quantum fluctuations at zero temperature and can be accessed by varying some physical parameters of the many-body system. Characterizing QPTs, which normally needs complicated theoretical calculations, becomes a fundamental problem to further study quantum matters. Here a group of physicists proposed to connect the geometrical properties of reduced density matrices (RDMs) of the physical system with its quantum phase transitions [2,3]     

A molecular quantum mechanics approach to evaluate molecular properties in liquid phase using statistical mechanics

A method for the calculation of molecular properties in liquid phase is proposed. The theoretical development links in a natural fashion Molecular Quantum Mechanics and Statistical Mechanics. The method can be applied to molecular liquids in thermodynamical equilibrium without strong interactions between molecules. After the Introduction (in which some fundamental ideas are discussed) the model is developed in its simplest form, giving a numerical application of interest for IR spectroscopy in liquid phase. Immediatly, this preliminar version is generalized for pure liquids taking into account the polarization effects on all the molecules of the sample. A numerical example on carbon disulphide shows the convergence of this method and its sensitivity to the structural changes in the liquid. In the last section, some future improvements are suggested and several topics related with the study of molecular liquids are considered.     

Geometrical aspects and generalizations of Newton-Cartan mechanics

Cartan's geometrical approach to Newtonian mechanics is discussed and generalized. The approach is geometrical and use is made of the theory of holonomy groups. The concept of a degenerate metric connection is formulated and discussed.     

普朗克论"不均匀介质中几何光学与量子力学定态薛定谔方程的关系"

介绍普朗克通过对不均匀介质中几何光学方程与经典静态守恒力场中自由质点运动方程的对比从而引入量子力学定态薛定谔方程的理论.     

Group-theoretic treatment of the axioms of quantum mechanics

This axiomatization is based on the observation that ifG is the group of automorphisms of the states (induced, e.g., by suitable evolutions), then we can define a spherical function by mapping each element ofG to the matrix of its transition probabilities. Starting from five physically conservative axioms, we utilize the correspondence between spherical functions and representations to apply the structure theory for compact Lie groups and their orbits in representation spaces to arrive at the standard complex Hilbert space structure of quantum mechanics.Supported by National Science Foundation under Grant MPS75-09371.     

Quantum phase for an electric quadrupole moment in noncommutative quantum mechanics

We study the noncoInmutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric qaudrupole moment, in the presence of an external magnetic field. First, by intro ducing a shift for the magnetic field, we give the Schrodinger equations in the presence of an external magnetic field both on a noncommutative space and a noncomlnutative phase space, respectively. Then by solving the SchrSdinger equations both on a noneommutative space and a noncommutative phase space, we obtain quantum phases of the electric quadrupole moment, respectively. Wc demonstrate that these phases are geometric and dispersive.     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

Geometrical spin dephasing in quantum dots

We study spin-orbit mediated relaxation and dephasing of electron spins in quantum dots. We show that higher order contributions provide a relaxation mechanism that dominates for low magnetic fields and is of geometrical origin. In the low-field limit relaxation is dominated by coupling to electron-hole excitations and possibly 1/f noise rather than phonons.     

Geometrical optics in Fresnel quantum holography

The geometrical optics in Fresnel quantum holography for quantum entangled two-photon source and classical thermal light source based on the second-order correlation measurement has been discussed. We theoretically prove that the Fresnel quantum hologram of the detected object can be obtained through second-order correlation measurement and the optical reconstruction process can be accomplished by making use of a point light source.