谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields

One of the most poorly understood subjects in physical optics is the origin of the Gouy phase (sometimes called “the phase anomaly near focus”). This is evident from the large number of publications on the subject, many of which attribute it to quite different causes. In this paper we show that the Gouy phase anomaly can be clearly understood from elementary properties of normal congruences of light rays and from the relationship between geometrical optics and physical optics. We also show that the Gouy phase anomaly may be regarded as a degenerate case of a rapid π/2 phase change that is found to occur at each focal line of an astigmatic pencil of rays. The intensity distribution in the region of the phase changes is also presented. Furthermore, symmetry relations for both the phase anomaly and the intensity distribution are derived.     

General semi-classical method for collective motion and its connection with the Rowe-Basserman and marumori theories,and adiabatic limits

In recent work, we have shown that in the adiabatic limit (large amplitude, small momentum), time-dependent Hartree-Fock theory (TDHF) yields a well-defined theory of large-amplitude collective motion which provides an essentially unique construction for a collective hamiltonian. An alternative theory, put forward by Rowe and Basserman and by Marumori is, apparently, not restricted to small momenta. We describe a general framework for the study of collective motion in the semi-classical limit without limitation on the size of coordinates or momenta, which includes all previous methods as limiting cases. We find it convenient, as in the past, to consider two general systems: first, a system with n degrees of freedom and no special permutation symmetry, and, second, a system of fermions described in TDHF. For both systems the problem can be formulated as a search for a hamiltonian flow confined to a finite-dimensional hypersurface in a phase space, which itself may be finite- or infinite-dimensional. Though, in general, there are no exact solutions to this problem, we can formulate consistent approximation schemes corresponding to both the adiabatic and Rowe-Basserman, and Marumori limits. We also show how to extend the momentum expansion, which underlies the adiabatic approximation, to higher orders in the momentum. We thereby confirm the structure of the theory found in our previous work.     

The modal interpretation of quantum mechanics and its generalization to density operators

We generalize the modal interpretation of quantum mechanics so that it may be applied to composite systems represented by arbitrary density operators. We discuss the interpretation these density operators receive and relate this to the discussion about the interpretation of proper and improper mixtures in the standard interpretation.     

The isothermal local resistivity and its connection to different theories

The purpose of the present paper is to discuss the theory of the isothermal local resistivity in the sense of linear response. Different methods, as the Langevin equation, the non-equilibrium density operator technique and the linear response theory of conduction, have been related with each other to clear up different ambiguities in the literature.The first two sections are devoted to introduce the hydrodynamic and linear response equations for the electron gas in a medium of scattering mechanisms (phonons, impurities, etc.). The inversion of the conductivity formula into the isothermal local resistivity is performed with help of a generalized Langevin equation in the isothermal limit (lim_{q → 0} lim_{ω → 0}A_qω). This result agrees with that of the non-equilibrium operator technique. Then the many-variable projection technique of Mori is used to establish the relations between microscopic theory of electrical conduction and the hydrodynamic equations. The relaxation matrix formulation of Fermi-liquid in a metal can describe sound wave propagation in the Fermi-liquid which corresponds to charge density waves. Further, the relation between the isothermal local resistivity and Köhler's variation principle is established for electron-phonon system on a general way, which allows one to make contact with the Boltzmann equation.In the one-electron approximation the isothermal local resistivity is discussed in terms of phase shifts of non-overlapping scatterers. The result is valid for a dense system of resonant scatterers.     

How conclusive is the scaling argument? The connection between local and global scale variations of finite action solutions of classical Euler-Lagrange equations

We analyze the argument that a critical point of the action is stationary under a global scale transformation. We establish a general criterion which allows one to prove rigorously the validity or nonvalidity of the argument in the various relevant classes of Euler-Lagrange equations. Furthermore, we give a priori estimates on solutions at infinity.     

Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras

We shall give a generalization of the Littlewood-Richardson rule forU _q (g) associated with, the classical Lie algebras by use of crystal base. This rule describes explicitly the decomposition of tensor products of given representations.     

The variational theory of adiabatic motions,and its applications to linear and non-linear oscillators

This paper presents a variational derivation of the adiabatic periodic motion theorems and related time-integral-of-energy results, including the virial theorem, and some of their applications to linear and non-linear oscillators. Specifically, (i) first, the Maupertuis-Euler-Langrange (MEL) action principle is formulated for the most general (scleronomic and holonomic) system; in the derivation the time-dependent system parameters are treated just like additional generalized co-ordinates and subjected to similar variations; (ii) next, combination of MEL's principle with the first law of thermodynamics yields the adiabatic theorem; subsequent specializations of it lead to additional energetic equations; (iii) the theory is then applied to the one d.o.f. linear and non-linear oscillator; the effects of linear friction and of a harmonic external force are also discussed; useful relations for the adiabatically varying system parameters are thus obtained.     

Harmonic oscillator in Snyder space: The classical case and the quantum case

The harmonic oscillator in Snyder space is investigated in its classical and quantum versions. The classical trajectory is obtained and the semiclassical quantization from the phase space trajectories is discussed. An effective cut-off to high frequencies is found. The quantum version is developed and an equivalent usual harmonic oscillator is obtained through an effective mass and an effective frequency introduced in the model. This modified parameters give us a modified energy spectrum also.     

On solutions of Coulomb system and its generalization to the Aharonov-Bohm effect

The paper numerically analyzes the Aharonov-Bohm effect of an infinitely thin magnetic flux for its influence on a two- or three-dimensional (3d) solutions of Coulomb system in momentum and coordinate spaces. For any definitive eigenstate, it is shown that the flux shifts the position of the most probable radius (MPR) of a probability distribution inward or outward in momentum or coordinate spaces, respectively. Moreover, the probability density of the shifted MPR is amplified in the momentum space, while reduced in the coordinate space. Since the Coulomb force among charged particles dominate the structure of matter, shifting of the MPR controlling by the flux effect may be beneficial to the construction of nanostructure by manipulating the atomic and molecular bonds.     

A generalization of tensor calculus and its applications to physics

Penrose's abstract index notation and axiomatic introduction of covariant derivatives in tensor calculus is generalized to fields with internal degrees of freedom. The result provides, in particular, an intrinsic formulation of gauge theories without the use of bundles.Work supported in part by the NSF Contract No. PHY-80-08155 and a grant from the Research and Equipment Fund, Syracuse University.Supported in part by Crédits Ministériels, Tranche de Type C.     

Quantum theory of orbital magnetization and its generalization to interacting systems

Based on standard perturbation theory, we present a full quantum derivation of the formula for the orbital magnetization in periodic systems. The derivation is generally valid for insulators with or without a Chern number, for metals at zero or finite temperatures, and at weak as well as strong magnetic fields. The formula is shown to be valid in the presence of electron-electron interaction, provided the one-electron energies and wave functions are calculated self-consistently within the framework of the exact current and spin-density functional theory.     

On the generalization of the Enskog equation to the case of steep repulsive potentials

For a classical homogeneous system of particles interacting via steeply repulsive potentials a generalization of the Enskog equation is proposed. This kinetic equation has the properties that it reduces to the usual Enskog equation in the limit of hard-sphere potentials and that the total instead of the kinetic energy is conserved in the system. The expression for the potential energy obtained is correct at arbitrary densities in equilibrium.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. II

We discuss some properties of a non-commutative generalization of the classical moment problem (them-problem) previously introduced. It is shown that there is a connexion between the determination of the problem and the self-adjointness properties in the corresponding Hilbert space. This generalizes the well-known connexion between the determination of the measure in the classical moment problem and the self-adjointness properties of the polynomials as operators in the correspondingL ²-space. The dependence of them-problem on the choice ofC*-semi-norms and on the action of *-homomorphisms is also investigated. As an application, it is shown that if a quantum field (in a very general sense) is essentially self-adjoint then them-problem for the Wightman functional is determined on the quasi-localizableC*-algebra and that the corresponding representation of the localizable algebra generates the bounded observables of the field. It is pointed out that (ultraviolet and spatially) cut-off fields fall in this class and, therefore, are in one to one correspondance with states on the quasi-localizableC*-algebra.Laboratoire associé au Centre National de la Recherche Scientifique.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I.

A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated. Laboratorie associé au Centre National de la Recherche Scientifique.     

A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

The statistical measure of spatial inhomogeneity for n points placed in χ cells each of size k×k is generalized to incorporate finite size objects like black pixels for binary patterns of size L×L. As a function of length scale k, the measure is modified in such a way that it relates to the smallest realizable value for each considered scale. To overcome the limitation of pattern partitions to scales with k being integer divisors of L, we use a sliding cell-sampling approach. For given patterns, particularly in the case of clusters polydispersed in size, the comparison between the statistical measure and the entropic one reveals differences in detection of the first peak while at other scales they well correlate. The universality of the two measures allows both a hidden periodicity traces and attributes of planar quasi-crystals to be explored.     

The Clausius-Mossotti formula and its nonlocal generalization for a dielectric suspension of spherical inclusions

Employing a recently developed cluster expansion for the effective dielectric constant of a suspension of spherical inclusions, we show which parts of the cluster integrals give rise to the Clausius-Mossotti formula. The same selection of terms is then used to obtain an approximate expression for the wave-vector-dependent effective dielectric tensor. For a system of hard spheres with only dipole polarizability this expression is evaluated in closed form. This last result is then used to derive the form of the electrostatic potential due to a point charge in the effective medium. For physically reasonable values of the polarizability, the potential has asymptotically the form corresponding to a medium with the Clausius-Mossotti dielectric constant, while at short range it oscillates about this form. For values of the polarizability beyond the physical range critical points are found at which the oscillations become long range.     

A quantum dot close to Stoner instability: The role of the Berry phase

The physics of a quantum dot with electron–electron interactions is well captured by the so called “Universal Hamiltonian” if the dimensionless conductance of the dot is much higher than unity. Within this scheme interactions are represented by three spatially independent terms which describe the charging energy, the spin-exchange and the interaction in the Cooper channel. In this paper we concentrate on the exchange interaction and generalize the functional bosonization formalism developed earlier for the charging energy. This turned out to be challenging as the effective bosonic action is formulated in terms of a vector field and is non-abelian due to the non-commutativity of the spin operators. Here we develop a geometric approach which is particularly useful in the mesoscopic Stoner regime, i.e., when the strong exchange interaction renders the system close to the Stoner instability. We show that it is sufficient to sum over the adiabatic paths of the bosonic vector field and, for these paths, the crucial role is played by the Berry phase. Using these results we were able to calculate the magnetic susceptibility of the dot. The latter, in close vicinity of the Stoner instability point, matches very well with the exact solution [I.S. Burmistrov, Y. Gefen, M.N. Kiselev, JETP Lett. 92 (2010) 179].     

Strain localization and its connection with the deformed state of the material

The detection of the passage of a shear band over an undeformed material poses a new question about the causal connection between the strain and the formation of shear bands. The collapse of a thick-walled tube is considered from the standpoint of the spall mechanism, according to which localized strain bands under pulsed loading are the result of interference of unloading waves; the negative stresses in the extension zone in this case do not exceed the strength of the material. It is found that radial cracks and their continuations in the form of shear bands appear at the unloading stage after the strained state of the material has already formed as a result of collapse. In other words, damageability is superimposed on the deformed material, and both these processes are independent and accompany each other.