Binary and multiparticle electron-electron interaction on the surface

It is shown for the first time that electron-electron scattering of slow electrons with an energy of 10–50 eV at the surface of some metals is mainly an event of binary scattering of particles with conserved total momentum and energy, while analogous scattering at the surface of a semiconductor (n-Si) and an insulator (MgO) is a multiparticle event. A model is proposed, in which the electron subsystem of a solid is characterized by short-range order. Each electron is at the center of a spherical cell and surrounded by nearest neighbors (electrons) with a coordination number of 12. The overlap of the fields of charges gives rise to a negative potential U _c (r) ≈ U _c , which is virtually constant along the coordinate and contains spherical cells with a central field U(r) of individual charges. The value of constant negative potential U _c depends on the extent of electron screening, which is high for metals and low for semiconductors and insulators. In metals, scattering governed by the binary mechanism may take place (i.e., scattering of a primary electron in the central field of an electron of the metal); this is ensured by a relatively small value of constant potential U _c . The electron subsystem of the metal behaves as a Fermi gas of weakly interacting quasiparticles. Electron screening in semiconductors and insulators is insignificant, and constant negative potential U _c is an order of magnitude higher than the analogous potential in metals. Slow primary electrons are scattered in the total field of many charges before they reach the central field of an individual electron. The electron subsystem of a semiconductor and an insulator in the excitation range studied here behaves as an ensemble of strongly interacting particles.     

Phase transition of ice precipitated in an aqueous ionic glassy matrix

In an easily forming glass aqueous electrolyte, an ice precipitation appears, that is variable with water to salt mole-ratio R. We here describe the 1 _c → 1 _h phase transition. The initial conditions of these transitions are small crystallites, a condensed form of water, with size varying from clusters to 250 Å. The transition to 1 _h can be considered as healing of the 1_h faults, which is accompanied by an apparent weakening of the cubic structure due to symmetry loss. Order parameter and specific effects, such as branching, are postulated in this growth which appears as a random walk in an ultrametric space.     

Characteristic Polynomials¶of Real Symmetric Random Matrices

The correlation functions of the random variables det(λ−X), in which X is an hermitian N×N random matrix, are known to exhibit universal local statistics in the large N limit. We study here the correlation of those same random variables for real symmetric matrices (GOE). The derivation relies on an exact dual representation of the problem: the k-point functions are expressed in terms of finite integrals over (quaternionic) k×k matrices. However the control of the Dyson limit, in which the distance of the various parameters λ's is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson–Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a finite number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large N limit. Received: 19 March 2001 / Accepted: 21 June 2001     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

Simplified interpretation of interference fringes obtained by time-average holography

A method of evaluating time-average holograms which gives the vibration amplitudes of an object is described. The method is based on the holodiagram, from which a value of sensitivity (k value) is obtained. For sinusoidal vibrations of an object, a_k values are calculated from the roots of the zero order Bessel function; these give the values of the vibration amplitude at the dark interface fringes directly by multiplication with the wave-lenght of the laser light and appropriate k value. a_k values are tabulated for the first twenty fringes. a_He values which incorporate the wavelenght of the commonly used He-Ne laser are also tabulated. The evaluation of a hologram of a metal sheet is described. Finally the difference between the position of fringes obtained from objects with sinusoidal and square-wave amplitudes of vibration is shown.     

Asymptotic equivalence of abstract impulsive differential equations

The notion of (h, k)-dichotomy is introduced, which is a generalization of the classical exponential dichotomy. By means of the Schauder-Tychonoff theorem an asymptotic equivalence is proved between a linear impulsive differential equation which is (h, k)-dichotomous and the corresponding perturbed nonlinear equation.     

A New Invariant For G-Invariant Star Products

The purpose of this Letter is to propose an invariant for a G-invariant star product on a G-transitive symplectic manifold which remains invariant under the G-equivalence maps. This invariant is defined by using a quantum moment map which is a quantum analogue of the moment map on a Hamiltonian G-space. On S ² regarded as an SO(3) coadjoint orbit in , we give an example of this invariant for the canonical G-invariant star product. In this example, there arises a nonclassical term which depends only on a class of G-invariant star products.     

Wigner function and the entanglement of quantized Bessel—Gaussian vortex state of quantized radiation field

A new kind of quantum non-Gaussian state with vortex structure, termed Bessel-Gaussian vortex state, is constructed, which is an eigenstate of sum of squared annihilation operators a² + b². The Wigner function of the quantum vortex state is derived and exhibits negativity which is an indication of nonclassicality. It is also found that quantized vortex state is always in entanglement. And a scheme for generating such quantized vortex states is proposed.     

Optimised polychromatic field-mediated suppression of H-atom tunnelling in a coupled symmetric double well: two-dimensional malonaldehyde model

The cis–cis isomerisation motion of malonaldehyde can be modelled as a symmetric double well coupled with an asymmetric double well, which includes the effect of the cis–trans out-of-plane motion on the cis–cis motion. We have presented an effective method for having control over the tunnelling dynamics of the symmetric double well which is coupled with the asymmetric double well by monitoring tunnelling splitting. When a suitable external field is allowed to interact with the system, the tunnelling splitting gets modified. As the external time perturbation is periodic in nature, the Floquet theory can be applied to calculate the quasi-energies of the perturbed system and hence the tunnelling splitting. The Floquet analysis is coupled with a stochastic optimiser in order to minimise the tunnelling splitting, which is related to slowering of the tunnelling process. The minimisation has been done by one of the stochastic optimisers, simulated annealing. Optimisation has been performed on the parameters which define the external polychromatic field, such as intensities and frequencies of the components of the polychromatic field. With the optimised sets of parameters, we have followed the dynamics of the system and have found suppression of tunnelling which is manifested by a much higher tunnelling time.     

Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl ₂ Witten–Reshetikhin–Turaev invariant, Z _K , at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K ⁻¹ which, for K an odd prime power, converges K-adically to the exact total value of the invariant Z _K at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S ³. The possibility of generalising such results is also discussed. Received: 26 October 1998 / Accepted: 1 March 1999     

Explicit form of the effectiveN-N potential from the quark cluster model

We derive an effectiveN-N potential from a microscopic quark Hamiltonian using the quark cluster model. We construct it in an explicit analytical form, which is expressed only by nuclear variables and which can be used in nuclear structure calculations. To this end we first solve the equation of motion for the six-quark system with a microscopic quark Hamiltonian that includes the quadratic-confinement, one-gluon-, and one-pion-exchange potentials. We then eliminate the quark (internal) degrees of freedom explicitly and express them implicitly in terms of an effectiveN-N potential. The equation of motion for the two-nucleon system is then described by a Schrödinger equation with an effectiveN-N potential. In addition to the one-pion-exchange potential, this effectiveN-N potential contains thequark-exchange potential, which represents the quark-exchange processes associated with a gluon or a pion exchange. This quark-exchange potential is incorporated into the effectiveN-N potential through nonlocal and isospin-dependent terms, which produce a short-range repulsion in theN-N interaction. We give the explicit analytical form of this quark-exchange potential so that it can be used in the nuclear structure calculations.Supported by the DFG under contract number Fa 67/10-5Dedicated to Prof. Erich Schmid on the occasion of his 60th birthday     

The excited states of 1,2,3,4-tetrachloro-7,8-diphenylcalicene; an investigation by a moments analysis of electro-optical spectra

To interpret electro-optical spectra a method based on moments analysis is described which may be particularly useful when an absorption band is associated with more than one electronic transition. An electro-optical study of the absorption spectrum of 1,2,3,4-tetrachloro-7,8-diphenylcalicene is reported. The near-ultraviolet absorption band contains three electronic transitions, an intense z-polarized one is flanked by two much weaker y-polarized transitions. Excited state dipole moments are determined. The dipole moment, which is large in the ground state, increases slightly upon excitation in the z-polarized band whereas it decreases drastically upon excitation in both y-polarized bands. These observations are explained and related to the unsubstituted calicene with the help of molecular orbital calculations.     

Implications of Space-Time Foam for Entanglement Correlations of Neutral Kaons

The role of CPT invariance and consequences for bipartite entanglement of neutral (K) mesons are discussed. A relaxation of CPT leads to a modification of the entanglement which is known as the ω effect. The relaxation of assumptions required to prove the CPT theorem are examined within the context of models of space-time foam. It is shown that the evasion of the EPR type entanglement implied by CPT (which is connected with spin statistics) is rather elusive. Relaxation of locality (through non-commutative geometry) or the introduction of an environment do not by themselves lead to a destruction of the entanglement. One model of the environment, which is based on non-critical strings and D-particle capture and recoil, leads to a specific momentum dependent stochastic contribution to the space-time metric and consequent change in the neutral meson bipartite entanglement. Although the class of models producing the omega effect is non-empty, the lack of an omega effect is demonstrated for a wide class of models based on thermal like baths which are often considered as generic models appropriate for the study of space-time foam.     

Equilibration,Generalized Equipartition,and Diffusion in Dynamical Lorentz Gases

We demonstrate approach to thermal equilibrium in the fully Hamiltonian evolution of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a d-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We analytically predict, and numerically confirm, that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature T, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics—they need not be in a state of thermal equilibrium. The temperature T to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy k _B T is equal to an appropriately defined average of the scatterers’ kinetic energy. In the equilibrated state, particle motion is diffusive.     

An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

Contribution to the theory of Lorentzian ionization

The probability w _L of Lorentzian ionization, which arises when an atom or ion moves in a constant magnetic field, is calculated in the quasiclassical approximation. The nonrelativistic (v≲e ²/ℏ=1, v is the velocity of the atom) and ultrarelativistic (v→c=137) cases are examined and the stabilization factor S, which takes account of the effect of the magnetic field on tunneling of an electron, is found. Pis’ma Zh. éksp. Teor. Fiz. 65, No. 5, 391–396 (10 March 1997)     

New nondestructive tests,from frames

The usefulness (Lubkin, 1976; Lubkin and Lubkin, 1979) ofB-plexes, an extension of Dirac's familiar observable (Dirac, 1947), is illustrated in a computation of all nondestructive tests over a system of finite ket-space dimensionn. These are tests whoseB outcomes are proportional to corresponding pure final states. The unitary motions of the system and aB-dimensional register, which effect such tests, depend very simply onn vectors from a frame inB-space. CaseB>n is not empty, and so presents nondestructive testing with nonorthogonal final states. The step where reduction of the wave packet happens, equivalently, computation of an Everett relative state, is given in an Appendix.     

GeneralizedM-diffusion model of molecular rotations

The widely usedM-model of rotational diffusion of molecules in fluid phases is generalized. The ordinaryM-model assumes that intermolecular collisions causeinstantaneous changes in the orientation of an otherwise free rotor. The present scheme takes cognizance of the ubiquitous intermolecular torques which should make the molecular orientation a continuously variable random function of time. It is assumed here that the component of the angular velocity, which is conjugate to the angle specifying the orientation of the molecule, is a stationary Gaussian-Markov process. The ordinaryM-model emerges then as a special case of the more general treatment presented here. The results derived for the dipole correlation function of a linear rotor on the basis of the generalized scheme are applied to a series of infrared data. The observed agreement is highly satisfactory. The present analysis affords a justification for the Gordon scheme which generalizes theM-model by assigning to the mean rate of collision anad-hoc dependence on the angular speed of the rotor. It is argued also that the model treated here incorporates certain memory effects which are ignored in the ordinaryM-model, and may yield, in some cases, results which are similar to those based on certain memory function formalisms.