Table of contents

We study the dispersion of the `temporally stable' coherent states for the hydrogen atom introduced by Klauder. These are states which under temporal evolution by the hydrogen atom Hamiltonian retain their coherence properties. We show that in the hydrogen atom such wavepackets do not move quasiclassically; i.e. they do not follow, with no or little dispersion, the Keplerian orbits of the classical electron. The poor quantum - classical correspondence does not improve in the semiclassical limit.

Voronoi tessellations of scale-invariant fractal sets are characterized by topological and metrical properties that are significantly different from those of natural cellular structures. As an example we analyse Voronoi diagrams of intermittent particle distributions generated by a directed percolation process in 2 + 1 dimensions. We observe that the average area of a cell increases much faster with the number of its neighbours than in natural cellular structures where Lewis' law predicts a linear behaviour. We propose and numerically verify a universal scaling law that relates shape and size of the cells in scale-invariant tessellations. An exponent, related to the topological properties of the tessellation, is introduced and estimated numerically.

We consider classical and quantum electromagnetic fields in a three-dimensional cavity and in a waveguide with oscillating boundaries of the frequency . The photons created by the parametric resonance are distributed in the wavenumber space around along the axis of the oscillation. When classical waves propagate along the waveguide in the one direction, we observe the amplification of the original waves and another wave generation in the opposite direction by the oscillation of side-walls. In the case of two opposite walls oscillating with the same frequency but with a phase difference, the interferences are shown to occur because of the phase difference in the photon numbers and in the intensity of the generated waves.

We investigate the statistics of the mean magnetization, of its large deviations and persistent large deviations in simple coarsening systems. In particular we consider more specifically the case of the diffusion equation, of the Ising chain at zero temperature and of the two-dimensional voter model. For the diffusion equation, at large times, the mean magnetization has a limit law, which is studied analytically using the independent interval approximation. The probability of persistent large deviations, defined as the probability that the mean magnetization was, for all previous times, greater than some level x, decays algebraically at large times, with an exponent continuously varying with x. When x = 1, is the usual persistence exponent. Similar behaviour is found for the Glauber-Ising chain at zero temperature. For the two-dimensional voter model, large deviations of the mean magnetization are algebraic, while the probability of persistent large deviations seem to behave as the usual persistence probability.

A difference equation is presented to describe traffic flow on a highway. The difference equation model is derived from the optimal velocity models formulated in terms of the differential equations. It is compared with the differential equation models. We investigate phase transitions among the freely moving phase, the coexisting phase in which jams appear, and the uniform congested phase. The linear stability theory is applied and the neutral stability line is obtained. We find the critical point below which no jams appear. To derive the modified Korteweg-de Vries equation near the critical point we apply the reductive perturbation method. We also compare the nonlinear analysis result with that of the optimal velocity model. It is shown that the critical point and the amplitude of the jam are different from those of the optimal velocity model.

In this paper, we propose an ultradiscrete Burgers equation of which all the variables are discrete. The equation is derived from a discrete Burgers equation under an ultradiscrete limit and reduces to an ultradiscrete diffusion equation through the Cole-Hopf transformation. Moreover, it becomes a cellular automaton (CA) under appropriate conditions and is identical to rule-184 CA in a specific case. We show shock wave solutions and asymptotic behaviours of the CA exactly via the diffusion equation. Finally, we propose a particle model expressed by the CA and discuss a mean flux of particles.

Exact dynamical properties are discussed for the ± J Ising model around the multicritical point (MCP). It is found that the relation of relaxations between the non-equilibrium remanent magnetization m(t) and the equilibrium autocorrelation function q(t) at the MCP is different from that at the pure critical point. The dynamic critical exponent for the ferromagnetic ordering defined by and that for the spin glass ordering defined by become identical at the MCP. Accurate numerical calculations for them are performed in two and three dimensions using the non-equilibrium relaxation analysis. The MCP is located at and the exponent is estimated as for the square lattice. They are estimated as and for the simple cubic lattice.

Tensor operators for the Jordanian quantum algebra are considered. Some explicit examples of them, which are obtained in the boson or fermion realization, are given and their properties are studied. It is also shown that the Wigner-Eckart theorem can be extended to .

Hertz-Debye potentials are generally used for time-harmonic fields. We are interested here in arbitrary time-dependent pulses tackled with the help of the Laplace transform, and in addition, we assume that propagation takes place in a chiral reciprocal medium. We show how the Hertz-Debye potentials work to solve the Laplace transformed Maxwell equations and illustrate this formalism with three applications. Finally, we discuss how to obtain the solution in the time domain through the inverse Laplace transform.

As is well known, the type 1 Lie superalgebra admits a one-parameter family of finite-dimensional irreducible representations. We have carried out an analytic Bethe ansatz related to this family of representations. We present formulae, which are deformations of previously proposed determinant formulae labelled by a Young superdiagram. These formulae will provide a transfer matrix eigenvalue in a dressed vacuum form related to the solutions of a graded Yang-Baxter equation, which depend not only on the spectral parameter but also on a non-additive continuous parameter. A class of transfer matrix functional relations among these formulae is briefly mentioned.