Interrelationship between kinetics and local hydrodynamics for states far from equilibrium

This study examines the kinetic theory of a nonrelativistic gas with internal degrees of freedom in the absence of hydrodynamic sources. The author showed previously for the case of weak sources that the kinetic theory is equivalent to nonlocal hydrodynamics. Here, this result is generalized to the case of sources of arbitrary intensity, when the theory is nonlinear. The nonlocality kernels that appear in the material relations are calculated from the kinetic equation. The methods of perturbation theory are used to find a procedure for successive calculation of the kernels that appear in the higher nonlinear terms. Institute of Earth Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, p. 67–71, August, 1996.     

Quantum chaos and properties of eigenstates

It is argued that the eigenstates of a Hamiltonian which gives rise to a GOE-type spectrum are only weakly correlated and that the individual eigenstates are expected to be highly sensitive against a generic perturbation. A simple mathematical model is used for demonstration.     

Exact phase space localized projectors from energy eigenstates

In investigations of the emergence of classicality from quantum theory, a useful step is the construction of quantum operators corresponding to the classical notion that the system resides in a region of phase space. The simplest such constructions are approximate projection operators. Here, we show how to construct exact projection operators which are localized on regions of phase. We elucidate the properties of such operators and explore their time evolution. For the harmonic oscillator we find sets of phase space localized histories which are exactly decoherent for any initial state and have probability 1 for classical evolution.     

Solution to the eigenstates of pairing Hamiltonian in finite nuclei

We apply the algebraic Bethe technique to the nuclear pairing problem with certain limits. We obtain the exact energies and eigenstates, and find the symmetry between the states corresponding to less and more than half full shell. We also proved that the problem of solving BAE can be transformed into the problem of finding the roots of a hypergeometric polynomial, which is much simpler.     

Quantum bound states with zero binding energy

After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the Schrödinger equation with E = 0 for the class of potentials V = −|γ|/r^ν, where −∞ < ν < ∞. For ν > 2, these solutions are normalizable and correspond to bound states, if the angular momentum quantum number l > 0. (These states are normalizable, even for l = 0, if we increase the space dimension, D, beyond 4; i.e for D > 4.) For ν < −2 the above solutions, although unbound, are normalizable. This is true even though the corresponding potentials are repulsive for all r. We discuss the physics of these unusual effects.     

Perturbative and nonperturbative parts of eigenstates and local spectral density of states: the wigner-band random-matrix model

The Wigner-band random-matrix model is studied by making use of a generalization of Brillouin-Wigner perturbation theory. Energy eigenfunctions are shown to be divided into perturbative and nonperturbative parts. Several perturbation strengths predicted by the perturbation theory are found to play important roles in the variation of the shape of the local spectral density of states with perturbation strength.     

Projection of a quantum state on eigenstates of average Hamiltonian

A scheme for projecting an arbitrary quantum state on eigenstates of average Hamiltonian is described. As an experimental example, projection on entangled Bell states, which are eigenstates of specially constructed average Hamiltonian, is demonstrated for a system of two dipolar-coupled nuclear spins. The results of a direct and time-reversed evolution are added to average out the coherences between different eigenstates and accomplish the projection.     

Two-sided bounds on the free energy from local states in Monte Carlo simulations

We show that a precise assessment of free energy estimates in Monte Carlo simulations of lattice models is possible by using cluster variation approximations in conjunction with the local states approximations proposed by Meirovitch. The local states method (LSM) utilizes entropy expressions which recently have been shown to correspond to a converging sequence of upper bounds on the thermodynamic limit entropy density (i.e., entropy per lattice site), whereas the cluster variation method (CVM) supplies formulas that in some cases have been proven to be, and in other cases are believed to be, lower bounds. We have investigated CVM-LSM combinations numerically in Monte Carlo simulations of the two-dimensional Ising model and the two-dimensional five-states ferromagnetic Potts model. Even in the critical region the combination of upper and lower bounds enables an accurate and reliable estimation of the free energy from data of a single run. CVM entropy approximations are therefore useful in Monte Carlo simulation studies and in establishing the reliability of results from local states methods.     

Quantum and classical relativistic energy states in stationary geometries

The positive and negative root states (E⁺, E^?) for a particle moving along a geodesic in a stationary background, introduced by Christodoulou and Ruffini, are here interpreted in the framework of a relativistic quantum field theory. It is shown how E⁺ and E^? have to be considered as the classical correspondent of the positive and negative energy states of a quantized field. It is explicitly shown that crossing between the states E⁺ and E^? can occur and consequently the necessary condition for particle creation as given by Klein, Sauter, Heisenberg and Euler can be encountered.     

Quantum Hamiltonian for the Rigid Rotator

An excess term exists when using hermitian form of Cartesian momentum p _i (i = 1,2,3) in usual kinetic energy 1/(2) p ² _i for the rigid rotator, and the correct kinetic energy turns to be 1/(2) (1/f _i ) p _i f _i p _i where f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics.     

Quantum tomograms from intermediate entangled states

We show that for quantum tomography there exist two mutually conjugating intermediate coordinate-momentum entangled states |η₁,η_2〉λ,ν and |?₁,?_2〉σ,τ. The Radon transforms of the Wigner operators are just the pure-state density matrices and , respectively. As a result, the tomogram of quantum states is the module-square of their wave function in these representations. A new convenient formalism of quantum tomogram is thus established.     

Non-integrability of the truncated Toda lattice Hamiltonian at any order

Two sufficient conditions for the non-existence of an additional analytic integral are given for Hamiltonian systems with non-homogeneous polynomial potential of an arbitrary degree. An application is made to the truncated three-particle Toda lattice, which is proved to be non-integrable at any order.     

Probing the photonic local density of states with electron energy loss spectroscopy

Electron energy loss spectroscopy performed in transmission electron microscopes is shown to directly render the photonic local density of states with unprecedented spatial resolution, currently below the nanometer. Two special cases are discussed in detail: (i) 2D photonic structures with the electrons moving along the translational axis of symmetry and (ii) quasiplanar plasmonic structures under normal incidence. Nanophotonics in general and plasmonics, in particular, should benefit from these results connecting the unmatched spatial resolution of electron energy loss spectroscopy with its ability to probe basic optical properties such as the photonic local density of states.