Microscopical structure of the states of deformed nuclei in the neighborhood of the yrast line

A simple model is derived which allows one to study the structure of the nuclear states in the neighborhood of the “yrast” band. In the present scheme the precession motion plays a role of one of the normal modes of oscillations. (The structure of the dispersion equation for this mode corresponds to the well known classical formula.) Vibrational states associated with quadrupole oscillations of the nuclear shape are determined from a general equation. At slow rotation this equation breaks up into the known equations for β-, Δ- and γ-vibrations and non-collectivized K^π = 1⁺ excitations.     

Quadrupole oscillations in a quantum degenerate Bose–Fermi mixture

We study the collective dynamics in a degenerate Bose–Fermi mixture of ¹⁷⁴Yb and ¹⁷³Yb atoms. We excite collective oscillations by a sudden reduction of the trapping confinement and observe low m=0 quadrupole oscillations of condensates in ¹⁷⁴Yb. First the oscillations in ¹⁷⁴Yb atoms alone are investigated, and they are well described by the time-dependent Gross–Pitaevskii equation in the Thomas–Fermi approximation. Using the same procedure the quadrupole oscillations are excited for a ¹⁷⁴Yb–¹⁷³Yb Bose–Fermi mixture. In comparing data taken with and without fermionic ¹⁷³Yb atoms, the oscillation frequency of the quadrupole mode in the condensate decreases with the presence of ¹⁷³Yb atoms.     

Instabilities and vortex-lattice formation in rotating conventional and dipolar dilute-gas Bose-Einstein condensates

A theoretical study of vortex-lattice formation in atomic Bose-Einstein condensates confined by a rotating elliptical trap is presented. For the conventional case of purely s-wave interatomic interactions, this is done through a consideration of both hydrodynamic equations and time-dependent simulations of the Gross-Pitaevskii equation. We discriminate three distinct, experimentally testable regimes of instability: ripple, interbranch, and catastrophic. Additionally, we generalize the classical hydrodynamical approach to include long-range dipolar interactions, showing how the static solutions and their stability in the rotating frame are significantly altered. This enables us to examine the routes towards unstable dynamics, which, in analogy to conventional condensates, may lead to vortex-lattice formation.     

Solution of the Vlasov equation for collective modes in nuclei

A semiclassical theory of giant resonances based on the Vlasov equation is developed. The linearized Vlasov equation is solved for the bound motion of particles in a central potential with an external time-dependent multipole field. The solution obeys an RPA-type integral equation. If the time-dependent part of the self-consistent field is neglected, the solution of the Vlasov equation has a simple analytical form. The strength function for each multipole can be expressed in terms of the natural frequencies of classical orbits and of radial integrals over the classical motion. The method is illustrated by studying the isoscalar monopole, quadrupole and octupole response in medium-heavy nuclei without residual interaction. There are remarkable similarities between the solutions of the semiclassical problem and the corresponding quantum problem. For a central potential with Saxon-Woods shape there is an interesting shift and concentration of strength in the quadrupole and octupole response functions.     

The domain structure of a nematic liquid crystal layer in an oscillating poiseuille flow

A spatially periodic structure arising in a nematic liquid crystal layer with planar orientation under the effect of an oscillating Poiseuille flow is described theoretically. The effect is analyzed on the basis of hydrodynamic equations of nematic liquid crystals, from which a self-consistent set of equations for perturbations of hydrodynamic variables is separated. It is demonstrated that the structure type and the threshold parameters of the effect depend on the frequency and the layer thickness through the scaling combination ωh ². The dependence of the configuration of arising distortions on the value of viscosity α₃ is analyzed.     

Nuclear viscosity and widths of giant resonances

We write down a set of coupled hydrodynamic equations of the Navier-Stokes type which describe the motion of two compressible, viscous nuclear fluids. The solutions of these equations give rise to giant resonances of both isoscalar and isovector type. The viscosity terms in the equations are responsible for the damping of these resonances. Within this framework we obtain expressions for the width of the resonances as a function of the mass number A, and relations between the widths and the excitation energies for various multipolarities (J = 0⁺, 1^?, 2⁺, 3^?, 4⁺), and isospins (T = 0,1). The A dependence of the calculated widths exhibit the experimental trends of the giant dipole and isoscalar quadrupole widths. Also, as a result of the calculation we obtain estimates of the values of viscosity coefficients in nuclei.     

Inertial mechanism of domain formation in a homeotropic nematic liquid crystal under periodic shear

Formation of a domain structure in a homeotropic liquid crystal layer under the action of a periodic Couette flow is studied. The effect is analyzed using hydrodynamic equations for the nematic liquid crystal. Comparison of the results of calculations with experimental data testifies that, at high frequencies, the formation of domains is governed by inertial effects. It is shown that the inertial mechanism leads to a scaling dependence of the threshold shear amplitude on frequency ω and layer thickness h in the form u _th ～ (ωh ²)^?1.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

A minimum photon “rest mass” — Using Planck's constant and discontinuous electromagnetic waves

Reasons for taking1/2h/c ² in cgs units as an equivalent in grams for the photon “rest mass” are given. Its numerical value of3.68×10 ^?48 g corresponds to the minimum mass equivalent energy for one half-cycle of an electromagnetic dipole field distribution, which is discontinuous. For the fluid models that are discussed, this field distribution corresponds somewhat to a hydrodynamic toroidal vortex which is stationary—if we use toroidal coordinates and assume that the ring origin has the radial velocity c, that the gauge is defined by the ring origin diameter, and that free space is represented by a two-fluid model (the fluids oppositely charged). There are mappings which can transform such toroidal entities (photons) into spherical ones. The toroidal entities are possible candidates for the role of “hidden variables.”     

Form-preserving Transformations for the Time-dependent Schrödinger Equation in (n + 1) Dimensions

We define a form-preserving transformation (also called point canonical transformation) for the time-dependent Schrödinger equation (TDSE) in (n+1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L ²-normalizability. We give a class of time-dependent TDSEs that can be mapped onto stationary Schrödinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton–Krylov–Schwarz

Numerical simulations of the four-field extended magnetohydrodynamics (MHD) equations with hyper-resistivity terms present a difficult challenge because of demanding spatial resolution requirements. A time-dependent sequence of r-refinement adaptive grids obtained from solving a single Monge–Ampère (MA) equation addresses the high-resolution requirements near the x-point for numerical simulation of the magnetic reconnection problem. The MHD equations are transformed from Cartesian coordinates to solution-defined curvilinear coordinates. After the application of an implicit scheme to the time-dependent problem, the parallel Newton–Krylov–Schwarz (NKS) algorithm is used to solve the system at each time step. Convergence and accuracy studies show that the curvilinear solution requires less computational effort than a pure Cartesian treatment. This is due both to the more optimal placement of the grid points and to the improved convergence of the implicit solver, nonlinearly and linearly. The latter effect, which is significant (more than an order of magnitude in number of inner linear iterations for equivalent accuracy), does not yet seem to be widely appreciated.     

Semiclassical stochastic trajectory approach to molecule-solid surface scattering

A semiclassical stochastic trajectory (SST) approach to the sudy of collision induced transitions in gas molecule-solid surface scattering is presented. The time-dependent Schrödinger equation provides the time-evolution of the transition amplitudes for the molecular internal states. Classical mechanics is used to describe the molecule's center of mass motion as well as the surface atoms' motion — the latter through the generalized Langevin equation (GLE) method which allows the treatment of non-rigid surfaces (i.e. surface temperature effects). These quantum and classical equations of motion are coupled through the use of a time-dependent interaction potential in the Schrödinger equation and the use of the expectation value of the interaction potential in the classical equations of motion. Advantages of the SST approach include: (1) flexibility in the choice of quantum versus classical coordinates; (c) strict energy conservation for non-dissipative system; and (3) realistic treatment of surface many-body effects within the GLE. The SST technique is applied to the study of vibrational and rotational inelasticity in a model H₂Pt(111) system. As an initial test, results obtained assuming a rigid, smooth surface with an exponentially repulsive potential are compared to exact quantal and quasi-classical trajectory values to determine the accuracy and utility of the SST approach. A limited practical application is presented for the same H₂Pt(111) system but for a non-rigid surface. These results, calculated at low gas kinetic energies, indicate that surface energy transfer and surface temperature effects should be minimal for this type of system, even though the energy gaps are quite similar for rotational and phonon degrees of freedom.     

Critical value computation for H∞-Riccati difference equations

In designing finite horizon discrete time H_∞ controllers, the associated H_∞-Riccati difference equations must be solved. But the Riccati equation has a non-negative solution only when γ⁻² is small enough. So it is important to get the upper bound of the parameter, i.e., the critical value that ensures the existence of the solution to the Riccati equation. The solution sequence of the Riccati difference equation can be constructed by the conjoined basis of an associated linear Hamiltonian difference system. Based on this expression and the Hamiltonian difference system eigenvalue theorems, the equivalence between the critical value and the first order eigenvalue of the linear Hamiltonian difference system is presented. Since the critical value is also shown to be the fundamental eigenvalue of a generalized Rayleigh quotient, an extended form of Wittrick-Williams algorithm is presented to search this value.     

Exact statements about many-particle systems with higher symmetries

A new method of calculating the energy spectrum of a system of A identical Fermi particles with translationally invariant interaction is developed under the assumption that there exists a high symmetry in the 3A-dimensional space of particle coordinates. For a special class of symmetries the many-body problem is split exactly into two sets of equations: one containing only totally symmetric combinations of the particle coordinates which are called “collective variables” and the other equation taking essentially into account the requirements of the Pauli principle and connected symmetry properties. In several cases it is possible to obtain the excitation spectra exactly showing qualitatively new features. They depend on “many-particle quantum numbers” varying independently of each other in an interval which sometimes depends on A. For special high symmetries the collective variables obey equations which are very similar to one-particle equations providing a new explanation of the “Independent-Particle Model” for arbitrary strength and form of the interaction potential. A manifold of unknown up to now excitation spectra of many-particle systems is obtained and discussed.     

Supersymmetry and classical solutions

We obtain classical solutions to the field equations of the massless supersymmetric Wess-Zumino model and to the field equations of the interacting SU(2) gauge supermultiplet. This is done by applying finite supersymmetry transformations to the known solutions of the scalar field equation with ?⁴ interaction and the Yang-Mills field equations. The relevance of supersymmetry to the solution of classical field equations involving anticommuting fermion fields is discussed.     

Geometrical relationship between the classical and quantum descriptions of arbitrary force interactions

The paper is a continuation of a series papers by the author in which it is shown that the equations of contemporary quantum theory (the Klein-Gordon equation and the Dirac integral equation) can be interpreted as conditions of harmonicity of linear form in the manifold^sV₄. In this paper, geometrization of classical and quantum interactions of a physical object is performed for the case of an arbitrary external force field in the manifold^kV₄. This manifold is constructed on the same set of surfaces as the manifold^sV₄. The equation representing conditions of harmonicity of linear form in^kV₄ is derived and is analogous to the KleinGordon equation.