Terms Of Collective Coordinates Research Articles

Stable Bloch Oscillations of Cold Atoms with Time-Dependent Interaction

We investigate Bloch oscillations of interacting cold atoms in a mean-field framework. In general, atom-atom interaction causes dephasing and destroys Bloch oscillations. Here we show that Bloch oscillations are persistent if the interaction is modulated harmonically with suitable frequency and phase. For other modulations, Bloch oscillations are rapidly damped. We explain this behavior in terms of collective coordinates whose Hamiltonian dynamics permits one to predict a whole family of stable solutions. In order to describe also the unstable cases, we carry out a stability analysis for Bogoliubov excitations. Using Floquet theory, we are able to predict the unstable modes as well as their growth rate, found to be in excellent agreement with numerical simulations.

Dynamics of Domain Walls in Magnetic Nanostrips

We express the dynamics of domain walls in ferromagnetic nanowires in terms of collective coordinates, generalizing Thiele's steady-state results. For weak external perturbations the dynamics is dominated by a few soft modes. The general approach is illustrated on the example of a vortex wall relevant to recent experiments with flat nanowires. A two-mode approximation gives a quantitatively accurate description of both the steady viscous motion of the wall in weak magnetic fields and its oscillatory behavior in moderately high fields above the Walker breakdown.

Magnetic vortex dynamics induced by spin-transfer torque

We theoretically study the dynamics of a magnetic vortex under spin-polarized electric current in ferromagnets. The equation of motion of the vortex in terms of collective coordinates is derived. We compare our theory with recent experiments for current-induced vortex displacement and resonance motion in a ferromagnetic nanodot. Our estimate for the displacement and the resonance frequency shows a good agreement with the experiment. We also study the current-induced motion of a vortex wall in a ferromagnetic thin wire.

Threshold Current of Domain Wall Motion under Extrinsic Pinning, β-Term and Non-Adiabaticity

Threshold current of domain wall motion under spin-polarized electric current in ferromagnets is theoretically studied based on the equation of motion of a wall in terms of collective coordinates. Effects of non-adiabaticity and a so-called $\beta$-term in Landau-Lifshitz equation, which are described by the same term in the equation of motion of a wall, are taken into account as well as extrinsic pinning. It is demonstrated that there are four different regimes characterized by different dependence of threshold on extrinsic pinning, hard-axis magnetic anisotropy, non-adiabaticity and $\beta$.

Nuclear alpha-emission—I. consistent quantum–mechanical approach in terms of collective coordinates

Alpha emitting is treated as a particular case of the nucleus decay. The classical Hamiltonian of the nuclear system is constructed, accounting for the Coulomb inner-nuclear interaction and strong interaction approximated by the "Yukava plus exponent" potential. The effective quantization in terms of the collective variable T (elongation) is performed. The effective mass, adjacent to T, depends on T and is of tensor character; its tensor character, is shown, can manifest itself in the collisional experiment as the "dependence of the potential barrier height" on the incident particle energy. The numerical calculations for 210Po show that the mass tensor character is of principal value for the spectrum of the emitted alpha-particles. The possible consequences for the secondary processes in the atomic electron shell and for the electron-positron pair production are discussed.

Scale symmetry of quantum solitons.

The nonlinear {sigma} model Lagrangian for a rotating and vibrating quantum soliton expressed in terms of collective coordinates is shown to possess a symmetry under scale transformation of the chiral field. By utilizing this symmetry, an integro-differential equation determining the chiral angle is obtained. A consistency condition between this equation and the Schroedinger equation for the quantum soliton is discussed, and a relation between the total soliton energy and that related to rotation and vibration is obtained without solving the Schroedinger equation. In the limiting case of only a vibrating or a rotating soliton, the integro-differential equation is reduced to a differential one, and the chiral angle becomes independent of eigensolutions of the relevant Schroedinger equation. The effects of chiral-symmetry breaking due to the pion mass are also examined.

Electromagnetic self-energy of proton and neutron in a chiral soliton model with vector mesons

We calculate the electromagnetic self-energy of the nucleons to lowest-order perturbation theory (one-photon exchange) in a chiral soliton model with vector mesons, including a sum over intermediate states. The matrix elements can be evaluated using wave functions for the baryons given explicitly in terms of collective coordinates. The dependence of the resulting electromagnetic mass splitting between proton and neutron on the values of the input parameters (in our case the pion decay constant f π and the ω coupling constant g ω ) as well as on the definition of the electromagnetic current (Noether method versus current field identity) is exhibited.

Field transformations, collective coordinates and BRST invariance

A very large class of general field transformations can be viewed as a field theory generalization of the method of collective coordinates. The introduction of new variables induces a gauge invariance in the transformed theory, and the freedom left in gauge fixing this new invariance can be used to find equivalent formulations of the same theory. First the Batalin-Fradkin-Vilkovisky formalism is applied to the Hamiltonian formulation of physical systems that can be described in terms of collective coordinates. We then show how this type of collective coordinate scheme can be generalized to field transformations, and discuss the Ward Identities of the associated BRST invariance. For Yang-Mills theory a connection to topological field theory and the background field method is explained in detail. In general the resulting BRST invariance we find hidden in any quantum field theory can be viewed as a consequence of our freedom in choosing a basis of coordinates φ( x) in the action S[ φ].

Surviving and Spurious Yukawa Couplings in the Skyrmion Model

1. It has been pointed out/H) within the present quantization algorithm of the Skyrmion model, that a pion-Skyrmion Yukawa coupling term vanishes from the pion-Skyrmion effective Lagrangian, where an asymptotically free pion field is introduced as quantum fluctuation about a classical Skyimion configuration. It was Skyrme himself who first pointed out this.4) If we have no ~oupling terms linear in the pion field at all, the iI state ~annot decay into N7r, and P-wave pion-nucleon scattering amplitudes do not develop nucleon poles. In order to construct a state with definite spin and isospin values such as Nand ii, we have to rotate the Skyrmion and project it on a definite spin and isospin state. The rotated Skyrmion described in terms of collective coordinates is not an exact solution of the equations of motion. Due to this breaking of the equations of motion at order Nc - a term linear in the pion field survives at order Nc -3/2. The surviving linear term has been regarded as the missing Yukawa coupling which gives rise to the iI decay,5)-8) while the present author9) has claimed that it is hard to regard such a surviving Yukawa coupling as the proper Yukawa couplings which are determined by experiments. On the other hand, there is an argumeneO),ll) that the Yukawa couplings of order INc exist also in a soliton model, but due to the cancellation between the leading terms the resultant Born terms become order 1. If this is the case, the defect of the vanishing Yukawa coupling is an illusion: But, the Born terms appear only as a result of the inadequate application of the reduction formula) to meson-soliton scattering amplitudes. In this paper we give additional arguments against the validity of the surviving Yukawa coupling and also of the spurious Born terms.

Confrontation of macroscopic and microscopic nuclear collective models

Since its start nuclear theory has lived with the dichotomy of viewing the nucleus microscopically, as a system of nucleons, or describing it macroscopically in terms of collective coordinates. In the last decade though, a point transformation has been introduced in which single particle coordinates can be expressed in terms of collective ones plus others, opening the possibility of deriving a microscopic collective model. In the present paper we confront the macroscopic and microscopic collective models, first in a space of two dimensions, in which we find explicitly the unitary representation in quantum mechanics of the canonical transformation that relates them. We then show how to extend every step of the analysis to the three-dimensional problem, though there some of the states required are not yet available in analytic form. One of the fundamental problems in collective models of the nucleus is that of shape. We indicate what are the operators whose expectation values give a reasonable description of the shape in the macroscopic and microscopic collective models and confront them critically.

Three-phonon vertex corrections to the excitation spectrum and liquid-structure function of liquid4He

The energy spectrum of elementary excitations and the liquid-structure function of liquid4He are calculated by means of the method of correlated basis functions in the approximation of second-order perturbation theory. The procedure is based on (i) the construction of phonon functions in terms of collective coordinates and the optimum Bijl-Dingle-Jastrow type of ground-state wave function and (ii) the evaluation of leading correction terms to the Bijl-Feynman excitation spectrum, which are generated by two types of three-phonon vertices. Numerical results are obtained using the optimum liquid-structure function computed by Campbell and Feenberg in the paired-phonon analysis.

The Collective Treatment of a Fermi Gas: I

The collective approach to a Fermi gas is discussed by considering the wave functions of non-interacting Fermi gases in one and three dimensions, and of the interacting Fermi gas. It is shown how to write the one-dimensional case exactly in terms of collective coordinates, and how these have a well defined existence for long wavelengths even without interaction. This is extended as far as possible to three dimensions, and it is shown how the statistics affect the collective motion in the presence of interaction. A brief discussion is given of how calculations may be performed when redundant sets of coordinates are used.

On the foundation of the collective model of the nucleus

An analysis is given of a number of assumptions underlying the collective model of the nucleus of A. Bohr. The connection between properties of the nucleus expressed in terms of collective coordinates and in terms of individual nucleon coordinates is studied. This is done both for the Hamiltonian and for the expressions for the magnetic moment, the quadrupole moment and matrix elements for nuclear transitions. A variational principle is used in order to derive the connection between the Bohr Hamiltonian for the motion of the surface of the core (and the extra nucleons) and the Hamiltonian expressed as an operator acting on the coordinates of the individual nucleons.