Rare transition event with self-consistent theory of large-amplitude collective motion

A numerical simulation method, based on Dang et al.’s self-consistent theory of large-amplitude collective motion, for rare transition events is presented. The method provides a one-dimensional pathway without knowledge of the final configuration, which includes a dynamical effect caused by not only a potential but also kinetic term. Although it is difficult to apply the molecular dynamics simulation to a narrow-gate potential, the method presented is applicable to the case. A toy model with a high-energy barrier and/or the narrow gate shows that while the Dang et al. treatment is unstable for a changing of model parameters, our method stable for it.     

Equations-of-motion approach to quantum mechanics: application to a model phase transition

We present a generalized equations-of-motion method that efficiently calculates energy spectra and matrix elements for algebraic models. The method is applied to a five-dimensional quartic oscillator that exhibits a quantum phase transition between vibrational and rotational phases. For certain parameters, 10 x 10 matrices give better results than obtained by diagonalizing 1000 x 1000 matrices.     

Biunitary representation of the selfconsistent collective-coordinate method for large-amplitude collective motion

A reformulation of the selfconsistent collective-coordinate method of Marumori, Maskawa, Sakata and Kuriyama is given in a biunitary representation for the purpose of studying the relation to the generalized Dyson boson mapping. It is shown that, in this reformulation, a special choice of representation gives the c-number version of the generalized Dyson boson mapping. It is also shown that this representation yields a powerful method to estimate the coupling effect between collective and non-collective phonon degrees of freedom in the large-amplitude collective motion.     

Microscopic derivation of a quantum hamiltonian for adiabatic collective motion

The generalized density matrix method is applied to the microscopic derivation of the adiabatic collective hamiltonian. By way of construction, the hamiltonian obtained is the quantum operator in the collective variable space. For the calculation of the hamiltonian parameters, the combination of the density matrix equations of motion and the collective motion saturation principle is used. The constraint (driving force) in the equations of motion is determined self-consistently. For illustrative purposes two simple solvable models are considered.     

Classical approach to quantum theory

A formulation of nonrelativistic, spinless, quantum mechanics is presented which is based on four postulates. Three of the postulates are very analogous to relations that hold in an operator formulation of classical mechanics, and the fourth is that the wave function evolves linearly in time. The conventional statistical assertions of quantum theory as well as the Schrödinger equation are recovered.     

Application of a self-consistent theory of large amplitude collective motion to the generalized meshkov-glick-lipkin model

A recent formulation of the theory of large amplitude collective motion in the adiabatic limit is applied to a generalized monopole shell model. Numerical calculations are carried out for the three-level model, approximately equivalent to a classical system with two degrees of freedom. Our results go somewhat beyond previous treatments of this system and provide substantiation for the validity of the method, in suitable parameter ranges, as a way of recognizing and decoupling the collective and the non-collective degrees of freedom.     

Semiclassical field theory approach to quantum chaos

We construct a field theory to describe energy averaged quantum statistical properties of systems which are chaotic in their classical limit. An expression for the generating function of general statistical correlators is presented in the form of a functional supermatrix non-linear σ-model where the effective action involves the evolution operator of the classical dynamics. Low-lying degrees of freedom of the field theory are shown to reflect the irreversible classical dynamics describing relaxation of phase space distributions. The validity of this approach is investigated over a wide range of energy scales. As well as recovering the universal long-time behavior characteristic of random matrix ensembles, this approach accounts correctly for the short-time limit yielding results which agree with the diagonal approximation of periodic orbit theory.     

On the theory of collective motion in a system of interacting particles

Collective variables are introduced: coordinates and momenta, which in fact are canonical variables. These variables are used to write a Hamiltonian describing collective motions in a classical system of interacting particles.The author is grateful to K. A. Valiev for his supervision and constant interest in the work.     

A study of collective paths in the time-dependent hartree-fock approach to large amplitude collective nuclear motion

In a simple two-dimensional landscape model the collective paths of Rowe-Bassermann and Marumori, of Villars, of Goeke-Reinhard, and of Baranger-Veneroni are determined and compared. The uniqueness of the solutions and the practicability of the methods regarding numerical applications are discussed.     

Quantum-trajectory approach to cavity quantum electrodynamics with up to three-atom collective effects

We present a quantum-trajectory treatment of the dynamics of a high-Q cavity mode interacting with a low-density atomic beam in the strong coupling regime. We consider up to three-atom collective effects, and evaluate their contribution to the destabilization of cavity field trapping states for mean number of atoms in the cavity both much smaller than one (micromaser or microlaser), and on the order of one (approaching a mesoscopic regime).     

Constrained quantum mechanics and a coordinate independent theory of the collective path

The settings for two formulations of quantum mechanics are, respectively, Hilbert spaces and symplectic manifolds. The former leads naturally to matrix mechanics and, for example, the shell model while the latter leads to hamiltonian mechanics, of which the time-dependent Hartree-Fock theory is a standard example. In order to obtain practical approximate theories one needs to restrict the dynamics in both cases to suitable finite-dimensional subspaces. This paper addresses the problem of constructing subspaces of the projective Hilbert space, the fundamental symplectic manifold of quantum mechanics. The collective paths of Villars, Goeke and Reinhard, the valley path and the collective path and submanifold of Rowe and Basserman are examined and phrased in a coordinate independent manner. In this way we expose the dynamical foundations and the essential geometrical structures upon which they are based.     

Uniqueness of the collective submanifold in the adiabatic theory of large amplitude collective motion

It is emphasized that the conditions for the existence of a collective submanifold which follow from adiabatic time-dependent Hartree-Fock theory are precisely the conditions for the existence of a manifold of solutions of Hamilton's equations confined to a surface of reduced dimensionality. A constructive procedure, valid in any number of dimensions and involving the concept of the multidimensional valley, is developed to determine whether a given system admits such a manifold. It is extended to include the idea of the approximate manifold, and an application to a generalized landscape model is described.