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Progress of Theoretical Physics

Generalization of Equation of Collective Submanifold: A Theory of Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom

M. Yamamura, A. Kuriyama

Published Apr 1, 1984• 2 citations
Publisher

Abstract

A theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.A canonical transformation is investigated in the full time•dependent Hartree•Fock theory, i.e., in the classical image of boson expansion theory.With the aid of the transfor• mation, the whole system is separated into collective and intrinsic degrees of freedom.Under the condition of the unique separation, two types of equations of collective submanifold, which are canonically invariant, are obtained.One is of the same form as that of the conventional equation of collective submanifold.A principle of the specification of coordinate system is discussed.

DOI
10.1143/ptp.71.752

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A theory based on the time•dependent Hartree•Fock theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.With the use of time•dependent canonical transformation, the whole system is separated into two types of degrees of freedom.Under the condition of unique separation, two kinds of equations of collective submanifold are obtained.One js for the collective motion and of the same form as that previously given by the present authors.The other is for the intrinsic degrees of freedom and consists of a set of linear partial differential equations.It is reduced to the old one, the form of which is non-linear.

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Progress of Theoretical Physics•Aug 1, 1983•20 citations
A theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.A canonical transformation is investigated in the full time•dependent Hartree•Fock theory, i.e., in the classical image of boson expansion theory.With the aid of the transfor• mation, the whole system is separated into collective and intrinsic degrees of freedom.Under the condition of the unique separation, two types of equations of collective submanifold, which are canonically invariant, are obtained.One is of the same form as that of the conventional equation of collective submanifold.A principle of the specification of coordinate system is discussed.
A theory based on the time•dependent Hartree•Fock theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.With the use of time•dependent canonical transformation, the whole system is separated into two types of degrees of freedom.Under the condition of unique separation, two kinds of equations of collective submanifold are obtained.One js for the collective motion and of the same form as that previously given by the present authors.The other is for the intrinsic degrees of freedom and consists of a set of linear partial differential equations.It is reduced to the old one, the form of which is non-linear.
From the viewpoint defferent from that of Part I, the time-dependent Harteree-Fock theory is formulated for describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. The basic idea consists of the introduction of an additional degree of freedom for the collective motion and the use of Dirac’s canonical theory with constraints. The formalism is essentially equivalent to the ordinary time-dependent Hartree-Fock theory presented in Part I. The result given in Part I is completely reproduced.
A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.
A systematic microscopic theory of large-amplitude collective motion, which has been recently developed within the framework of the time-dependent Hartree-Fock (TDHF) theory under INS-TSUKUBA joint research project, is reformulated in terms of classical mechanics in the TDHF symplectic manifold. This reformulation clarifies an essential concept in the theory, i.e., an approximate integral surface in the TDHF manifold, which is identified as a dynamical collective submanifold for the large-amplitude collective motion. It is shown that the approximate integral surface must satisfy three dynamical conditions; (a) a stationary condition (i.e., a maximal-decoupling condition), (b) a stability condition and (c) a separability condition between collective and “local” non-collective modes of motion. These three dynamical conditions make clear the concept of “local” intrinsic modes of motion and guarantee that the TDHF-trajectory representing the large-amplitude collective motion under consideration is successfully reproduced on the approximate integral surface.
With the purpose of clarifying characteristic difference of the optimum collective submanifolds between in nonresonant and resonant cases, we propose an improved method of solving the basic equations of the self-consistent collective-coordinate (SCC) method, which describes optimum (“maximally-decoupled”) large-amplitude collective motion within the time-dependent Hartree-Fock theory. It is shown that, in the resonant cases, there inevitably arise essential coupling terms which break the maximal-decoupling property of the collective motion, so that we have to extend the optimum collective submanifold so as to properly treat the degrees of freedom bringing about the resonances. An illustrative example is given with a simple model Hamiltonian.
The geometry of the self-consistent collective-coordinate (SCC) method formulated within the frame-work of the time-dependent Hartree-Fock (TDHF) theory is investigated by associating the variational parameters with a symplectic manifold (a TDHF manifold). With the use of a canonical-variable parametrization, it is shown that the TDHF equation is equivalent to the canonical equations of motion in classical mechanics in the TDHF manifold. This enables us to investigate geometrical structure of the SCC method in the language of the classical mechanics. The SCC method turns out to give a prescription how to dynamically extract a “maximally-decoupled” collective submanifold (hypersurface) out of the TDHF manifold, in such a way that a certain kind of trajectories corresponding to the large-amplitude collective motion under consideration can be reproduced on the hypersurface as precisely as possible. The stability of the hypersurface at each point on it is investigated, in order to see whether the hypersurface obtained by the SCC method is really an approximate integral surface in the TDHF manifold or not.