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Progress of Theoretical Physics

Optimum Collective Submanifold in Resonant Cases by the Self-Consistent Collective-Coordinate Method for Large-Amplitude Collective Motion

Yukio Hashimoto, Toshio Marumori, Fumihiko Sakata

Published Dec 1, 1987• 4 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.78.1364

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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 applicability of the concept of “optimal” collective submanifold determined by the self-consistent collective-coordinate (SCC) method is investigated by using a simple model Hamiltonian. Long-time behavior of the SCC-method trajectories on the optimal collective submanifold is analyzed, by comparing the geometrical structure of the SCC-method trajectories with that of the time-dependent Hartree-Fock (TDHF) trajectories in terms of the Poincaré-mapping method. The concept of the optimal collective submanifold determined by the SCC method turns out to be applicable for a fairly large domain of the TDHF manifold where the separatrix is not included, in the sense that it can represent an approximate time-averaged trajectory of the complicated TDHF trajectories.
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.
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.
The starting point for most studies of large amplitude collective motion in nuclear physics, the time-dependent Hartree-Fock equations, can be mapped to a problem in classical Hamiltonian mechanics, which is the form of the problem studied in this work. For a system with N degrees of freedom, collective motion is identified with motion completely confined to a surface in K<N dimensions. Conditions for the existence of such decoupled surfaces are worked out and a procedure for constructing them is formulated. For most practical problems, where such surfaces do not strictly exist, a concept of approximate decoupling is developed and a test of its accuracy is described. Our results generalize the method of maximum global decoupling found in the earlier literature.
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.
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.