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Physical Review C

Determination of the collective Hamiltonian in a self-consistent theory of large amplitude adiabatic motion

G. Do Dang, Aurel Bulgac, Abraham Klein

Published Dec 1, 1987• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physrevc.36.2661

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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 classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.
Villars's equations, first derived from the adiabatic limit of time-dependent Hartree-Fock theory, are presented here as a set of exact conditions for the decoupling (in the adiabatic limit) of a sub-system from a classical Hamiltonian system. It is emphasized that these conditions do not incorporate all the required decoupling conditions. By combining the additional requirements with Villars's equations, we obtain a mathematically complete system, which will yield exact solutions where such exist, but which can also be applied to cases of approximate decoupling.
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.
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.
The theory of large amplitude adiabatic motion is applied to a recently proposed model of tunneling in many body systems. We construct the corresponding collective Hamiltonian and study the applicability of the decoupling procedure, both by comparing the results of the large amplitude theory with exact results and by studying different measures of the decoupling procedure, including the stability of the collective motion against noncollective vibrations. We also analyze briefly another approximation scheme especially suited for the problem at hand. The theory of large amplitude collective motion provides an extremely accurate description of the collective motion and of the tunneling process.
By placing emphasis on conceptual correspondence to the “classical” theory which has been developed within the framework of the time-dependent Hartree-Fock theory, a full quantum theory appropriate for describing large-amplitude collective motion is proposed. A central problem of the quantum theory is how to determine an optimal representation called a dynamical representation; the representation is specific for the collective subspace where the large-amplitue collective motion is replicated as satisfactoryly as possible. As an extension of the classical theory where the concept of an approximate integral surface plays an important role, the dynamical representation is properly characterized by introducing a concept of an approximate invariant subspace of the Hamiltonian.