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Progress of Theoretical Physics

A Canonical Coordinate System Suitable for Adiabatic Treatment of Collective Motion: An Illustrative Model

A. Kuriyama, M. Yamamura

Published Feb 1, 1983• 2 citations
Publisher

Abstract

With the aid of the canonicity condition, we specify a canonical coordinate system suitable for the adiabatic treatment of “large” amplitude collective motion. We introduce a double transformation in order to specify a time-dependent Slater determinantal state. This specification follows that given by Baranger and Veneroni. The utility of the canonicity condition is shown with the use of an illustrative model.

DOI
10.1143/ptp.69.681

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With the aid of the canonicity condition, we specify a canonical coordinate system suitable for the adiabatic treatment of “large” amplitude collective motion. We introduce a double transformation in order to specify a time-dependent Slater determinantal state. This specification follows that given by Baranger and Veneroni. The utility of the canonicity condition is shown with the use of an illustrative model.
It is shown that the ordinary framework of (adiabatic) time-dependent Hartree-Fock method can uniquely specify the collective submanifold and its canonical coordinate system, with the aid of the RPA and the canonicity condition. Some criticism is given to the pesimistic viewpoint concerning the unique specification of collective submanifold within the adiabatic time-dependent Hartree-Fock method. A decisive foundation of the approach developed by Marumori, Maskawa, Sakata and one of the present author (A. K.) is given.
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.
We propose a new set of equations to determine the collective Hamiltonian including the second-order collective-coordinate operator on the basis of the adiabatic self-consistent collective-coordinate (ASCC) theory. We illustrate, with the two-level Lipkin model, that the collective operators including the second-order one are self-consistently determined. We compare the results of the calculations with and without the second-order operator and show that, without the second-order operator, the agreement with the exact solution becomes worse as the excitation energy increases, but that, with the second-order operator included, the exact solution is well reproduced even for highly excited states. We also reconsider which equations one should adopt as the basic equations in the case where only the first-order operator is taken into account, and suggest an alternative set of fundamental equations instead of the conventional ASCC equations. Moreover, we briefly discuss the gauge symmetry of the new basic equations that we propose in this paper.
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.