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arXiv:nucl-th/9307021 · Physical Review C

Classical mappings of the symplectic model and their application to the theory of large-amplitude collective motion

Abraham Klein, Niels R. Walet

Published Feb 1, 1994• 7 citations

Abstract

We study the algebra Sp(n,R) of the symplectic model, in particular for the cases n=1,2,3, in a new way. Starting from the Poisson-bracket realization we derive a set of partial differential equations for the generators as functions of classical canonical variables. We obtain a solution to these equations that represents the classical limit of a boson mapping of the algebra. We show further that this mapping plays a fundamental role in the collective description of many-fermion systems whose Hamiltonian may be approximated by polynomials in the associated algebra, as is done in the simplest versions of the symplectic model. The relationship to the collective dynamics is formulated as a theorem that associates the mapping with an exact solution of the time-dependent Hartree approximation. This solution determines a decoupled classical symplectic manifold, thus satisfying the criteria that define an exactly solvable model in the theory of large amplitude collective motion. The models thus obtained also provide a test of methods for constructing an approximately decoupled manifold in fully realistic cases. We show that an algorithm developed in one of our earlier works reproduces the main results of the theorem.

DOI: 10.1103/physrevc.49.840

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A theory of large amplitude collective motion of a many-particle system is presented, which is relevant, for example, to nuclear fission. The theory is a combination of techniques used in many areas of physics and mathematics. The starting point is the application of the time-dependent Schrödinger equation to generate invariant subspaces of the Hamiltonian in the Hartree–Fock approximation. This is a generalization of the group-theoretical device of generating orbits of a group in the construction of reduced representations. It is shown how solutions of the time-dependent Schrödinger equation can be expressed as instantaneous stationary states of a constrained static Hamiltonian. Thus contact is made with the traditional cranking models and constrained Hartree–Fock theories of large amplitude collective motion. The collective motion is quantized using the Hill–Wheeler–Griffin method of generator coordinates in a basis of generalized coherent states. One is thereby able to exploit much of the theory of harmonic oscillator coherent states, which have been so successfully used in the quantum theory of the laser. The resulting Schrödinger equation for the collective dynamics is expressed both in the Bargmann representation and in the more familiar Schrödinger representation. It is shown that solution of the Schrödinger equation in the small amplitude harmonic approximation reproduces the well-known RPA result. A pilot calculation for 28Si shows that application in large amplitude is also feasible.

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