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Physical Review C

Generalized valley approximation applied to a schematic model of the monopole excitation

Niels R. Walet, Abraham Klein, G. Do Dang, Aurel Bulgac

Published Jan 1, 1990• 9 citations
Publisher

Abstract

In recent years we have developed a new mathematical treatment of large amplitude collective motion in the adiabatic limit and formulated a successful approximation method, called the generalized valley approximation. In this paper we discuss its application to adiabatic time-dependent Hartree theory, for which the method is ideally suited. We apply the method first to an exactly solvable limiting case (the Suzuki model), for which we have shown in a previous paper that the usual form of adiabatic time-dependent Hartree theory is not general enough to yield the exact solution. We introduce an extended theory that remedies this deficiency. The modified theory has also been applied to monopole models close to the Suzuki model that are not exactly solvable. The algorithm developed for this case is sufficiently general to serve as a prototype for those necessary to study more complex realistic models of collective motion.

DOI
10.1103/physrevc.41.318

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In recent years we have developed a new mathematical treatment of large amplitude collective motion in the adiabatic limit and formulated a successful approximation method, called the generalized valley approximation. In this paper we discuss its application to adiabatic time-dependent Hartree theory, for which the method is ideally suited. We apply the method first to an exactly solvable limiting case (the Suzuki model), for which we have shown in a previous paper that the usual form of adiabatic time-dependent Hartree theory is not general enough to yield the exact solution. We introduce an extended theory that remedies this deficiency. The modified theory has also been applied to monopole models close to the Suzuki model that are not exactly solvable. The algorithm developed for this case is sufficiently general to serve as a prototype for those necessary to study more complex realistic models of collective motion.
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.
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.
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 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.