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Canadian Journal of Physics

Coherent state theory of large amplitude collective motion

DAVID J. ROWE, R. Bassermann

Published Oct 1, 1976• 107 citations

Abstract

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.

DOI: 10.1139/p76-234

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The time-dependent variational principle of the Schr\"odinger equation is applied to a formally exact solution of the Schr\"odinger equation whose variational elements are operators which define a collective subspace of the many-body system under study. This generalizes the procedure employed to derive time-dependent Hartree-Fock theory. Four distinct formally exact time-independent variational principles, including several familiar forms, are derived. Application to a class of exactly soluble models, studied in the vibrational regime, illustrates the different modes of implementation of two of the principles. The general theory of large amplitude collective motion is derived. It is shown that the principles can be applied equally well, using either boson or fermion pair degrees of freedom. Some aspects of the relation to the semiclassical limit are discussed.[NUCLEAR STRUCTURE Time-dependent variational principle for collective motion. Invariance principle of Schr\"odinger equation. Generalized coherent state trial function. Exact time-independent variational principles. Variational principle for intrinsic state. Cranking variational principle. Large amplitude collective motion. Semiclassical limit.]

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.

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.

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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.

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