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AIP conference proceedings

The time-dependent Hartree–Fock method and chaos in nuclear dynamics

Michael R. Strayer

Published Jan 1, 1987• 0 citations
Publisher

Abstract

The long‐time classical behavior of many‐body systems is examined using time‐dependent Hartree–Fock (TDHF) theory. Detailed computer simulations for the formation of a nuclear molecule in low‐energy 12C+12C collisions are discussed at length. The representation of effective classical coordinates, and their correspondence to physical observable for systems which undergo large‐amplitude quasi‐periodic and chaotic motion is briefly summarized. The relationship between these calculations and new methods, which allow the calculation of quasiperiodic solutions to the TDHF equations under more general circumstances, is summarized.

DOI
10.1063/1.36897

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The long‐time classical behavior of many‐body systems is examined using time‐dependent Hartree–Fock (TDHF) theory. Detailed computer simulations for the formation of a nuclear molecule in low‐energy 12C+12C collisions are discussed at length. The representation of effective classical coordinates, and their correspondence to physical observable for systems which undergo large‐amplitude quasi‐periodic and chaotic motion is briefly summarized. The relationship between these calculations and new methods, which allow the calculation of quasiperiodic solutions to the TDHF equations under more general circumstances, is summarized.
We investigate the validity of the time-dependent Hartree-Fock (TDHF) approximation for a finite fermion model system. With the help of coherent states we derive equations of motion for the one-particle density, which are of classical Hamiltonian form and can be interpreted in terms of generalized coordinates and momenta. The corresponding trajectories in phase space show transitions between regular and chaotic regimes depending on the parameters of the model. A chaotic behavior indicates a failure of the TDHF approximation because the exact solutions are always regular due to the linearity of the Schr\"odinger equation.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.
The title originally planned for this series of lectures was “Time-dependent Hartree Fock Theory of Nuclear Motions and Collisions”. In fact, in what follows, the TDHF method will be considered for collective motion only. This is because a new, more practical theory of collisions has recently emerged. These lectures thus contain two main and distinct chapters, the first dedicated to this new theory of collisions and the second to collective motion. As a distraction to the reader, a last and small chapter recalls elementary ideas about some conjectured mechanisms for the dynamics of a very special system with collective motions, namely a living organ in the nervous system. This Summer School being dedicated to the subject of order and chaos, it is hoped that the considerations which follow never go too far from that fascinating subject indeed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We discuss some aspects of implementing the time-dependent Hartree-Fock method in the case of nuclear physics. Topics discussed include implementation of the time-stepping algorithm, considerations involving the effective interaction, and the use (or not) of particular optional terms in the energy density functional, and boundary conditions. Examples of application of the technique to giant resonances and reactions are given, concentrating on issues to do with numerical and conceptual interpretation.
Aiming at exploring microscopic dynamics responsible for the dissipative large-amplitude collective motion, the dynamic response and correlation functions are introduced within the general theory of nuclear coupled-master equations. The theory is based on the microscopic theory of nuclear collective dynamics which has been developed within the time-dependent Hartree-Fock (TDHF) theory for disclosing the complex structure of the TDHF manifold. A systematic numerical method for calculating the dynamic response and correlation functions is proposed. By performing numerical calculation for a simple model Hamiltonian, it is pointed out that the dynamic response function gives important information in understanding the large-amplitude dissipative collective motion which is described by an ensemble of trajectories within the TDHF manifold.