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Physical Review C

Mean-field approximation for inclusive observables

Y. Alhassid, Berndt Müller, S. E. Koonin

Published Jan 1, 1981• 11 citations
Publisher

Abstract

The excitation of a many-body system by a one-body perturbation is considered. The stationary phase approximation to a functional-integral representation of the many-body evolution operator shows that the optimal mean field for describing the final expectation values of one- and two-body operators is that given by the time-dependent Hartree-Fock method. The forced Lipkin model is considered as a test of the mean-field approximation.NUCLEAR REACTIONS Mean-field approximation, inclusive observables. Time-dependent Hartree-Fock.

DOI
10.1103/physrevc.23.487

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A theory is presented in which approximations to quantum observables are obtained by applying the stationary-phase approximation to an exact functional integral representation of the many-body evolution operator. The requirement that the leading correction to the stationary-phase approximation vanish leads to a time-dependent Hartree-Fock mean field. Application of the theory to the Fourier transform of the trace of the evolution operator and the study of its poles yields quantized bound states with large amplitude. The theory is shown to reproduce the familiar static Hartree-Fock and random-phase approximations in the appropriate limits and yields an excellent approximation to the entire spectrum of the exactly solvable Lipkin model.NUCLEAR STRUCTURE Derivation of time-dependent mean-field theory from functional integral representation. Quantum theory of large amplitude collective motion. Application to Lipkin model.
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.
A summation of selected graphs in the perturbation expansion for the statistical operator and the operator of state leads to a simplified equation of motion for the nuclear many-body problem. It contains as a special limit the time-dependent Hartree-Fock and the quantum-statistical master equation. In a pictorial way one could label the approach as "Hartree-Fock plus collision terms." The formalism allows one to understand the progression of the nucleus from the boiling of the particle-hole excitations to the collective state of motion. The work is restricted presenting the formalism. Various applications are possible and indicated, but apart from an illustrative example no concrete comparison with the experiment has been performed.NUCLEAR REACTIONS Formalism developed for unifying collective and single-particle aspects; mean field theory, Hartree-Fock plus collision terms.