Time-independent Mean Field Research Articles

Calculation of the propagation amplitude in the case of a Coulomb interaction via time-independent mean-field theory

The exact study of many-body microscopic systems is impossible when the number of particles is large (N ≥ 3). Approximate methods are then used. The time-independent mean-field (TIMF) approximation has been proposed for the description of collisions in many-body systems. Collision amplitudes are derived by the use of a variational principle and the choice of trial functions as products of single-particle orbitals. Resulting mean-field equations with a nonvanishing right-hand side turn out to be a generalization of the traditional Hartree or Hatree–Fock type equations. These TIMF equations are successfully solved numerically for the case of short-range forces. In this paper, we test the validity of this theory for the Coulomb interaction between two particles, that is, a long-range interaction. A numerical comparison between the exact and the mean-field solutions is conducted PACS Nos.: 31.15.Ne, 31.15.Pf, 21.45.+v,25.10.-i

Continuum singularities of a mean-field theory of collisions

Consider a complex energy z for an N-particle Hamiltonian H and let χ be any wave packet accounting for any channel flux. The time-independent mean-field (TIMF) approximation of the inhomogeneous, linear equation (z−H)|Ψ〉=|χ〉 consists of replacing Ψ by a product or Slater determinant φ of single-particle states φi. This results, under the Schwinger variational principle, in self-consistent TIMF equations (ηi−hi)|φi〉=|χi〉 in single-particle space. The method is a generalization of the Hartree–Fock (HF) replacement of the N-body homogeneous linear equation (E−H)|Ψ〉=0 by single-particle HF diagonalizations (ei−hi)|φi〉=0. We show how, despite strong nonlinearities in this mean-field method, threshold singularities of the inhomogeneous TIMF equations are linked to solutions of the homogeneous HF equations.

Inhomogeneous Random Phase Approximation for Nuclear and Atomic Reactions

A random phase approximation (RPA) of the inhomogenous time-independent mean field (TIMF) equations for reactions is developed. The TIMF method is based on a general variational principle for calculating matrix elements of operator inverses as, for example, matrix elements of the resolvent of a Hamiltonian or the scattering T-operator. In the case of nuclear or atomic reactions, these matrix elements are calculated with respect to the asymptotic channel wave functions which also define the inhomogeneous terms in the variational equations. For inhomogeneous equations a direct RPA-like treatment, analogous to the standard RPA used for diagonalization problems, is not possible. Hence the starting point to obtain an RPA extension for the inhomogeneous mean field equations is a transformation of the inhomogeneous into homogeneous variational equations. The problem of operator inversion is then transformed into the inversion of a function which is obtained by the diagonalization of an auxilliary operator. This auxilliary operator consists of the operator to be inverted and an additional term incorporating the inhomogeneities. Taking into account particle-hole correlations for the nonhermitian diagonalization problem, the standard random phase approximation for hermitian operators can be generalized to a nonhermitian RPA built on top of TIMF. For that purpose, the antisymmetric TIMF theory for operator inversion is formulated in second quantization. Complications are discussed, which result from the use of several biorthogonal bases and of only intra-fragment antisymmetrized channel wave functions, as needed in antisymmetric scattering theories. In this framework it is shown how particle-hole correlations can be used to generalize the inhomogeneous TIMF method, thus going beyond the mean field approximation.

Inhomogeneous Random Phase Approximation: A Solvable Model

A recently developed method to include particle-hole correlations into the time-independent mean field theory for scattering (TIMF) by an inhomogeneous random phase approximation (IRPA) is applied to a numerically solvable model. Having adapted the procedure according to numerical requirements, IRPA calculations turn out to be tractable. The obtained results improve TIMF results.

Beyond the time independent mean field theory for nuclear and atomic reactions: Inclusion of particle-hole correlations in a generalized random phase approximation.

The time independent mean field method (TIMF) for scattering defines biorthonormal sets of single-particle wave functions and corresponding creation and annihilation operators. 2p - 2h correlations can be introduced through a generalized random phase approximation; 1p - 1h contributions vanish (Brillouin theorem). While the general variational method for scattering by Giraud and Nagarajan solves inhomogeneous Euler equations by inversion of the standard, hermitean Hamiltonian, the present approach diagonalizes a non-hermitean Hamiltonian, which carries the information about entrance and exit channels.

Time Independent Mean Field Theory of Collisions(Microscopic Cluster Models of Light Nuclei and Related Topics)

Time Independent Mean Field Theory of Collisions(Microscopic Cluster Models of Light Nuclei and Related Topics)

Symmetry breaking by propagation

Matrix elements of propagators are calculated variationally. Two analytically soluble models confirm the validity of the time independent mean field variational approximation. The models have many solutions (7 and 85, respectively), the physical relevance of which can be completely analyzed and screened. One of the models shows solutions with a ‘‘spontaneous’’ symmetry breaking. The restoration of the broken symmetry is implemented. It describes correlations occurring during propagation.

Generator coordinate representation of the time independent mean field theory of collisions

We show how matrix elements of theT-matrix can be easily estimated on a basis of Slater determinants, with a mean field approximation. Linear superpositions of these Slater determinants then generate plane waves, or distorted (Coulomb) waves. This provides physical matrix elements ofT.

Time-independent mean field theory of collisions: the case of long range forces

Instead of diagonalizing the many-body Hamiltonian H, the authors invert E-H, where E is a complex energy, which is eventually made real. All the traditional approximations to diagonalization may be adopted for the evaluation of (E-H)-1. Mean field methods are specially investigated. Previous papers reported several numerically soluble cases where a time-independent mean field approximation (TIMF) was tested with success for the calculation of collision amplitudes with short range forces. They test the validity of TIMF for a long range interaction: the Coulomb Green function for two particles is found to be successfully approximated by TIMF.

Time-independent mean field description of collisions

Products of single-particle orbitals are used as trial functions in a variational principle for the calculation of multistep nuclear collisions. The resulting dynamical equations are time-independent and generalize Hartree equations to nondiagonal mean fields with inhomogeneous source terms. These equations are solved for a p + 3H → p + n + d breakup reaction.

Time-independent mean field description of collisions: B. G. Giraud, M. A. Nagarajan, and I. J. Thompson. Daresbury Laboratory, Warrington WA4 4AD, UK

Time-independent mean field description of collisions: B. G. Giraud, M. A. Nagarajan, and I. J. Thompson. Daresbury Laboratory, Warrington WA4 4AD, UK