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Columbia University Press eBooks

I: INTRODUCTION AND SUMMARY

Paul Wiers

Published Dec 31, 1944• 1 citation
Publisher

Abstract

The time-dependent variational principle for many-body trial states is used to discuss the relation between the approaches of different molecular dynamics models to describe indistinguishable fermions.Early attempts to include effects of the Pauli principle by means of nonlocal potentials as well as more recent models which work with antisymmetrized many-body states are reviewed under these premises.

DOI
10.7312/wier90946-005

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The time-dependent variational principle for many-body trial states is used to discuss the relation between the approaches of different molecular dynamics models to describe indistinguishable fermions.Early attempts to include effects of the Pauli principle by means of nonlocal potentials as well as more recent models which work with antisymmetrized many-body states are reviewed under these premises.
classical many body models supplemented by repulsive momentum-dependent potentials to simulate the Pauli and Heisenberg principles have been use with some success for nuclear and atomic bound state and collision problems. They are capable of describing mean ground state properties, hydrodynamics, shocks (if warranted by the physics), viscosity, correlations, clustering, fragmentation, etc. We have become interested in the Feldmeier Gaussian packet formulation since it is based on a variational principle using trial wave functions. We discuss some limitations of the model and discuss further directions of investigation.
For a given statistical state at time t0{t}_{0}t0​, the expectation value of some observable at a later time t1{t}_{1}t1​ is expressed in a variational form. Different trial choices for the quantities to be varied (state and observable) generate different approximations, in which the evolution of the state is optimally fitted to the measured quantity. Examples are given in the context of mean-field theories.
The usual perturbation treatment of the many-body problem is modified by eliminating the two-body interaction and introducing in its place a ttt matrix. The Watson, Brueckner, and Bethe (W.B.B.) treatment of the many-body problem is shown to be a special case of the modified perturbation treatment. The modified perturbation treatment involves no assumption except that of supposing the convergence of the perturbation expansion. Thus to each order in the expansion the results must be meaningful and no unphysical "unlinked cluster" terms can appear. Within the framework of the modified perturbation treatment it is easy to interpret the concepts of the Pauli principle for intermediate states and the self-consistent choice of the comparison potential which play such an important role in the W.B.B. treatment. This work is similar to that of J. Goldstone in that we relate the W.B.B. treatment to a perturbation treatment of the many-body problem. We differ from Goldstone in that we use the usual time-independent perturbation theory instead of the time-dependent one. The time-independent approach permits a direct comparison with the W.B.B. treatment and facilitates discussion of the Pauli principle for intermediate states and of the self-consistent potential.
Fermionic Molecular Dynamics (FMD) models a system of fermions by means of many-body states which are composed of antisymmetrized products of single-particle states. These consist of one or several Gaussians localized in coordinate and momentum space. The parameters specifying them are the dynamical variables of the model. As the repulsive core of the nucleon-nucleon interaction induces short range correlations which cannot be accommodated by a Slater determinant, a novel approach, the unitary correlation operator method (UCOM), is applied. The unitary correlator moves two particles away from each other whenever their relative distance is within the repulsive core. The time-dependent variational principle yields the equations of motion for the variables. Energies of the stationary ground states are calculated and compared to exact many-body results for nuclei up to Ca 48. Time-dependent solutions are shown for collisions between nuclei.