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Journal of Physics A Mathematical Nuclear and General

Many-particle systems VIII. Saturation and a lower-bound shell model

G M Horton

Published May 1, 1973• 2 citations
Publisher

Abstract

For pt V11 see abstr. A62988 of 1971. The lower-bound shell model developed by Carr and Post is applied to a certain class of non-local two-body interactions and shows saturation. The case of the Yamaguchi interaction is worked out in detail.
DOI
10.1088/0305-4470/6/5/006

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The programme of reducing N -particle problems to one-particle problems is extended to include the Pauli principle. To calculate the ground-state energy of a system of N fermions interacting by pair forces, a rigorous lower-bound shell model is derived. This shell model follows a building-up principle, and tends to improve with increasing number of particles as well as with increasing strength of interaction. For twenty particles antisymmetric in ordinary space interacting by square-well interaction of strength V 0 a 2 = 200 hslash 2 /2 m the shell model differs by less than 8% from the calculated upper bound, and hence a fortiori from the exact energy. In order to test the quality of the approximation for various interactions, it was necessary to calculate upper bounds.
We study isolated quantum systems with two-body interactions after a quench. In these systems, the energy shell is a Gaussian of width $\ensuremath{\sigma}$, and it gives the maximum possible spreading of the energy distribution of the initial states. When the distribution achieves this shape, the fidelity decay can be Gaussian until saturation. This establishes a lower bound for the fidelity decay in realistic systems. An ultimate bound for systems with many-body interactions is also derived based on the analysis of full random matrices. We find excellent agreement between numerical and analytical results. We also provide the conditions under which the short-time dynamics of few-body observables is controlled by $\ensuremath{\sigma}$. The analyses are developed for systems, initial states, and observables accessible to experiments.