Many-Particle Systems: Derivation of a Shell Model

R J M Carr, H R Post

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.

I. I. Ivanchik

E. B. Balbutsev

A detailed derivation of the energy lower bound model (SHRIMP model) for a translation-invariant many-fermion system has recently been proposed (Carr 1978). The author asserts that an N-particle shell model retaining antisymmetry in all N particles is obtained. The present author wishes to point out the error in this derivation which makes the SHRIMP model wrong leaving correct the related RIP and HIP models.

Joseph N. Ginocchio

Using the theory of groups the many-particle average of the product of one-body operators is determined in terms of the one-body average of the products of these operators. With this all the energy moments of the noninteracting system are derived in terms of the one-particle moments. For the interacting systems the third and fourth moments are derived.

H R Post