The many-particle shell model

H R Post

D.M. Brink, G.R. Satchler

K. Heyde

Werner Kutzelnigg, Vedene H. Smith

Abstract The distinction between open‐ and closed‐shell states is quite important in few‐particle quantum mechanics‐especially in view of the correlation problem. The current definitions of such states and of shells in general are, however, only meaningful if one can assume that the independent particle model is a valid description of the state under consideration. Definitions are given in terms of rigorous quantum‐mechanical concepts which allow a classification of states irrespective of any model assumptions. Closed‐shell states and different types of open‐shell states are distinguished. Sufficient criteria for a state to be a closed‐shell state are derived in terms of unitary invariants of the one‐particle density matrix of this state.

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.