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Journal of Physics A Mathematical and General

Many-particle systems VI. N-fermion systems: derivation of a lower-bound shell model

R J M Carr, H R Post

Published Sep 1, 1968• 10 citations

Abstract

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.

DOI: 10.1088/0305-4470/1/5/314

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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.

A method by Post, which has been used to calculate lower bounds on the energies of the lowest spatially symmetric or antisymmetric states of many-identical-particle systems, is shown to apply quite generally to the ground states of systems of bosons or fermions for any pair interactions whatever. In particular, the N -nucleon system is considered. No ordinary central pair potential was found to give the empirical binding energies of 2 H, 3 H and 4 He. For the exponential and square-well potentials the calculated bounds on the energies are closer than 1% over wide ranges of the potential parameters. The method justifies and generalizes certain `equivalent two-body' calculations originated by Wigner.

Two methods which give the lower bound for the ground-state energy of a system with N interacting particles are described. The methods exploit the solution of the Schrodinger equation for the comparison system with K<N particles. An illustrative example of fermions interacting with the harmonic force is discussed. More realistic interactions used in nuclear physics are treated numerically with K=2, 3 and 4, showing that the lower bound tends to the exact value of the energy when K increases. The lower bound is strongly dependent on the choice of the interaction.

A restriction on the choice of translation-invariant coordinates for a system of N identical fermions, interacting by pair forces, allows a lower bound to be established on the ground-state binding energy. This bound is the sum of the first N -1 eigenvalues of a two-particle problem. The quality of the method is found to depend on the choice of coordinates consistent with the initial restraint, the optimum set giving a bound which, in the case of Hooke's law pair forces, is [radical]3( N -1)/2( N +1) times the exact energy.

Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.