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Proceedings of the Physical Society

Many-particle systems: IV. Short-range interactions

Richard L. Hall, H R Post

Published Feb 1, 1967• 98 citations
Publisher

Abstract

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.

DOI
10.1088/0370-1328/90/2/309

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Proceedings of the Physical Society•May 1, 1967•32 citations
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.
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.
Studies the ground-state energy of a system of N identical bosons, each having mass m, which interact in one dimension via the pair potential V(x)= gamma f(x/a) and obey non-relativistic quantum mechanics. A recent energy lower bound based on the known exact solution for the delta potential is compared to an earlier bound provided by the equivalent two-body method. The delta bound is good whenever the potential is very narrow and deep. For any bounded potential, the earlier energy bound is always better for large N. Detailed results and graphs are given for the sech-squared potential and the square-well potential.
Alternative lower bounds for the binding energy of a quantum-mechanical system of interacting particles are presented. These bounds are expressed in terms of two-particle quantities and improve the conventional bounds of the Hall-Post type. They are constructed by considering not only the energy in the two-particle system, but also the structure of the pair wave function. We apply the formal results to various numerical examples, and show that in some cases dramatic improvement over the existing bounds is reached.
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.
Close upper and lowei bounds are found on the binding energy of a system of N identical bosons interacting in one dimension by an attractive square-well pair potential. In the limit in which the potential has the shape of a δ function the bounds are found to be consistent with McGuire's exact solution. In three dimensions the limit of vanishingly small range, in which the two-particle binding energy is kept constant, gives infinite binding to the N -particle system, in agreement with Thomas's theorem Consideration of other limiting Proc.esses eliminates an apparent contradiction between Thomas and Svartholm.
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.