© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Improved lower bounds for the N-fermion problem

Martin Manning

Published May 1, 1978• 7 citations

Abstract

Lower bound energy formulae for the N-fermion problem of the type used by Hall (1967) and Post (1956) are considered in terms of a reduced density operator for the interparticle separation. In bounds of this type the N-body energy is bounded below by a sum of eigen-energies of states of relative motion weighted by occupations for the relative states. It is shown that the wavefunctions of relative motion are subject to constraints and in particular must be zero when the particle separation is zero. This modifies the energy spectrum of the reduced Hamiltonian of relative motion and improves the lower bound due to Hall. The improvement is demonstrated for some simple model problems. Furthermore it is shown that the bounds of this type currently known do not exhibit saturation regardless of the interaction details. This leads to some speculation on improved lower bounds.

DOI: 10.1088/0305-4470/11/5/016

Related papers

Improved lower bounds for the ground-state energy of many-body systems

Dimitri Van Neck, Y. Dewulf, Michel Waroquier

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.

arXiv:nucl-th/0102061•Physical Review A•May 10, 2001•6 citations

Lower bound on the ground-state energy and one-dimensionalN-fermion problem

Srećko Kilić, K. Mikelić

We derive an expression for a lower bound of the ground-state energy of N identical fermions. In principle, this relation can describe any order of multiplicity of many-body forces and represents a generalization of a Hall inequality. Applying the Hall lower-bound inequality to the Calogero and Sutherland potentials, we show the great importance of spin states. Furthermore, using a generalized formula, we obtain a lower bound for a system with three-body forces.

Physical Review A•Oct 1, 1993•3 citations

Testing Hall-Post inequalities with exactly solvable N -body problems

Avinash Khare, J. M. Richard

The Hall-Post inequalities provide lower bounds on N -body energies in terms of N '-body energies with N ' < N . They are rewritten and generalized to be tested with exactly solvable models of Calogero-Sutherland type in one and higher dimensions. The bound for N spinless fermions in one dimension is better saturated at large coupling than for noninteracting fermions in an oscillator potential.

arXiv:quant-ph/0105130•Journal of Physics A Mathematical and General•Aug 10, 2001•5 citations

The role of model systems in the few‐body reduction of the N‐fermion problem

Jerome K. Percus

Abstract The problem of upper and lower ground state energy bounds for many‐fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.

International Journal of Quantum Chemistry•Jan 1, 1978•147 citations

Lower Bounds to the Many-Body Problem Using Density Matrices

Louis J. Kijewski, Jerome K. Percus

Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.

Physical Review•Dec 5, 1967•15 citations

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

Journal of Physics A Mathematical and General•Sep 1, 1968•10 citations

Lower bound energy formulae for the N-fermion problem of the type used by Hall (1967) and Post (1956) are considered in terms of a reduced density operator for the interparticle separation. In bounds of this type the N-body energy is bounded below by a sum of eigen-energies of states of relative motion weighted by occupations for the relative states. It is shown that the wavefunctions of relative motion are subject to constraints and in particular must be zero when the particle separation is zero. This modifies the energy spectrum of the reduced Hamiltonian of relative motion and improves the lower bound due to Hall. The improvement is demonstrated for some simple model problems. Furthermore it is shown that the bounds of this type currently known do not exhibit saturation regardless of the interaction details. This leads to some speculation on improved lower bounds.

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.

Abstract The problem of upper and lower ground state energy bounds for many‐fermion systems is considered from the viewpoint of reduced density matrices. Model density matrices are used for upper bounds to, first uncoupled, then coupled fermions. Model Hamiltonians are developed for lower bounds in corresponding fashion. Both mathematical and physical models are constructed for setting up universally valid inequalities on density matrices. These are joined by both inequalities and equalities in which the explicit form of the system at hand is used. A few illustrative examples are presented.

Variation principles for obtaining lower bounds by means of density matrices are applied to a number of many-body problems. This is done by means of restrictions derived from the study of exacly solvable models. Two classes of problems are considered: (a) obtaining a lower bound to the ground-state energy of a system of particles, and (b) obtaining a lower bound to the Helmholtz free energy. For the first case an exact lower bound is obtained for the ground-state energy of the particle-conserving Bogoliubov Hamiltonian. Of particular significance in the nonzero temperature case is a rigorous lower bound to the free energy of the Ising model with small external field at low temperatures.

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.