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arXiv:quant-ph/0008048 · Physical review. B, Condensed matter

Lower bound on fermion binding energies

Olivier Juillet, Sonia Fleck, L. Theußl, J. M. Richard, Kálmán Varga

Published Jan 22, 2001• 4 citations
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Abstract

We derive a lower bound for the ground-state energy ${E}^{F}(N,S)$ of N fermions with total spin S in terms of binding energies ${E}^{F}(N\ensuremath{-}1,S\ifmmode\pm\else\textpm\fi{}1/2)$ of $(N\ensuremath{-}1)$ fermions. Numerical examples are provided for some simple short-range or confining potentials.

DOI
10.1103/physrevb.63.073102

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We derive a lower bound for the ground-state energy EF(N,S){E}^{F}(N,S)EF(N,S) of N fermions with total spin S in terms of binding energies EF(N\ensuremath−1,S\ifmmode±\else\textpm\fi1/2){E}^{F}(N\ensuremath{-}1,S\ifmmode\pm\else\textpm\fi{}1/2)EF(N\ensuremath−1,S\ifmmode±\else\textpm\fi1/2) of (N\ensuremath−1)(N\ensuremath{-}1)(N\ensuremath−1) fermions. Numerical examples are provided for some simple short-range or confining potentials.
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.
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Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin 1/21/21/2 fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.
Recent developments in the physics of low density trapped gases make it worthwhile to verify old, well known results that, while plausible, were based on perturbation theory and assumptions about pseudopotentials. We use and extend recently developed techniques to give a rigorous derivation of the asymptotic formula for the ground state energy of a dilute gas of NNN fermions interacting with a short-range, positive potential of scattering length aaa. For spin 1/2 fermions, this is E∼E0+(ℏ2/2m)2πNρaE \sim E^0 + (\hbar^2/2m) 2 πN ρaE∼E0+(ℏ2/2m)2πNρa, where E0E^0E0 is the energy of the non-interacting system and ρρρ is the density.