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Journal of Mathematical Physics

Lower energy bounds for translation-invariant atomlike systems

Richard L. Hall

Published Sep 1, 1978• 5 citations
Publisher

Abstract

A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).

DOI
10.1063/1.523903

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A system of N identical particles with masses m1 interact with each other via a pair potential V and with an additional particle of mass m0 by the pair potential U, and the system obeys nonrelativistic quantum mechanics. A general lower bound formula is derived for the ground-state energy, which reduces to Coleman’s result in the special case m0=∞. The lower bound is compared with the exact energy for the harmonic oscillator problem V (x) =k21x2 and U (x) =k02x2, which has recently been solved exactly (Hall).
We derive a set of inequalities which relate the translation invariant problem of N identical particles with pairwise interactions to an independent particle problem. These inequalities apply to attractive power law potentials V(r)=rq, 1≤q≤∞, and superpositions of such potentials; they become identities in the harmonic oscillator case q=2. We use the inequalities to derive new upper and lower bounds for the ground state energies of fermion systems, which interact through these potentials. These bounds improve all previous results in the range 1≤q≤∞; they reduce to the exact answer in the harmonic oscillator case.
Suppose EN{E}_{N}EN​ is the lowest energy of a translation-invariant system of NNN identical bosons which interact by attractive central pair potentials and obey nonrelativistic quantum mechanics. Sufficient conditions are prescribed on the potential to prove that EN{E}_{N}EN​ is bounded above by $({N}{2}){({K}{2})}^{\ensuremath{-}1}{E}_{K}$, where $N\ensuremath{\ge}\ensuremath{\mu}K$. The $\ensuremath{\mu}$ numbers are specified for various classes of potential shape. In cases where $\ensuremath{\mu}\ensuremath{\le}\frac{3}{2}$, it follows that ${E}_{N}\ensuremath{\le}\frac{1}{2}N(N\ensuremath{-}1){E}_{2}$ for $N\ensuremath{\ge}3$.
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.
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.
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.