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Journal of Physics A Mathematical and General

The equivalent two-body and delta energy lower bounds for N-boson systems

Richard L. Hall

Published Jan 21, 1989• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/2/007

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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.
We study 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)=\ensuremath{\gamma}\ensuremath{\Vert}x\ensuremath{\Vert} and obey nonrelativistic quantum mechanics. It is shown that the energy is given by the formula \ensuremath{\varepsilon}=C(N)(N-1)(${\ensuremath{\Elzxh}}^{2}$/ m)1/3{)}^{1/3})1/3(\ensuremath{\gamma}N/2)2/3{/2)}^{2/3}/2)2/3, where 1.01879(N)1.02333 for all N\ensuremath{\ge}2. The lower bound is provided by the ``equivalent two-body method'' whereas the upper bound is derived by the use of collective field theory. The general relation between these complementary theories is investigated.
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 study a bound system of N identical bosons interacting by model pair potentials of the form $V(r)=\ensuremath{\lambda}\mathrm{sgn}{(p)r}^{p}+\ensuremath{\mu}{/r}^{2},$ $\ensuremath{\lambda}>0,$ $\ensuremath{\mu}>~0.$ By using a variational trial function and the ``equivalent two-body method,'' we find explicit upper and lower bound formulas for the N-particle ground-state energy in arbitrary spatial dimensions d> 3d>~3d> 3 for the two cases p=2p=2p=2 and $p=\ensuremath{-}1.$ It is demonstrated that the upper bound can be systematically improved with the aid of a special large-N limit in collective field theory.
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.
We consider a system of N identical bosons which interact in one dimension via attractive pair potentials and obey nonrelativistic quantum mechanics. This system is studied from two complementary viewpoints. The equivalent two-body method approximates the system as a collection of (N-1) independent two-particle systems with coupling constants enhanced by the factor N/2, and this yields energy lower bounds. The collective-field method approaches the problem from the standpoint of the limit as N\ensuremath{\rightarrow}\ensuremath{\infty} and leads to energy upper bounds. We find that Gaussian trial functions for the N-particle Hamiltonian and Gaussian trial densities in the collective-field theory lead to precisely the same energy upper bounds. The upper bounds provided by field theory allow for systematic improvement via a variational principle. Details are worked out for two exactly soluble problems, namely, the harmonic oscillator and the attractive \ensuremath{\delta} potential.
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.