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(\ensuremath{\gamma}N
/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.
We study a system composed of N identical bosons interacting in three dimensions via attractive Yukawa pair potentials V(r)=-${\mathit{e}}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}\mathit{r}}$/r. Four approaches to the problem are related: the ``equivalent two-body method,'' translation-invariant Gaussian trial functions, collective field theory, and Hartree trial functions. Upper and lower energy bounds are given and are compared to some recent results which were obtained independently by an optimized Hartree method [M. Membrado, F. Pacheco, and J. Sa\~nudo, Phys. Rev. A 39, 4207 (1989)]. In the pure gravitational limit \ensuremath{\lambda}\ensuremath{\rightarrow}0 the N-body energy is determined for all N and all values of the coupling with an error of less than 7.2%.
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.
A simple upper bound to the ground-state energy of a quantal system composed of
N equal particles obeying Boltzmann or Bose statistics and interacting through the interparticle potential
V(r) is established. This bound applies in the limit of large
N; different expressions obtain depending on whether the potential is or is not singularly attractive at the origin. In the former case, the bound reads - $C{N}^{\frac{(4\ensuremath{-}q)}{(2\ensuremath{-}q)}}$, -
q being the (negative) exponent characterizing the behavior of the pair potential at the origin through ${\mathrm{lim}}_{r\ensuremath{\rightarrow}0}[{r}^{q}V(r)]={V}_{0}$, $\ensuremath{-}\ensuremath{\infty}<{V}_{0}<0$ (of course,
q<2 to prevent two-body collapse); in the latter case, it reads $\ensuremath{-}B{N}^{2}$. Explicit expressions for the constants
C and
B in terms of the pair potential
V(r) are obtained; it is expected (and, in some cases, demonstrated) that the associated upper bound to the ground-state energy of the system approximates closely the actual value. It is also noted that, if for some non-negative value of
p the quantity $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dr}{r}^{2} \mathrm{exp}(\ensuremath{-}{p}^{2}{r}^{2})V(r)$ is negative, the
N-body system is unstable in the sense that, at large
N, its total energy is negative and increases in modulus at least as
N2 (so that the binding energy per particle increases in modulus at least as
N). Examples illustrating the nontrivial nature of this conclusion are displayed.
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.
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.