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).
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.
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.
Two methods which give the lower bound for the ground-state energy of a system with N interacting particles are described. The methods exploit the solution of the Schrodinger equation for the comparison system with K<N particles. An illustrative example of fermions interacting with the harmonic force is discussed. More realistic interactions used in nuclear physics are treated numerically with K=2, 3 and 4, showing that the lower bound tends to the exact value of the energy when K increases. The lower bound is strongly dependent on the choice of the interaction.