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Physical Review

Ground State of a Bose System of Hard Spheres

Tai Tsun Wu

Published Sep 15, 1959• 297 citations
Publisher

Abstract

It is shown that the pseudopotential method can be extended to yield further terms in the low-density expansion of the ground-state energy of a system of Boltzmann or Bose particles with hard-sphere interaction. Two terms beyond the known result are found, and the expansion is no longer a power series in ${({a}^{3}\ensuremath{\rho})}^{\frac{1}{2}}$. Other related properties of the system are discussed.

DOI
10.1103/physrev.115.1390

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The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where $a$ is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.

Physical Review•Jun 15, 1963•106 citations
It is shown that the pseudopotential method can be extended to yield further terms in the low-density expansion of the ground-state energy of a system of Boltzmann or Bose particles with hard-sphere interaction. Two terms beyond the known result are found, and the expansion is no longer a power series in (a3\ensuremathρ)12{({a}^{3}\ensuremath{\rho})}^{\frac{1}{2}}(a3\ensuremathρ)21​. Other related properties of the system are discussed.
A many-body pseudopotential for a hard-sphere system which is mathematically equivalent to the boundary conditions imposed on the wave functions is obtained as a generalization of Huang and Yang's binary pseudopotential, and the lowest order term of the ground state energy resulting from the triple collision is calculated for the Bose system. Instead of the direct perturbational treatment of the pseudopotential, the use of the ρk-representation for the Bose system is proposed and the ground state energy is calculated in this representation. It is shown that the energy thus calculated is of the form of an expansion in powers of (ρa3)1/2, where ρ is the average density and a the hard-sphere diameter, which is agreement with Lee and Yang's conclusion drawn from the binary collision expansion method. Some discussions associated with the roton spectrum of liquid He4 are given.
The ground state energy of the Bose particle system at low densities is investigated by using the method which is based on summing up the terms of the conventional perturbation series giving rise to the lowest order with respect to the density. The summation is shown to be greatly simplified if the scattering matrix is introduced, and a simple example is demonstrated for the repulsive square well potential by calculating a few terms of the series first directly and secondly with the aid of scattering matrix. The final result is shown to be valid both for weak and strong interactions and the calculation is carried out analytically for some types of potential function. The connection of the present method with the pseudopotential method and with the Brueckner theory is investigated and some problems associated with the attractive interactions are discussed.
The system under consideration is an NNN-particle quantum-mechanical system enclosed in a volume VVV, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter aaa. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the NNN-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of aaa. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order a3{a}^{3}a3 and that for the Fermi system, to order a2{a}^{2}a2. The physical interpretation and validity of these results are discussed.
A method developed by the authors for treating Bose-Einstein particles at low temperature is specialized to the case of hard sphere interactions. It is shown that the integral equations determining the energy can in this case be replaced by an integral involving the unknown excitation spectrum. The energy is first determined as a power series in ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the hard-sphere radius. This solution is shown to diverge badly at high density. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low density and to a good approximation at high density. The excitation spectrum of the system is shown to have a peculiar nonmonotonic relation between energy and momentum very similar to that deduced by Landau from the low-temperature properties of He4{\mathrm{He}}^{4}He4.
The well-known first two terms in the asymptotic density series for the ground-state energy of a Bose gas, ${E}_{0}=2\ensuremath{\pi}N\ensuremath{\rho}a[1+(\frac{128}{15\ensuremath{\surd}\ensuremath{\pi}}){(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}]$, where aaa is the scattering length of the pair potential, is ordinarily obtained by summing an infinite set of graphs in perturbation theory. We show here how this same series may be obtained by elementary methods. Our method offers the advantages of simplicity and directness. Another advantage is that the hard-core case can be handled on the same basis as a finite potential, no pseudopotential being required. In fact, the analysis of the hard-core potential turns out to be simpler than for a finite potential, as is the case in elementary quantum mechanics. In an Appendix we discuss the high-density situation and show that for a certain class of potentials Bogoliubov's theory is correct in this limit. Thus, Bogoliubov's theory, which is never correct at low density unless a pseudopotential is introduced, is really a high-density theory.