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Progress of Theoretical Physics

Many-Body Pseudopotential for Hard-Sphere Interaction

Ryuzo Abe

Published Jan 1, 1958• 7 citations

Abstract

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.

DOI: 10.1143/ptp.19.1

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Excitation Spectrum of an Impurity in the Hard-Sphere Bose Gas

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The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.

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LOW-TEMPERATURE BEHAVIOUR OF A BOSE GAS WITH HARD-SPHERE INTERACTION

Raj K. Pathria, Adarsh Singh

Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)

•Sep 26, 1960•0 citations

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 system under consideration is an

N

-particle quantum-mechanical system enclosed in a volume

V

, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter

a

. 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

N

-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of

a

. 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

{a}^{3}

and that for the Fermi system, to order

{a}^{2}

. The physical interpretation and validity of these results are discussed.

The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.

Using the energy-momentum relation derived by Brueckner and Sawada, the low-temperature thermodynamical properties of a Bose gas with hard-sphere interaction are investigated. The two cases, (1) when this energy spectrum is similar to that observed in liquid helium II and (2) when it becomes similar to that derived by Lee, Huang, and Yang, are investigated separately. In the first case (where the spectrum exhibits a nonmonotonic character), the analytical derivation of the expressions for the various thermodynamical properties is rendered possible by dividing the momentum range into three suitable intervals such that within each interval the epsilon -p relation can be approximated to a simple algebraic one. The numerical magnitudes of the various parameters characterizing the algebraic relations are calculated for the case of a gas with density equal to that of liquid helium(II) and for three plausible values of the hard-sphere diameter. The calculated results agree qualitatively with those obtained from the experimental study of this liquid. For the second case, expressions are derived for the various macroscopic properties of the assembly in terms of suitable mathematical functions and then are written down in the form of series expansions for the two particular cases of interest. (auth)