Many-Body Pseudopotential for Hard-Sphere Interaction

Ryuzo Abe
1958 Progress of theoretical physics  
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 Ilk-representation for the Bose system is proposed and the ground
more » ... d 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 (pa 3)li2, where p is the average density and a the hard-sphere diameter, which is in 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. § I. Introduction 1 In order to treat the hard-sphere interaction, Huang and Y ang 1 l have proposed the method of the pseudopotential and calculated the energy eigenvalue and the wave function of the many-body system in a form of the power series of the hard-sphere diameter a. The same method has been applied by Huang, Yang and Luttinger 2 J to the theory of the virial expansion and of the Bose-Einstein condensation of an imperfect Bose gas. The great merit of such a method consists in being able to transform the problem for the hard-sphere system into a form where the usual perturbational calculation can be applied of which use is impossible for the original form of the potential function. In spite of these successes, however, there can be pointed out two difficult points in Huang and Yang's original -treatment of the pseudo potentiaL One of them is found in their neglect of the triple and ·higher order collision terms. Since the triple collision term is estimated to be of the order a 4 by a dimensional analysis, it is not possible to calculate exactly the terms of order a 4 by their pseudopotentiaL Another difficulty lies in the direct perturbational treatment of the pseudopotentiaL According to Huang and Yang the ground state energy E 0 of N Bose particles is given by up to the terms of order a 3 , where V ( = £3) is the volume of the system, C and ~ are numerical constants. If one keeps the density p=N/V constant and makes N tend to infinity, the above expression diverges as N 113 • Of course, such a behaviour of the energy Downloaded from
doi:10.1143/ptp.19.1 fatcat:ll3jl3hmj5a55mjncqzimplyzq