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Modern Physics Letters B

WAVE FUNCTION AND PAIR DISTRIBUTION FUNCTION OF A DILUTE BOSE GAS

Bo-Bo Wei

Published Jun 20, 2009• 0 citations
Publisher

Abstract

The wave function of a dilute hard sphere Bose gas at low temperatures is revisited. Errors in an early 1957 paper are corrected. The pair distribution function is calculated for two values of [Formula: see text].
DOI
10.1142/s0217984909020229

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Physical Review•Dec 1, 1958•193 citations
The wave function of a dilute hard sphere Bose gas at low temperatures is discussed, emphasizing the formation of pairs. The pair distribution function is calculated for two values of ρa3\sqrt{ρa^3}ρa3​.
The pair distribution function of a hard-sphere Bose gas is evaluated using a modified cluster expansion formula developed in the preceding papers of this series [H. Ichimura, Prog. Theor. Phys. 31 (1964), 538; ibid. 37 (1967), 484]. The contributions for the pair distribution function are expressed in terms of diagrams, and a partial summation of the chain-type diagrams is carried out. The temperature dependence of the resultant expression is investigated for low temperatures, at short and long distances. The result is compared with the expression for the pair distribution function which has been obtained for a weakly interacting Bose gas. The two- and three-particle distribution functions for a degenerate ideal Bose system are derived.
The pair distribution function for a quantum Bose gas is expressed as a power series in terms of the fugacity, the coefficients of which are temperature dependent. For the hard sphere case, these coefficients have been evaluated to the first order in a/λ, (a being the scattering length and λ the thermal wavelength) by using torons with two fixed points, or alternatively U cluster functions. The result gives the first order correction to the ideal gas formula of London and Placzek introduced by the interactions between the particles.
With the pair Hamiltonian model as the starting point, perturbation theory calculations are performed for the radial distribution function and the structure factor through second order in the two-body interaction potential for a many-boson system below the Bose-Einstein transition temperature. For the special case of a dilute hard-sphere Bose gas at T=0° K, the structure factor is calculated in the low-momentum limit keeping terms through second order in the hard-sphere gas parameter (n a3)1/2, where n is the gas density and a is the hard-sphere diameter. The resulting expression for the structure factor is shown to satisfy the Feynman-Bijl relation explicitly.
By a generalization of the method used in a previous paper the distribution of energy levels of a dilute Bose system of hard spheres is found. These energy levels are then used to compute the statistical properties for the system. A phase transition is found, and the transition point is calculated to the lowest order of aaa, the diameter of the hard spheres. Furthermore, the thermodynamical functions of the system in both the gas phase and the degenerate phase are obtained, provided $(\frac{a}{\ensuremath{\lambda}})\ensuremath{\ll}1$ and $\ensuremath{\rho}{a}^{3}\ensuremath{\ll}1$, where $\ensuremath{\lambda}$ is the thermal wavelength and $\ensuremath{\rho}$ is the particle density.