The pair distribution function for a hard sphere Bose system has been calculated by using a method in which the hard sphere potential is replaced by the so-called pseudo-potential. The problem is carried through using the quasi-particle formalism of Bogoliubov. In this calculation one considers a system of bosons interacting via the pseudo-potential which can be regarded as a weak interaction when the system is very dilute. The Hamiltonian of the system is diagonalized by means of a canonical transformation which has the effect of separating the energy into two parts, one of which is the ground state energy and the other corresponds to an ideal Bose gas composed of ``elementary excitations'' or ``quasi-particles.'' This formalism is applied to the calculation of the pair distribution function. This quantity is calculated by averaging over a grand canonical ensemble constructed with the total number of elementary excitations. A result which is seen to be valid both for the condensed and gaseous phases of the system and also for any distance r between the particles is obtained. A discussion of the difference between these results and the ones obtained in a previous calculation and a comparison between both results and the experimental ones is also given.
The ground-state energy of a system of interacting N-bosons is investigated in a collective formalism. Ultraviolet divergences in perturbation calculations are eliminated by the introduction of quasi-particle operators which are related by a nonlinear canonical transformation to the collective variables, i.e., the density fluctuation operators and their canonical conjugates. The transformation is presented in terms of a series in powers of N−1/2. The ground-state energy appears as a series in powers of N−1. The perturbation contributions of order N−2 to the ground-state energy are calculated. Most of these contributions can be incorporated into the lower order ones which agree with the previously obtained results.
We obtain thermodynamical functions for a dilute Bose gas of hard spheres by using the efffective Hamiltonian which is obtained by a quasi-linear transformation proposed previously and may be recognized as an improvement of the model Hamiltonian of Lee and Yang which is not correct about the terms of second order in the diameter of hard spheres when it is applied to the problem of the phase transition. The Bose condensation occurs at a temperature below which the degenerate phase of the quasi-particles appears. We evaluate the amount of the gap of the specific heat and that of the compressibility at the transition point.
The method of point transformations is used to provide a Hamiltonian for strongly interacting particles that is equivalent to the usual Hamiltonian. The new Hamiltonian is Hermitian, Fourier analyzable, velocity dependent, and free of the difficulties of pseudopotentials. The present work is restricted to the case that two-body encounters are dominant. The theory is expressed in the quantized wave formalism so that number-violating approximations can be applied. The well known results for the spectrum and ground-state energy of the dilute hard-sphere gas are then derived as an elementary application.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at
T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons
N, and that the additional trace over
N that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature
T goes past the critical transition temperature
Tc is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons
N is taken to be a constant that does not depend on
T. Consequences on the physics of interacting bosons are briefly discussed.