A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with
F the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of
F and in this range the deviation of
W from the Bogoliubov value is small. There is, however, a transition point
FT, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by
FT. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
The density fluctuation excitation spectrum is continuous, since all particle transitions allowed by the Pauli principle will contribute. This chapter shows that the quasi-particle, as a bare particle moving together with the surrounding distortion brought about by particle interactions. The importance, or indeed the existence, of the mechanism, depends on the quasi-particle energies and the phase space available for the decay process. Higher order collision processes can also limit the lifetime of a quasi-particle. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary. For an arbitrary particle interaction, there is no a priori region of momenta for which quasi-particles exist as well-defined elementary excitations of the system. In contrast to the single quasi-particle excitations, the multi-particle excitations will be spread over a continuous range of energies, as one allows the momentum of any single quasi-particle component to vary.
The infrared divergence of the interaction among Bogoliubov's quasi· particles in a degener ate boson system can be avoided by introducing improved quasi-particles. A formula is given which relates the single· particle operators ap directly to the improved quasi-particle operators bp ' which have been defined in terms of collective variables in a preceding paper. The operators ap are expressed in terms of bp as a series using N112 as a formal expansion parameter, where N is the number of particles in the system. Terms up to order N312 are given together with a tenn of order N2 which is necessary for the calculation of the ground· state energy to order N2 • The fonnula makes the reason clear why the operators b p enable us to eliminate the ultraviolet divergence haunting collective theories including the quantum hydrodynamics.
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.