Analysis of the Bogoliubov Method Applied to a Simple Boson Model

Abstract

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 ${F}_{T}$, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by ${F}_{T}$. 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.