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Progress of Theoretical Physics

Single Particle Aspects in a Collective Formalism for an Interacting Boson System

Shuichiro Yamasaki, T. Kebukawa, Sigenobu Sunakawa

Published Aug 1, 1976• 1 citation
Publisher

Abstract

A basic formula to describe the single particle aspects of the interacting Boson system is derived in the framework of the collective formalism developed so far by the present authors. The formula is a generalization of the well-known Bogoliubov transformation. On the basis of the formula, the momentum distribution of the Boson system is evaluated and the result coincides precisely with those obtained by Berdahl, Grest and Nishiyama.

DOI
10.1143/ptp.56.415

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A basic formula to describe the single particle aspects of the interacting Boson system is derived in the framework of the collective formalism developed so far by the present authors. The formula is a generalization of the well-known Bogoliubov transformation. On the basis of the formula, the momentum distribution of the Boson system is evaluated and the result coincides precisely with those obtained by Berdahl, Grest and Nishiyama.
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.
Presently there are two distinct classes of microscopic theories which describe a system of many identical spinless bosons. One is the Bogoliubov theory, which places special emphasis on the occupation of the zero-momentum state and as such depends explicitly on the number of particles in this state, N0{N}_{0}N0​. The other depends on density fluctuations to describe the fluid. In this work, we derive a modified version of the Bogoliubov theory which results in a number-conserving Hamiltonian with no explicit dependence on N0{N}_{0}N0​. The ground- and first-excited-state energies evaluated through second order in 1N12\frac{1}{{N}^{\frac{1}{2}}}N21​1​ are equivalent to the results of a variational-perturbative procedure based on the method of correlated basis functions in the uniform limit.
An occupation-number formulation is considered for a many-particle boson system when the ground state is described by the Bijl-Dingle-Jastrow (BDJ) type of variational wave function. A nonunitary operator acting on the state function of a noninteracting boson system is introduced to convert the BDJ function in configuration space into an equivalent state function in second-quantized space. The momentum distribution function is derived in an infinite series in terms of the liquid-structure function and the correlation function. In the case of the optimum variational description under the uniform limit, the leading term of the obtained result is found to agree with the Bogoliubov formula evaluated in the weak-coupling limit. The same analysis in the occupation-number space also reproduces the mean kinetic energy calculated in configuration space.
The ground-state energy of an interacting many-boson system obtained by Bogoliubov and Zubarev using the collective-coordinate method is shown to be equivalent through second order to the variation-perturbation energy evaluated in the uniform limit by means of the method of correlated basis functions. It is also shown that the wave function derived with a slight modification of the Bogoliubov-Zubarev approach is equivalent to that determined formally from the Rayleigh-Schr\"odinger perturbation theory in the uniform limit.