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Proceedings of the Physical Society

Integrable Many Body Wave Functions for a System of Bose Particles

Sam F Edwards, M.A. Naqvi

Published Feb 1, 1961• 2 citations
Publisher

Abstract

It is shown that, by means of a parametric representation, a variational calculation for a system of Bose particles can be carried through using explicit trial wave functions which describe a state with a fixed number of particles and include pair correlations. For the ground state the results of previous work are obtained as the first term in a steepest descent calculation. An effective Schrodinger equation is obtained for the relative motion of two particles. The application of the method to excited states is discussed; in particular for the case of a vortex line in the system it is shown that the method leads to an analysis which is similar to that for the ground state and could be carried through in the same way.

DOI
10.1088/0370-1328/77/2/326

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It is shown that, by means of a parametric representation, a variational calculation for a system of Bose particles can be carried through using explicit trial wave functions which describe a state with a fixed number of particles and include pair correlations. For the ground state the results of previous work are obtained as the first term in a steepest descent calculation. An effective Schrodinger equation is obtained for the relative motion of two particles. The application of the method to excited states is discussed; in particular for the case of a vortex line in the system it is shown that the method leads to an analysis which is similar to that for the ground state and could be carried through in the same way.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.
Quantum field theoretical methods are discussed to evaluate the spatial distribution in many-body systems in equilibrium. The pair and triplet distribution functions are explicitly evaluated for noninteracting cases. A general method to evaluate the pair distribution function in the chain and ring diagram approximations for interacting systems is presented in detail. The result is expressed in terms of the eigenvalues of the propagators characteristic of the diagrams. This result generalized the Gell-Mann and Brueckner formula for the equation of state, the derivation of which is also given. The eigenvalues are evaluated for Boltzmann, Bose and Fermi statistics. Application of the result to a hard-sphere Bose gas is made, and the effect of condensation is studied. It is found that the phonon velocity depends on the number of particles in the lowest state. The distribution functions of a degenerate Fermi system are also calculated.
The many-boson system with repulsive interactions is treated by a variational method based on a variational trial state of an exponential pair-excitation type obtained by a generalization of that of Bogoliubov; the treatment is closely related to an intermediate-coupling approximation with respect to pairs. The nonlinear integral equation which determines the variational ground state is derived, and various properties of this ground state and the associated excited states are examined. The resultant low-lying spectrum lies below that of Bogoliubov by an amount proportional to the total number of particles. The variational principle is shown to produce rigorous energy eigenvalue differences for the pair part of the Hamiltonian. The variational states, however, still exhibit unphysical features characteristic of pair-excitation states: The pair correlation function does not go to zero at zero particle separation and the phonon spectrum exhibits a gap above the ground state. It is suggested that these features can be removed by using states which take into account excitation of momentum-conserving groups of more than two particles.
We apply Feynman's path integral theory to study Bose-Einstein condensates in a double well potential and in a random potential. The main idea of this thesis is to perform the mean field approximation in two body interaction by the path integral approach. Bose particles are confined in the double well which is taken as a harmonic potential with Gaussian or cosine barrier. Performing the variational calculations we obtain analytical results of the ground state energy, wave functions, and overlap integrals. The overlap integrals enable us to calculate the tunneling rates which are in good agreement with numerical values for the two-mode model of Gross-Pitaevskii's equation. We also consider the model of a Bose system consisting of N particles with finite interacting pair potentials under the influence of random potentials. The porosity or nonhomogeneity of the system can be represented by the density of porosity with amplitude of the fluctuation, and a correlation length. We obtain analytical results of the effective mass and the ground state energy which are in good agreement with that of Bugoliubov approach. The propagator of the system is used to calculate the partition function, specific heat, the critical temperature, condensate density, and the superfluid density of the system.
A correlated wave function for interacting Bose systems with correlation factors up to five-particle parts is suggested. The vehicle used is an analysis for the weakly interacting Bose gas by a new method. Terms up to $O({\ensuremath{\lambda}}^{2})$, where $\ensuremath{\lambda}$ measures the strength of the interaction, for the two-particle (Jastrow) factor and the three-particle factor, as well as ground-state energy up to $O({\ensuremath{\lambda}}^{4})$, are shown to agree with exact results obtained previously with other methods. New results include all multiparticle correlation factors up to $O({\ensuremath{\lambda}}^{4})$. The wave function will be most useful for analytically soluble model systems, and more importantly, for Monte Carlo or molecular-dynamics calculations dealing with realistic systems such as superfluid 4He^{4}\mathrm{He}4He.