Perturbation method for low states of a many-particle boson system

Abstract

A theoretical description of the ground state and low excited states of liquid He 4 is developed in terms of a set of correlated basis functions. We start from a simple correlated trial function ψ 0 suitable for an approximate description of the ground state under the assumption of a strong repulsive force when two particles approach closely. The function ψ 0 and a set of model functions ϕ n are used to construct a set, ψ n = ψ 0 ϕ n , of linearly independent correlated basis functions. Matrix elements of the identity and Hamiltonian operator are evaluated by systematic application of a generalized Kirkwood type superposition approximation. A normalized, orthogonal basis | e n > is constructed from linear combinations of the functions ψ m ; the associated matrix elements 〈 e n | H| e m 〉 vanish everywhere except on the three diagonals m = n, n ± 1. This result is a consequence of an appropriate choice of the model functions and use of the superposition approximation in evaluating the matrix elements. At this point is it proper to speak of a free phonon description. A final approximate diagonalization, neglecting phonon-phonon interaction, yields explicit formulas for the ground state energy and the momentum dependence of the phonon energy. The analogy with Bogoliubov's treatment of the boson system, using uncorrelated basis functions is very close as is also the relation to Feynman's theory of the excitation energies. A parallel analysis is successful with ψ 0 taken to be the correct ground-state eigenfunction. In this case the matrix elements of the phonon-phonon interaction can be expressed completely in terms of the elementary liquid structure function as given by the analysis of x-ray diffraction at low temperatures. The way is open to an accurate evaluation of the phonon energy as a function of momentum (the Landau curve) and a corresponding accurate evaluation of the thermodynamic properties of the liquid at low temperatures.