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Progress of Theoretical Physics

Variational Study of Ground State of a Bose-Einstein Fluid

Kazuo Hiroike

Published Feb 1, 1962• 39 citations
Publisher

Abstract

Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorously expressed in terms of the pair distribution function. An integro-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integro-differential equation become identical with those derived by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigated by other theories. It is suggested that the present theory may in general predict the phonon-like excitations.

DOI
10.1143/ptp.27.342

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Under the assumption that the ground-state wave function is expressed as a product of pair functions, the expectation value of the ground-state energy is rigorously expressed in terms of the pair distribution function. An integro-differential equation is derived for the pair distribution function from the variational principle that the expectation value of the energy is minimum. It is pointed out that the expression for the energy and the integro-differential equation become identical with those derived by Abe if the hyper-netted chain (HNC) approximation is introduced. It is proved that the HNC approximation is compatible with the virial theorem. The nature of the formulas is investigated by applying them to some cases that have been investigated by other theories. It is suggested that the present theory may in general predict the phonon-like excitations.
"A theory is developed to allow the systematic calculation of the properties of quantum fluids given the microscopic two-body potential. Physical arguments are used to construct a correlation operator that approximately transforms the states of a non-interacting system into the corresponding states of the interacting-system. The theory can be regarded as a microscopic realization of Landau's assumption of a one-to-one correspondence between the non-interacting and interacting states. It is variational since the correlation operator is determined by minimizing E(p) the expectation value of the ground-state energy. We first give a survey of various other many-body theories including the simpler Jastrow variational method. Differential equations are derived to calculate the correlation operator and the related Jastrow correlation"" function. A comparison of results in several Bose systems with various methods is given. The Fermi hypernetted chain integral equations are generalized to allow the calculation of E(p) in Fermi systems, using the correlation operator wave function. The method is applied to liquid 3He and the v2-model of nuclear matter. In 3He the correlation operator gives results in much closer agreement with experiment than that with the Jastrow wave function. A new set of hypernetted chain equations are then derived to allow the calculation of the excited st;ate energies in Bose systems. The theory is then applied to the calculation of the single quasi-particle excitation spectrum of liquid 4He. The spectrum is found to be a stringent test on the form of the correlation operator, deficiencies in the correlation operator giving corresponding errors in the spectrum. Good results are obtained for the density dependence of the roton energy. A variational principle for the free energy is used to generalize the correlation operator technique to finite temperatures. Results are given for the finite temperature properties of the v8-model of nuclear matter."
Abstract A modification of the Rayleigh—Ritz variational principle is described which makes possible a calculation of the energy, wave function, and pair distribution function f12≡f(x1,x2) of a Bose fluid, such as liquid 4He, at absolute zero. The assumptions made are: (i) two-body interactions with potential Uij, (ii) trial wave functions of the form f12≡f(x1,x2) and (iii) the Kirkwood ‘superposition' approximation. Under these approximations, the expectation energy is E=12n2∫∫d3X1d3X2f12U12−h2m−1(∇18ϕ12)+(∇1ϕ12)2+n∫d3X2f13f23ϕ12⋅∇1ϕ13, where n ≡ N/V. It is shown here that making E stationary with respect to independent variations in f and ɸ corresponds to simultaneously applying the ordinary Rayleigh-Ritz principle and solving the Born-Green-Yvon integral equation for f. The method is illustrated by reproducing Bogolyubov’s results for the case where U is small. The case where U is large must be dealt with numerically, but transformations for simplifying the computations are given here.
Bose-Einstein condensation is described in terms of the condensate wave function and the pair-excitation function, the latter being responsible for the existence of phonons. This minimal description in terms of these two functions is generalized to the case with an external potential. For a dilute gas with short-range pairwise repulsive interaction and at very low temperatures when the Bose-Einstein condensation is nearly complete, a partial differential equation is obtained for the condensate wave function and an integro-differential equation for the pair excitation. Experimentally, the external potential is used to trap the atoms, i.e., to keep them together. Since the trap is of macroscopic dimensions, the resulting external potential is often slowly varying. In these cases and when the condensate is in the lowest state, the partial differential equation for the condensate wave function and the integro-differential equation for the pair excitation are solved approximately for the case of a time-independent trap.
A series expansion for the ground-state energy of a multispecies quantum fluid is obtained. This expansion is derived by assuming that the ground-state wave function for the fluid can be approximated by the symmetrized ground-state wave function for a mixture of quantum ideal gases times a product of pair functions. The lowest-order approximation to the energy is minimized with respect to variations in these pair functions in order to obtain an approximation to the ground-state wave function for the system. In principle, this variational procedure can be carried out to any desired order in the energy expansion, and the machinery for doing so is explicitly exhibited. The results of this variation in the lowest order are applied to a two-species system consisting of electrons and nuclei. The results are consistent at high densities.