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Physical Review A

Three-Particle Variational Wave Function for Bose Liquids

Chia-Wei Woo

Published Dec 1, 1972• 27 citations
Publisher

Abstract

A new variational wave function is proposed for the ground state of Bose liquids. It consists of a Jastrow function multiplied by exponentials of three-particle functions. The form of the wave function is deduced from a formal analysis carried out for the weakly interacting Bose gas characterized by a strength parameter $\ensuremath{\lambda}$. By using the wave function proposed here, the energy obtained is exact to $O({\ensuremath{\lambda}}^{4})$. Several useful and straightforward applications of the new method to liquid helium and Coulomb gases are suggested.

DOI
10.1103/physreva.6.2312

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•Aug 1, 1979•0 citations
A new variational wave function is proposed for the ground state of Bose liquids. It consists of a Jastrow function multiplied by exponentials of three-particle functions. The form of the wave function is deduced from a formal analysis carried out for the weakly interacting Bose gas characterized by a strength parameter \ensuremathλ\ensuremath{\lambda}\ensuremathλ. By using the wave function proposed here, the energy obtained is exact to O(\ensuremathλ4)O({\ensuremath{\lambda}}^{4})O(\ensuremathλ4). Several useful and straightforward applications of the new method to liquid helium and Coulomb gases are suggested.
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.
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part v(r>d)v(r>d)v(r>d) and a constant part $\ensuremath{\lambda}$ at r<dr<dr<d of the bare interaction are assumed to contribute to the average field. The correlation function f(r)f(r)f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance ddd is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of ddd in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present f(r)f(r)f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.
A trial wave function describing the ground state of a quantum system of NNN interacting bosons is written in the Jastrow form, a product of pair functions. With the interaction potential chosen to represent liquid He4{\mathrm{He}}^{4}He4, and with the parametrized form of the pair function chosen to include a long-range term which has been found necessary to represent the zero-point motion of the long-wavelength density oscillations, a variational calculation has been performed using a new approximate integral equation for the pair distribution function. This equation, which can also be used for classical fluids, is found to be more accurate for repulsive potentials than the Percus-Yevick equation and comparable to (but much simpler than) the Percus-Yevick 2 equation. The essential results are that including the zero-point motion in the wave function tends to lower the energy, raises the equilibrium density, corrects the behavior of the structure function and the momentum distribution of the particles in the low-wave-number region, and slightly decreases the Bose-Einstein condensate fraction. The value of the lower limit on the wavelength of the density oscillations was determined variationally to be about three interparticle spacings.
Some properties of a new class of variational wave functions for boson systems are studied. This study extends the Jastrow class, and many-body correlations are implicitly introduced by a coupling of the particle coordinates to subsidiary shadow variables. We prove that the new wave function has Bose-Einstein condensation both in the liquid and in the solid phase. The maximum-overlap criterion for this wave function with the exact ground state is developed. The kind of Jastrow correlations implicitly contained in the new wave function is studied and we formulate a theory of correlations based on the reference interaction site model.
"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."