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Physical Review

Properties of One-Dimensional Correlated Gaussian Wave Functions

T. R. Koehler

Published Jan 7, 1966• 7 citations
Publisher

Abstract

The properties of a certain class of unsymmetrized one-dimensional correlated Gaussian wave functions---those which are ground-state eigenfunctions of some coupled harmonic-oscillator Hamiltonian---are investigated in detail. It is shown that a properly symmetrized wave function constructed from these may be used to calculate the expectation value ${E}_{0}$ of the Hamiltonian appropriate to a system of interacting one-dimensional atoms and that this energy is, to a high degree of accuracy, equal to the value obtained when the unsymmetrized wave function is used. A method is given by which correction terms to ${E}_{0}$ may be obtained. In addition, it is found that even though the number of particles be very large, the necessary multivariable integrals may be performed quite simply.

DOI
10.1103/physrev.141.281

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The properties of a certain class of unsymmetrized one-dimensional correlated Gaussian wave functions---those which are ground-state eigenfunctions of some coupled harmonic-oscillator Hamiltonian---are investigated in detail. It is shown that a properly symmetrized wave function constructed from these may be used to calculate the expectation value E0{E}_{0}E0​ of the Hamiltonian appropriate to a system of interacting one-dimensional atoms and that this energy is, to a high degree of accuracy, equal to the value obtained when the unsymmetrized wave function is used. A method is given by which correction terms to E0{E}_{0}E0​ may be obtained. In addition, it is found that even though the number of particles be very large, the necessary multivariable integrals may be performed quite simply.
A two-parameter many-body correlated Gaussian wave function is used to calculate the expectation value of the Hamiltonian for NNN one-dimensional particles interacting through the one-dimensional analog of an atomic potential. Using a potential equal to four times the Slater-Kirkwood He-He potential, we calculate the optimum average interparticle separation, the ground-state energy, and the velocity of sound of the system as a function of particle mass. It is found that the correlated Gaussian always gives an improvement in energy compared with an uncorrelated Gaussian.
The method of correlated basis functions is used in the study of the ground-state properties of a one-dimensional system of bosons interacting through a repulsive $\ensuremath{\delta}$-function potential. Ground-state energies are determined numerically for intermediate values of the coupling constant in three steps: (1) a trial form of the radial distribution function generated by the Bijl-Dingle-Jastrow (BDJ) type of wave function is introduced to obtain approximate ground-state energies, (2) the paired-phonon analysis is applied to improve the BDJ description of the ground state, and (3) leading corrections to the improved approximate energies are evaluted from the second-order perturbation energy generated by the three-phonon vertex. The obtained energy values are found to agree closely with the exact results calculated by Lieb and Liniger.
We study the correlations in a family of ground-state wave functions for one-dimensional quantum many-body systems. For a particular ordering of the M particles, these wave functions are given as the absolute value of the free-fermion ground-state wave function, raised to an arbitrary power \ensuremath{\lambda}. For other permutations of the particles, we use the particle statistics---boson or fermion---to construct the wave function. This prescription works equally well on the continuum line of length L or on the discrete lattice of size N. The theory of random matrices singles out the ``special'' values of \ensuremath{\lambda}=1/2, 1, and 2, and provides techniques for calculating all correlations. Results obtained in this work include analytic expressions for the lattice momentum distributions, accurate numerical results for all momentum distributions and their critical exponents, and explicit results for the lattice pair-correlation functions. We emphasize that this is only for the stated special values of \ensuremath{\lambda}. Finally, these wave functions are evaluated as candidates for variational wave functions.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.