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Annalen der Physik

Construction of Propagators for Quantum Crystals

Vyacheslav I. Yukalov

Published Jan 1, 1981• 21 citations
Publisher

Abstract

Abstract We present here a mathematical basis of the iterative procedure developed earlier by the author for strongly anharmonic crystals. We generalize the approach to the case of any strongly nonideal nonuniform system. The convergence of this procedure is defined as a convergence in the mean with respect to averages of observable‐quantity operators. The definitions yield self‐consistent conditions for a zero‐order iteration in a natural way. Divergences in high‐order iterations are considered and two ways to eliminate them are proposed. The method is applied to a strongly‐anisotropic strongly‐anharmonic quantum crystal.

DOI
10.1002/andp.19814930606

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Abstract We present here a mathematical basis of the iterative procedure developed earlier by the author for strongly anharmonic crystals. We generalize the approach to the case of any strongly nonideal nonuniform system. The convergence of this procedure is defined as a convergence in the mean with respect to averages of observable‐quantity operators. The definitions yield self‐consistent conditions for a zero‐order iteration in a natural way. Divergences in high‐order iterations are considered and two ways to eliminate them are proposed. The method is applied to a strongly‐anisotropic strongly‐anharmonic quantum crystal.
Using the unsymmetrized self-consistent field method the quasi-classical expansion of the one-particle density matrices is considered. The integral equations for the first quantum corrections to the classical one-particle probability densities are derived. The exact solution for the perfect crystals with the strong anharmonicity of the fourth order is found and used to calculate the first quantum correction to the effective amplitudes of the strongly anharmonic vibrations of atoms. The solution is obtained also for the semi-infinite linear chain as the simplest model of the anharmonic crystal with the surface.
Abstract It is well known that quantum-mechanical perturbation theory often gives rise to divergent series that require proper resummation. Here I discuss simple ways in which these divergences can be avoided in the first place. Using the elementary technique of relaxed fixed-point iteration, I obtain convergent expressions for various challenging ground-states wave functions, including quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing Rayleigh-Schrödinger coefficients), all at arbitarily strong coupling. These results challenge the notion that non-analytic functions of coupling constants are intrinsically “non-perturbative”. A possible application to exact diagonalization is briefly discussed.
This is one of the two papers where the optimized perturbation theory was first formulated. The other paper is published in Theor. Math. Phys. 28, 652--660 (1976). The main idea of the theory is to reorganize the perturbative sequence by introducing control functions, defined by optimization conditions, so that the reorganized approximation sequence be convergent. In the present paper, the theory of perturbations is suggested for statistical systems in the absence of small interaction parameters. A new form is advanced for self-consistent conditions defining the optimal parameters for trial Green functions in iterating nonlinear propagator equations. Superharmonic, semiharmonic, and pseudoharmonic approximations for a molecular crystal are considered as examples.