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arXiv:0910.5798

Energy and Eigenstate Using First Order Perturbation Theory

Raghu Ramachandran, Rama Komaragiri

Published Oct 30, 2009• 0 citations
arXivHTMLPDF

Abstract

The first order estimate of energy and eigenstate of a perturbed system are found by summation over its basis.
DOI
10.48550/arxiv.0910.5798

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The WKB quantization condition can be used to estimate the quantized bound state energies, En, in a potential V(x). If a small perturbation of the form λV(x) is added to this potential, we show that the resulting shift in the quantized energy eigenvalues is given by a form which is quite similar to standard first-order perturbation theory. Several examples are presented in which the large n form of the lowest-order energy shift is more easily evaluated using WKB-perturbation techniques than by using standard first-order perturbation methods.
1D spin-1/2 Fermi gas with a self-interaction allows a non-perturbative solution. Using the algebraic Bethe ansatz approach eigenvectors and spectrum of the system can be written. Computation of the observables, such as transport coefficients,  however still very complicated problem. The first problem, studied in this thesis, is a calculation of the form factors of the monodromy matrix, that can be used for the computation of form factor of the ultralocal operators. The correlation functions can be expressed via a form factors series. In order to perform summation of this infinity series the numerical algorithm that allows to scan the basis of the eigenstates and choose only the important contributions to the from factor sum was developed. The developed approach allows to study system at zero-temperature and can be generalised for a similar model of Bose gas and lattice supersymmetric t-J system of electron gas.
Some methods for the convergence acceleration of the Moller-Plesset perturbation series for the correlation energy are discussed. The order-by-order summation is less effective than the Feenberg series. The latter is obtained by renormalizing the unperturbed Hamilton operator by a constant factor that is optimized for the third order energy. In the fifth order case, the Feenberg series can be improved by order-dependent optimization of the parameter. Alternatively, one may use Pad{e} approximants or a further method based on effective characteristic polynomials to accelerate the convergence of the perturbation series. Numerical evidence is presented that, besides the Feenberg-type approaches, suitable Pad{e} approximants, and also the effective second order characteristic polynomial, are excellent tools for correlation energy estimation.