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Quantum mechanical perturbation theory in terms of characteristic functions Gomberg, Martin Godfrey Luis

Abstract

A quantum mechanical perturbation theory, for finite dimensional cases, based not on the perturbed Hamiltonian operator itself, but on the characteristic function f(z,λ) = det|z-H(λ)| is developed. A perturbation hierarchy in terms of derivatives of the characteristic function is constructed. From this hierarchy, perturbation series for individual eigenvalues are found. Various cases of degeneracy and degeneracy lifted in various orders are examined in detail. This perturbation theory for individual eigenvalues is generalized. Perturbation theory is developed for a set of eigenvalues considered together. Here the perturbation series are for the coefficients of a 'reduced characteristic function' for this set of eigenvalues. These perturbation series are found by a contour integral method and by an algebraic method. The expressions for the individual eigenvalues and their generalization, the expressions for the reduced characteristic function, both of which are in terms of derivatives of the (full) characteristic function, correspond, respectively, to the familiar matrix element expressions in Rayleigh-Schroedinger, and Van Vleck perturbation theories. Some illustrations and applications of the characteristic function perturbation formulae are given. General expressions are found, to second order, for the perturbed Hückel π-molecular orbital energy levels, of any perturbed even-membered ring of carbon atoms. The familiar Rayleigh-Schroedinger perturbation formulae are rederived from their corresponding characteristic function expressions. The relationship between energy derivatives and physical properties is discussed with particular reference to simple spin systems. Expressions for the dipole and guadrupole spin polarizations and for spin polarizabilities in simple spin systems are found from the characteristic functions of the spin systems. These properties are useful in connection with weak hyperfine coupling, and for predicting the intensity of peaks occurring in polycrystalline spectra.

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