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Reviews of Modern Physics

Generalized Perturbation Theory in Operator Form

Hans Primas

Published Jul 1, 1963• 218 citations
Publisher

Abstract

It is proposed to formulate any quantum mechanical perturbation theory in operator form and to take advantage of the fact that such a theory can be completely formulated in the domain of a Lie algebra. The concept of a T- operator is generalized and used to compose a solution of a complicated quantum mechanical problem from subproblems connected therein. (R.E.U.)

DOI
10.1103/revmodphys.35.710

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It is proposed to formulate any quantum mechanical perturbation theory in operator form and to take advantage of the fact that such a theory can be completely formulated in the domain of a Lie algebra. The concept of a T- operator is generalized and used to compose a solution of a complicated quantum mechanical problem from subproblems connected therein. (R.E.U.)
>A generalized ford of a perturbation theory for a (nonrelativistic) quantum mechanical many-body Hamiltonian is given that can be useful for problems of quantum chemistry and other problems with a mcderate number of particles. A modification of Watson's t-operator allows a perturbational development of a many body problem in terms of simpler subproblems. Some examples of such cluster approximations are given. The whole theory is formulated in terms of state vectors and there are no assumptions about the degeneracy of the Hamiltonian. The given approximations to the level shift transformation are both unitary and Lie functions in every order of the development. Every step of the calcalation can be done in the domain of a Lie algebra and it is recomdended that full advantage be taken of this fact in practical calculations. The use of the diagram technique is avoided and there are no explicit partial summations of the perturbation series, but similar results are gained by the systematic use of the unitary conditions, of the Lie character, and of a modified t-operator. (auth)
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of a^†\hat{a}^\daggera^†, a^\hat{a}a^, 1^\hat{1}1^ operators, the second one has been based in the angular momemtum algebra of L^+\hat{L}_+L^+​, L^−\hat{L}_-L^−​ and L^0\hat{L}_0L^0​ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.
The quantum damped harmonic oscillator is described by the master equation with usual Lindblad form. The equation has been solved completely by us in arXiv: 0710.2724 [quant-ph]. To construct the general solution a few facts of representation theory based on the Lie algebra su(1,1) were used. In this paper we treat a general model described by a master equation with generalized Lindblad form. Then we examine the algebraic structure related to some Lie algebras and construct the interesting approximate solution.