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Physical Review A

Projection-operator approach to perturbation theory

Lyman Mower

Published Sep 1, 1980• 17 citations
Publisher

Abstract

A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.

DOI
10.1103/physreva.22.882

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A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.
The connection between the projection operator approach and the continued fraction approach to perturbation theory is investigated. A concise solution to the linear operator equation A mod x>= mod b> is found in terms of the level shift operator using projection operator techniques. The analogous process to the use of projection operators in the continued fraction method of solving the same problem is identified, and a parallel development performed. The connection between the two approaches is thereby established, and continued fraction expressions for the level shift operator obtained. The abstract equation is then specialised to deal with (i) the eigenvalue problem and (ii) the calculation of transition probabilities for quantum mechanical systems described by a time-independent Hamiltonian. Particular attention is paid to the problem of degeneracy and it is shown that the most convenient expressions are found by a hybrid of the two approaches.
We present a new derivation of Time-Independent Perturbation Theory (PT) that is shorter and more general than the usual ones presented in Quantum Mechanics (QM) texts. Expressing the task of PT in terms of Green's functions, the key to our approach is to write the bound-state equation for the perturbed wave function directly in terms of the unperturbed Green's function. Not only does this greatly simplify the derivation of PT expressions for energies and wave functions, but it does so without using the orthogonality or completeness properties of unperturbed Hamiltonian eigenstates, which form the basis of standard derivations of PT. As a result, our approach does not require the potentials to be energy independent as is required in standard derivations, nor does it require the inverse free Green's operator G0−1(E) to be a linear function of energy, as is the case in QM. Consequently, our approach is applicable not only to QM but also to various extensions of QM, including Quantum Field Theory. For students, our approach provides an easy-to-understand introduction to the use of Green's functions in QM while at the same time presenting a description of PT that is shorter and more fundamental than the standard ones currently advocated.
A perturbation theory is described which leads to two alternative methods for calculating the eigenvalues and normalized eigenstates of a perturbed Hamiltonian, the results being given to all orders in the perturbation. The expansions obtained are formulated in terms of two sets of partial sums which have simple, easily characterized forms, and they combine some of the advantages of both the Rayleigh-Schr\"odinger and Brillouin-Wigner perturbation methods while avoiding some of the difficulties of each. In particular, one form of the perturbation expansions given here will in most cases of interest have a larger radius of convergence than the Rayleigh-Schr\"odinger series. As is usual for infinite-order treatments, the explicit form of the final result applies only to nondegenerate levels. However, some of the intermediate relations obtained in the derivation are of general validity. Since certain classes of unlinked diagrams appear in the expansions, the method described here is not suitable for application to systems containing a large number of particles.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.
Abstract The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schrödinger equation allows the derivation of an alternative expression for the evolution operator, which differs from the traditional chronological exponent. An appropriate choice of projection operators results in the possibility of studying the diagonal and non-diagonal elements of the evolution operator separately. The suggested expression implies a particular form of perturbation expansion, which leads to a new formula for the short time dynamics. The new kind of perturbation expansion can be used to improve the accuracy of the usual chronological exponent significantly. The evolution operator for any arbitrary time can be efficiently recovered using the semigroup properties. The method is illustrated by two examples, namely the dynamics of a three-level system in two nonresonant laser fields and the calculation of the partition function of a finite XY-spin chain.