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The Journal of Chemical Physics

Convergence of Magnus and Magnus-like expansions in the Schrödinger representation

W. R. Salzman

Published Oct 15, 1986• 52 citations
Publisher

Abstract

General and greatly simplified methods are presented for calculating terms in the exponent in Magnus and Magnus-like expansions to first order in a ‘‘small’’perturbation and to infinite order in the overall Hamiltonian in the Schrödinger representation. These techniques are applied to four simple but important models and it is shown that in each case the Magnus exponent diverges for some range of the parameters of the model. This result casts serious doubt on the utility of the Magnus expansion in the Schrödinger representation. Several of the problems are also treated in the interaction representation giving results which converge throughout the useful range of parameters. There is no evidence that a similar question exists in this representation.

DOI
10.1063/1.451781

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The Journal of Chemical Physics•May 15, 1987•51 citations
General and greatly simplified methods are presented for calculating terms in the exponent in Magnus and Magnus-like expansions to first order in a ‘‘small’’perturbation and to infinite order in the overall Hamiltonian in the Schrödinger representation. These techniques are applied to four simple but important models and it is shown that in each case the Magnus exponent diverges for some range of the parameters of the model. This result casts serious doubt on the utility of the Magnus expansion in the Schrödinger representation. Several of the problems are also treated in the interaction representation giving results which converge throughout the useful range of parameters. There is no evidence that a similar question exists in this representation.
It is shown by calculations on several model systems of physical importance that exponential time-dependent perturbation theory (the Magnus expansion and similar expansions) does not converge in the Schr\"odinger representation for time intervals larger than the natural period of the system. This result has serious implications for the application of exponential perturbation theory in the Schr\"odinger representation to scattering and adiabatic turn-on problems. It is argued that this result also applies generally to more realistic systems.
The convergence of the Magnus expansion in the Schr\"odinger representation is investigated with the aid of two exactly solvable models. A baffling observation regarding the dynamics of a spin-(1/2 system driven by a superposition of a constant and a rotating magnetic field is elucidated. Perturbation theory is applied to the exponential solution to the Schr\"odinger equation for a time-dependent harmonic oscillator. The first terms of the perturbation expansion for the singularity of the exponent are exactly calculated. The scope and limitations of perturbation theory in obtaining the range of validity of the Magnus expansion are discussed.
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.
This paper derives criteria for the convergence of the Magnus expansion, applied to an arbitrary time dependent two level system, in terms of the eigenvalues of the Hamiltonian for that system. When a weak to moderate intensity field perturbs a two level system the resonance frequency of the system must lie below the frequency of the applied field for the series to converge. An alternative solution for the exponential form of the evolution operator is developed that converges for all values of the expansion parameter and clearly shows the singularities that lead to the failure of the Magnus solution. Finally, a simple unitary transformation is presented that extends considerably the radius convergence of the Magnus series, as illustrated by a spin 1/2 in a linearly polarized field.