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Physical Review A

New criterion for convergence of exponential perturbation theory in the Schrödinger representation

W. R. Salzman

Published Nov 1, 1987• 34 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.36.5074

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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.
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.
A time-dependent perturbation theory for the solution of the time-dependent Schrödinger equation is developed and is applied to systems having two states with pulse and Lorentzian types of perturbations. It is shown in the case of the pulse perturbation that the expression corresponding to the Nth order of the perturbation theory converges to the exact solution when N tends to infinity. The results obtained from this perturbation theory, the Magnus approximation, and the time-dependent perturbation theory developed by Dirac are compared with the exact results. This theory is found to be good for larger values of the perturbations and for larger intervals of time.
We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.