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arXiv:2401.12955

Exponential perturbative expansions and coordinate transformations

Ana Arnal, Fernando Casas, Cristina Chiralt

Published Jan 23, 2024• 0 citations
arXivHTMLPDF

Abstract

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 can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.

DOI
10.48550/arxiv.2401.12955

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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 can be formulated for any time-dependent linear system of ordinary differential equations. All the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.
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.
An alternative to the Magnus expansion in time-dependent quantum mechanical perturbation theory is presented. An exponential form of the time-development operator, given as the exponential of a sum of products of integrals well known in time-dependent quantum mechanics, is derived from the standard perturbation expansion. The derivation is simple and the form of the terms in the exponential is so straightforward that the Nth order term can be written by inspection. The standard Magnus terms can be obtained by rearrangement of the terms of the new expansion probably easier than they can be obtained by iteration of the original equations. A proof by induction is presented to establish the form of the Nth term and it is shown how the new expansion terms relate to the Magnus terms. Some apparent differences between Magnus terms in the literature are resolved.
Abstract The time-dependent quantum perturbation theory developed by Born, Heisenberg and Jordan in 1926 is revisited. We show that it not only reproduces the standard theory formulated in the interaction picture, but also allows one to construct more accurate approximations if time averaging techniques are employed. The theory can be rendered unitary even if the expansion is truncated by using a transformation previously suggested by Heisenberg. We illustrate the main features of the procedure on a simple example which clearly shows its advantages in comparison with the standard perturbation theory.
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 Magnus expansion offers a powerful geometric method to express a time-ordered exponential operator solution, typical of time-dependent linear differential equations, as an ordinary operatorial exponential. This representation possesses advantageous theoretical and qualitative features, such as preserving the underlying Lie algebraic structure of the system while remaining an exact solution to the original differential equation. For any finite-dimensional Hermitian Hamiltonian governing a quantum system, the standard Magnus expansion guarantees a manifestly unitary representation at every finite order of truncation. However, this foundational property is no longer preserved if the underlying Hamiltonian is non-Hermitian. In this work, we derive a generalized, unitarized expansion framework that systematically restores and maintains the property of geometric unitarity for all bounded finite-dimensional non-Hermitian Hamiltonians.