[0810.5488] The Magnus expansion and some of its applications
2008年10月30日 · When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory.
2018年3月2日 · Magnus proposed that the exponent can be expanded in a series in parameter (t;t 0) = X1 k=1 k k(t;t 0) = 1 1(t;t 0) + 2 2(t;t 0) + :::: (12) Substituting this series expansion in the di erential equation for (t;t 0) and comparing terms order by order (in ): @ @t 1(t;t 0) = H~(t) ) 1(t;t 0) = Z t t 0 dt0H~(t0) (13) @ @t 2(t;t 0) = 1 2 [1(t;t 0 ...
However, in 1954, Wilhelm Magnus provided another method to approximate the fundamental matrix of an initial value problem that is of exponential form, called the Magnus Expansion. Later in this article, the Picard iterative method will be used to make a …
Time-dependent perturbation theory as a tool to compute approximate solutions of the Schr ̈odinger equation does not preserve unitarity. Here we present, in a simple way, how the Magnus expansion (also known as exponential perturbation theory) provides such …
The Magnus expansion and some of its applications
2009年1月1日 · When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory .
A general formula for the Magnus expansion in terms of iterated ...
2018年3月14日 · The Magnus expansion is an extremely useful device when dealing with time-dependent linear differential equations of the form Y' = A(t) Y. It yields the solution of such equations in exponential form, the exponent defined as an infinite series whose terms can be constructed in a recursive way as multiple integrals of nested commutators of the ...
[2312.16674] What is the Magnus Expansion? - arXiv.org
2023年12月27日 · The Magnus expansion, introduced by Wilhelm Magnus in 1954, is an infinite Lie series employed to express solutions for first-order homogeneous linear differential equations involving a linear operator. Since its discovery it has evolved into a pivotal tool used across diverse disciplines, including physics, chemistry, and engineering.
The Magnus expansion Numerical integrators based on Magnus expansion Some properties of Magnus expansion If A(t) belongs to some Lie algebra g, then (t) (and any truncation of the Magnus series) also stays in g and therefore exp() 2G, where Gis the Lie group whose corresponding Lie algebra is g. 1 Symplectic group in classical mechanics
In this brief review we describe the coming of age of Magnus expansions as an asymp-totic and numerical tool in the investigation of linear differential equations in a Lie-group and homogeneous-space setting.
At the heart of the Magnus expansion is the idea of solving Eq. (3) by using the quantum propagator U(t;t 0) that connects wave functions at di erent times, e.g. (t) = U(t;t 0) (t 0) (4) Furthermore, the Magnus expansion assumes that U(t;t 0) can be represented as an expo-nential, U(t;t 0) = exp((t;t 0)) (5) This yields the modi ed TDSE @ @t U ...
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