Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Comment on: The Magnus expansion for the harmonically driven harmonic oscillator

Francisco Marcelo Fernández

Published Jan 1, 1988• 19 citations
Publisher

Abstract

The exponential time-evolution operator for the harmonically driven harmonic oscillator is exactly obtained from the equations of motion for the coordinate and momentum in the Heisenberg representation. The convergence of the Magnus expansion is discussed.
DOI
10.1063/1.454483

Related papers

Exponential time-evolution operator for the time-dependent harmonic oscillator

Francisco Marcelo Fernández

The time-evolution operator for the time-dependent harmonic oscillator H= (1)/(2) {α(t)p2 +β(t)q2} is exactly obtained as the exponential of an anti-Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

Journal of Mathematical Physics•Dec 1, 1987•16 citations

Driven harmonic oscillators in the adiabatic Magnus approximation

Serge Klarsfeld, José Ángel Oteo

The time evolution of driven harmonic oscillators is determined by applying the Magnus expansion in the basis set of instantaneous eigenstates of the total Hamiltonian. It is shown that the first-order approximation already provides transition probabilities close to the exact values even in the intermediate regime.

Physical Review A•Mar 1, 1993•6 citations

Convergence of the Magnus expansion

Francisco Marcelo Fernández

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.

Physical Review A•Mar 1, 1990•31 citations

On the convergence of the Magnus expansion in the Schrödinger representation

Francisco Marcelo Fernández, Julián Echave, Eduardo Alberto Castro

The convergence properties of the Magnus expansion in the Schrödinger representation are investigated. A quite general result is rigorously derived from first-order perturbation theory. A finite matrix representation is presented for obtaining the exponential time-evolution operator more easily. Two time-dependent models, an oscillator and a spin system, are considered as illustrative examples.

Journal of Mathematical Physics•Feb 1, 1990•4 citations

Operator techniques applied to the driven harmonic oscillator

P. Croxson

For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.

Annual Meeting Optical Society of America•Jan 1, 1989•0 citations

Heisenberg equation and constants of the motion for an anharmonic oscillator in the high-frequency limit

L DeFalco, R. Mignani, Roberto Scipioni

Using the best approximation of the time evolution operator of a quartic oscillator in the high-frequency limit, we derive its equation of the motion in the Heisenberg picture and show that such an anharmonic oscillator admits a wider class of constants of the motion than the standard harmonic oscillator.

•Jan 1, 1997•1 citation
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.
For many problems a solution in the Heisenberg picture is straightforward, but the Schrodinger picture solution is difficult, although presumably it is possible. A good example is the driven harmonic oscillator. The equations of motion are easy to solve when the driving force is periodic. The Hamiltonian can be reduced to time-independent form and the evolution operator is easy to write down as a single exponential. To work out the transition probability from one number state to another, it is necessary to decompose the exponential. As well as the Baker-Hausdorff formula, other operator techniques are used together with the coherent-state projection operator, to solve the problem.