Application of a folding transformation to the Magnus solution for the evolution of periodically time dependent systems
- 1 November 1986
- journal article
- Published by AIP Publishing in The Journal of Chemical Physics
- Vol. 85 (9) , 5167-5174
- https://doi.org/10.1063/1.451710
Abstract
Two difficulties arise in the Magnus expansion for the evolution operator of a periodic time dependent Hamiltonian when the eigenvalues of the Hamiltonian are separated by more than 2π/τ, where τ is the period: (1) The Magnus expansion generally does not converge, and (2) the equilibrium properties of the system cannot be determined by statistical mechanical techniques analogous to those for time independent systems. A transformation is introduced that folds the eigenvalue spectrum into the interval −π/τ<λi<π/τ, yet retains the periodicity of the Hamiltonian, thereby alleviating the above difficulties. The method is compared to the approach taken by Shirley [Phys. Rev. B 138, 979 (1965)] for the Floquet solution to the evolution operator. It is applied to a dipole coupled spin system in an oscillating transverse field and the implications of the folding transformation on the spin thermodynamics of this system and the ‘‘Magnus paradox’’ are discussed.Keywords
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