An alternative to the magnus expansion in time‐dependent perturbation theory

1985 ◽  
Vol 82 (2) ◽  
pp. 822-826 ◽  
Author(s):  
W. R. Salzman
Keyword(s):  
Perturbation Theory ◽  
Time Dependent ◽  
Magnus Expansion

scholarly journals Exponential Perturbative Expansions and Coordinate Transformations

2020 ◽  
Vol 25 (3) ◽  
pp. 50
Author(s):  
Ana Arnal ◽  
Fernando Casas ◽  
Cristina Chiralt
Keyword(s):  
Perturbation Theory ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Periodic Systems ◽  
Quantum Mechanical ◽  
Perturbation Techniques ◽  
Unified Approach ◽  
Magnus Expansion ◽  
Theory Lead ◽  
Standard Perturbation Theory

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.

Canonical Perturbation of a Fast Time-Periodic Hamiltonian via Liapunov-Floquet Transformation

1997 ◽  
Author(s):  
Eric A. Butcher ◽  
S. C. Sinha
Keyword(s):  
Perturbation Theory ◽  
Classical Method ◽  
Time Dependent ◽  
Fast Time ◽  
Quadratic Part ◽  
Canonical Perturbation ◽  
Internal Excitation ◽  
Quadratic Hamiltonian ◽  
Unperturbed Part ◽  
Time Periodic

Abstract In this study a possible application of time-dependent canonical perturbation theory to a fast nonlinear time-periodic Hamiltonian with strong internal excitation is considered. It is shown that if the time-periodic unperturbed part is quadratic, the Hamiltonian may be canonically transformed to an equivalent form in which the new unperturbed part is time-invariant so that the time-dependent canonical perturbation theory may be successfully applied. For this purpose, the Liapunov-Floquet (L-F) transformation and its inverse associated with the unperturbed time-periodic quadratic Hamiltonian are computed using a recently developed technique. Action-angle variables and time-dependent canonical perturbation theory are then utilized to find the solution in the original coordinates. The results are compared for accuracy with solutions obtained by both numerical integration and by the classical method of directly applying the time-dependent perturbation theory in which the time-periodic quadratic part is treated as another perturbation term. A strongly excited Mathieu-Hill quadratic Hamiltonian with a cubic perturbation and a nonlinear time-periodic Hamiltonian without a constant quadratic part serve as illustrative examples. It is shown that, unlike the classical method in which the internal excitation must be weak, the proposed formulation provides accurate solutions for an arbitrarily large internal excitation.

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